Frank Lübeck   

Conjugacy Classes and Character Degrees of 2E6(2)sc

[Character table][Classes][Characters]

The complex character table of the finite simple Chevalley group of type 2E6(2) is known. This group has an exceptional Schur multiplier of type 3x2x2. Character tables of central extensions by a cyclic group of order 2 or a 2-group of form 2x2 are also known. These tables are available via GAP, in ATLAS notation they are called "2.2E6(2)" and "2^2.2E6(2)".

The finite group of Lie type 2E6(2)sc arising as fixed points of a Frobenius map of a simple simply connected algebraic group yields the central extension of the finite simple group by a cyclic group of order 3. Until recently, its ordinary character table was not known. But a former version of this page contained at least information on conjugacy classes, centralizer orders and character degrees. We leave this information below.

NEW 08/2005: CharacterTables of 3.2E6(2) and 6.2E6(2)

I have computed these tables of central extensions of the simple group of type 2E6(2) by cyclic groups of order 3 and 6 using a combination of Deligne-Lusztig theory, the known table for the simple quotient, standard tricks like tensoring and inducing known characters, and some combinatorics. A paper giving more details is in preparation.

Using the tables: You can download the tables and use them via GAP. Assuming that you have GAP installed and you are using a UNIX system: download the file cto2e62.tbl (1020 kB) or cto2e62.tbl.gz (132 kB) and copy it to, say, a subdirectory ~/ctblextra of your home directory; then reading this file notify.g into GAP (or copying its content in your ~/.gaprc file) you can access them like other library tables, e.g.,
t3 := CharacterTable("3.2E6(2)");
t6 := CharacterTable("6.2E6(2)");

These tables will be included in future versions of GAPs character table library CTblLib. But note that then the classes and characters will be reordered according to the conventions of that library.

Conjugacy Classes and Centralizer Orders

The following table lists a parameterization of the conjugacy classes of 2E6(2)sc. It is obtained by considering the group as group of fixed points under a Frobenius morphism inside a connected reductive algebraic group G of simply connected type. Each element g in G has a Jordan decomposition g=su=us with s semisimple and u unipotent.

Actually, the information given below was computed generically, that is for all groups of type 2E6(q)sc, q an arbitrary prime power, and then specialized to the case q=2.

We give the following information: each row stands for a set of classes which have representatives with the same centralizer in G. The column "# classes" tells how many classes are in this set. The column "|C(su)(q)|, q=2" tells the order of the centralizer of elements in these classes. The next two columns describe the centralizer of the semisimple part s of an element in these classes; "type of C(s)" gives the semisimple part of the centralizer of s in G under the restricted Frobenius morphism, and "|Z0(C(s))(q)|" gives the number of rational points in the radical of the centralizer of s (generically, as polynomial in q (= 2), the polynomials are factorized into cyclotomic polynomials, phiN means the evaluation of the N-th cyclotomic polynomial at q). Finally, in column "type of u" a label for the class of the unipotent part u is given; we don't give precise explanations of that labeling here.

(Here is a GAP-readable file containing the group order and the sequence of centralizer orders (with multiplicities) as given below.)

There are 346 conjugacy classes.

 # classes|C(su)(q)|, q=2 type of C(s) |Z0(C(s))(q)|type of u
1 3 2^36*3^10*5^2*7^2*11*13*17*19 2E6(q) 1 -
2 3 2^36*3^7*5*7*11 2E6(q) 1 A1
3 3 2^33*3^5*5*7 2E6(q) 1 2A1
4 3 2^31*3^4 2E6(q) 1 3A1
5 3 2^27*3^6 2E6(q) 1 A2
6 3 2^27*3^3*5*7 2E6(q) 1 A2
7 3 2^26*3^4 2E6(q) 1 A2+A1
8 3 2^22*3^4*7 2E6(q) 1 2A2
9 3 2^22*3^4*7 2E6(q) 1 2A2
10 3 2^22*3^4*7 2E6(q) 1 2A2
11 3 2^25*3^2 2E6(q) 1 A2+2A1
12 3 2^19*3^3*5 2E6(q) 1 A3
13 3 2^22*3^2 2E6(q) 1 2A2+A1
14 3 2^22*3^2 2E6(q) 1 2A2+A1
15 3 2^22*3^2 2E6(q) 1 2A2+A1
16 3 2^19*3^2 2E6(q) 1 A3+A1
17 3 2^19*3^3 2E6(q) 1 D4(a1)
18 3 2^19*3 2E6(q) 1 D4(a1)
19 3 2^18*3^2 2E6(q) 1 D4(a1)
20 3 2^15*3^2 2E6(q) 1 A4
21 3 2^14*3^3 2E6(q) 1 D4
22 3 2^14*3^3 2E6(q) 1 D4
23 3 2^15*3 2E6(q) 1 A4+A1
24 3 2^13*3 2E6(q) 1 D5(a1)
25 3 2^12*3^2 2E6(q) 1 A5
26 3 2^12*3^2 2E6(q) 1 A5
27 3 2^12*3^2 2E6(q) 1 A5
28 3 2^13*3 2E6(q) 1 A5+A1
29 3 2^13*3 2E6(q) 1 A5+A1
30 3 2^13*3 2E6(q) 1 A5+A1
31 3 2^13*3 2E6(q) 1 A5+A1
32 3 2^13*3 2E6(q) 1 A5+A1
33 3 2^13*3 2E6(q) 1 A5+A1
34 3 2^10*3 2E6(q) 1 D5
35 3 2^10*3 2E6(q) 1 D5
36 3 2^8*3 2E6(q) 1 E6(a1)
37 3 2^8*3 2E6(q) 1 E6(a1)
38 3 2^8*3 2E6(q) 1 E6(a1)
39 3 2^7*3 2E6(q) 1 E6
40 3 2^7*3 2E6(q) 1 E6
41 3 2^7*3 2E6(q) 1 E6
42 3 2^7*3 2E6(q) 1 E6
43 3 2^7*3 2E6(q) 1 E6
44 3 2^7*3 2E6(q) 1 E6
45 1 2^9*3^9 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 1, 1, 1 ], [ 1, 1, 1 ], [ 1, 1, 1 ] ]
46 1 2^9*3^7 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 1, 1, 1 ], [ 1, 1, 1 ], [ 2, 1 ] ]
47 1 2^8*3^6 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 1, 1, 1 ], [ 1, 1, 1 ], [ 3 ] ]
48 1 2^9*3^7 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 1, 1, 1 ], [ 2, 1 ], [ 1, 1, 1 ] ]
49 1 2^9*3^5 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 1, 1, 1 ], [ 2, 1 ], [ 2, 1 ] ]
50 1 2^8*3^4 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 1, 1, 1 ], [ 2, 1 ], [ 3 ] ]
51 1 2^8*3^6 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 1, 1, 1 ], [ 3 ], [ 1, 1, 1 ] ]
52 1 2^8*3^4 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 1, 1, 1 ], [ 3 ], [ 2, 1 ] ]
53 1 2^7*3^4 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 1, 1, 1 ], [ 3 ], [ 3 ] ]
54 1 2^7*3^4 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 1, 1, 1 ], [ 3 ], [ 3 ] ]
55 1 2^7*3^4 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 1, 1, 1 ], [ 3 ], [ 3 ] ]
56 1 2^9*3^7 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 2, 1 ], [ 1, 1, 1 ], [ 1, 1, 1 ] ]
57 1 2^9*3^5 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 2, 1 ], [ 1, 1, 1 ], [ 2, 1 ] ]
58 1 2^8*3^4 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 2, 1 ], [ 1, 1, 1 ], [ 3 ] ]
59 1 2^9*3^5 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 2, 1 ], [ 2, 1 ], [ 1, 1, 1 ] ]
60 1 2^9*3^3 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 2, 1 ], [ 2, 1 ], [ 2, 1 ] ]
61 1 2^8*3^2 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 2, 1 ], [ 2, 1 ], [ 3 ] ]
62 1 2^8*3^4 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 2, 1 ], [ 3 ], [ 1, 1, 1 ] ]
63 1 2^8*3^2 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 2, 1 ], [ 3 ], [ 2, 1 ] ]
64 1 2^7*3^2 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 2, 1 ], [ 3 ], [ 3 ] ]
65 1 2^7*3^2 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 2, 1 ], [ 3 ], [ 3 ] ]
66 1 2^7*3^2 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 2, 1 ], [ 3 ], [ 3 ] ]
67 1 2^8*3^6 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 1, 1, 1 ], [ 1, 1, 1 ] ]
68 1 2^8*3^4 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 1, 1, 1 ], [ 2, 1 ] ]
69 1 2^7*3^4 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 1, 1, 1 ], [ 3 ] ]
70 1 2^7*3^4 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 1, 1, 1 ], [ 3 ] ]
71 1 2^7*3^4 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 1, 1, 1 ], [ 3 ] ]
72 1 2^8*3^4 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 2, 1 ], [ 1, 1, 1 ] ]
73 1 2^8*3^2 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 2, 1 ], [ 2, 1 ] ]
74 1 2^7*3^2 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 2, 1 ], [ 3 ] ]
75 1 2^7*3^2 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 2, 1 ], [ 3 ] ]
76 1 2^7*3^2 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 2, 1 ], [ 3 ] ]
77 1 2^7*3^4 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 3 ], [ 1, 1, 1 ] ]
78 1 2^7*3^4 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 3 ], [ 1, 1, 1 ] ]
79 1 2^7*3^4 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 3 ], [ 1, 1, 1 ] ]
80 1 2^7*3^2 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 3 ], [ 2, 1 ] ]
81 1 2^7*3^2 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 3 ], [ 2, 1 ] ]
82 1 2^7*3^2 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 3 ], [ 2, 1 ] ]
83 1 2^6*3^2 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 3 ], [ 3 ] ]
84 1 2^6*3^2 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 3 ], [ 3 ] ]
85 1 2^6*3^2 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 3 ], [ 3 ] ]
86 1 2^6*3^2 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 3 ], [ 3 ] ]
87 1 2^6*3^2 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 3 ], [ 3 ] ]
88 1 2^6*3^2 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 3 ], [ 3 ] ]
89 1 2^6*3^2 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 3 ], [ 3 ] ]
90 1 2^6*3^2 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 3 ], [ 3 ] ]
91 1 2^6*3^2 2A2(q) + 2A2(q) + 2A2(q) 1 [ [ 3 ], [ 3 ], [ 3 ] ]
92 3 2^15*3^8*5*7*11 2A5(q) phi2 [ 1, 1, 1, 1, 1, 1 ]
93 3 2^15*3^6*5 2A5(q) phi2 [ 2, 1, 1, 1, 1 ]
94 3 2^14*3^4 2A5(q) phi2 [ 2, 2, 1, 1 ]
95 3 2^12*3^4 2A5(q) phi2 [ 2, 2, 2 ]
96 3 2^11*3^5 2A5(q) phi2 [ 3, 1, 1, 1 ]
97 3 2^11*3^3 2A5(q) phi2 [ 3, 2, 1 ]
98 3 2^9*3^3 2A5(q) phi2 [ 3, 3 ]
99 3 2^9*3^3 2A5(q) phi2 [ 3, 3 ]
100 3 2^9*3^3 2A5(q) phi2 [ 3, 3 ]
101 3 2^8*3^3 2A5(q) phi2 [ 4, 1, 1 ]
102 3 2^8*3^2 2A5(q) phi2 [ 4, 2 ]
103 3 2^6*3^2 2A5(q) phi2 [ 5, 1 ]
104 3 2^5*3^2 2A5(q) phi2 [ 6 ]
105 3 2^5*3^2 2A5(q) phi2 [ 6 ]
106 3 2^5*3^2 2A5(q) phi2 [ 6 ]
107 3 2^6*3^3*5*7^2 A2(q2) phi3 [ 1, 1, 1 ]
108 3 2^6*3*7 A2(q2) phi3 [ 2, 1 ]
109 3 2^4*3*7 A2(q2) phi3 [ 3 ]
110 3 2^4*3*7 A2(q2) phi3 [ 3 ]
111 3 2^4*3*7 A2(q2) phi3 [ 3 ]
112 1 2^12*3^7*5^2*7 D4(q) phi2^2 [ [ 1, 1, 1, 1, 1, 1, 1, 1 ], [ -1 ] ]
113 1 2^12*3^5 D4(q) phi2^2 [ [ 2, 2, 1, 1, 1, 1 ], [ -1, 0 ] ]
114 1 2^10*3^4*5 D4(q) phi2^2 [ [ 2, 2, 1, 1, 1, 1 ], [ -1, 1 ] ]
115 1 2^10*3^4*5 D4(q) phi2^2 [ [ 2, 2, 2, 2 ], [ -1, 0 ], '+' ]
116 1 2^10*3^4*5 D4(q) phi2^2 [ [ 2, 2, 2, 2 ], [ -1, 0 ], '-' ]
117 1 2^10*3^3 D4(q) phi2^2 [ [ 2, 2, 2, 2 ], [ -1, 1 ] ]
118 1 2^9*3^2 D4(q) phi2^2 [ [ 3, 3, 1, 1 ], [ -1, -1, -1 ] ]
119 1 2^9*3^4 D4(q) phi2^2 [ [ 3, 3, 1, 1 ], [ -1, -1, -1 ] ]
120 1 2^6*3^3 D4(q) phi2^2 [ [ 4, 2, 1, 1 ], [ -1, 1, -1, 1 ] ]
121 1 2^6*3^3 D4(q) phi2^2 [ [ 4, 4 ], [ -1, -1, -1, 0 ], '+' ]
122 1 2^6*3^3 D4(q) phi2^2 [ [ 4, 4 ], [ -1, -1, -1, 0 ], '-' ]
123 1 2^6*3^2 D4(q) phi2^2 [ [ 4, 4 ], [ -1, -1, -1, 1 ] ]
124 1 2^5*3^2 D4(q) phi2^2 [ [ 6, 2 ], [ -1, 1, -1, -1, -1, 1 ] ]
125 1 2^5*3^2 D4(q) phi2^2 [ [ 6, 2 ], [ -1, 1, -1, -1, -1, 1 ] ]
126 3 2^6*3^3*5^2*7 A3(q) phi2 phi4 [ 1, 1, 1, 1 ]
127 3 2^6*3^2*5 A3(q) phi2 phi4 [ 2, 1, 1 ]
128 3 2^5*3^2*5 A3(q) phi2 phi4 [ 2, 2 ]
129 3 2^4*3*5 A3(q) phi2 phi4 [ 3, 1 ]
130 3 2^3*3*5 A3(q) phi2 phi4 [ 4 ]
131 3 2^2*3^5 A1(q) + A1(q) phi2^2 phi6 [ [ 1, 1 ], [ 1, 1 ] ]
132 3 2^2*3^4 A1(q) + A1(q) phi2^2 phi6 [ [ 1, 1 ], [ 2 ] ]
133 3 2^2*3^4 A1(q) + A1(q) phi2^2 phi6 [ [ 2 ], [ 1, 1 ] ]
134 3 2^2*3^3 A1(q) + A1(q) phi2^2 phi6 [ [ 2 ], [ 2 ] ]
135 3 2^2*3^3*5^2 A1(q2) phi2^2 phi4 [ 1, 1 ]
136 3 2^2*3^2*5 A1(q2) phi2^2 phi4 [ 2 ]
137 3 2^3*3^2*7^2 A2(q) phi3 phi6 [ 1, 1, 1 ]
138 3 2^3*3*7 A2(q) phi3 phi6 [ 2, 1 ]
139 3 2^2*3*7 A2(q) phi3 phi6 [ 3 ]
140 3 2*3^3*5 A1(q) phi1 phi2^2 phi4 [ 1, 1 ]
141 3 2*3^2*5 A1(q) phi1 phi2^2 phi4 [ 2 ]
142 6 2*3^2*11 A1(q) phi2 phi10 [ 1, 1 ]
143 6 2*3*11 A1(q) phi2 phi10 [ 2 ]
144 1 2*3^4 A1(q) phi2 phi6^2 [ 1, 1 ]
145 1 2*3^3 A1(q) phi2 phi6^2 [ 2 ]
146 6 3^2*11 A0(q) phi2^2 phi10 [ [ 1 ], 1 ]
147 1 3^2*7 A0(q) phi1^2 phi2^2 phi3 [ [ 1 ], 1 ]
148 6 3*19 A0(q) phi18 [ [ 1 ], 1 ]
149 3 3*13 A0(q) phi6 phi12 [ [ 1 ], 1 ]
150 3 3^2*7 A0(q) phi1 phi2 phi3 phi6 [ [ 1 ], 1 ]
151 6 3*17 A0(q) phi1 phi2 phi8 [ [ 1 ], 1 ]
152 6 3*5*7 A0(q) phi1 phi2 phi3 phi4 [ [ 1 ], 1 ]

Character Degrees

The following table lists the degrees of the complex irreducible characters of 2E6(2).

The irreducible characters are parameterized via Lusztig-series and these correspond to classes of semisimple elements in the "dual group". The rows of the following table correspond to types of such Lusztig-series. The column "type of series" gives the Dynkin type of the centralizer of the semisimple element parameterizing the series. The column "# series" is the number of series of that type. The column "# chars in series " shows the number of characters in each series of that type and in "degrees" the corresponding character degrees are listed.

(Here is a GAP-readable file containing the sequence of character degrees (with multiplicities) as given below.)

There are 346 irreducible characters.

 type of series # series # chars in series degrees
1 2E6(q) 1 30
1
1938
48620
554268
815100
1828332
2089164
2956096
4331600
20155200
22170720
62741952
137225088
145411200
145411200
221707200
278555200
289697408
497306304
1003871232
1289932800
1418926080
2217779200
2270281728
3338649600
7488847872
8557215744
12745441280
32514244608
68719476736
2 (2A2(q) + 2A2(q) + 2A2(q)).3 1 17
56581525
56581525
56581525
339489150
452652200
452652200
452652200
678978300
1357956600
2715913200
2715913200
5431826400
10863652800
21727305600
28969740800
28969740800
28969740800
3 2A2(q3).3 2 9
34459425
34459425
34459425
1929727800
1929727800
1929727800
17643225600
17643225600
17643225600
4 2D5(q) 2 20
46683
7189182
15872220
34918884
104756652
244432188
359459100
499134636
698377680
1066613184
1676106432
1955457504
2793510720
3910915008
5751345600
7986154176
8939234304
16253153280
29446889472
48950673408
5 2A4(q) + A1(q) 2 14
27776385
55552770
277763850
555527700
1222160940
2444321880
3333166200
4888643760
6666332400
8888443200
9777287520
17776886400
28443018240
56886036480
6 2A5(q) 1 11
1322685
29099070
333316620
581981400
740703600
814773960
4655851200
6518191680
10666131840
14898723840
43341742080
7 A2(q2) 1 3
505076715
10101534300
32324909760
8 D4(q).3 1 26
2909907
2909907
2909907
81477396
81477396
81477396
145495350
145495350
145495350
733296564
872972100
872972100
872972100
2036934900
2036934900
2036934900
2828429604
2828429604
2828429604
9311702400
9311702400
9311702400
11732745024
11918979072
11918979072
11918979072
9 3D4(q).3 2 24
7194825
7194825
7194825
187065450
187065450
187065450
374130900
374130900
374130900
1410185700
1410185700
1410185700
2331123300
2331123300
2331123300
3367178100
3367178100
3367178100
11972188800
11972188800
11972188800
29470003200
29470003200
29470003200
10 2A2(q) + A1(q) 2 6
4583103525
9166207050
9166207050
18332414100
36664828200
73329656400
11 A3(q) 3 5
707107401
9899503614
14142148020
39598014456
45254873664
12 A1(q) + A1(q) 1 4
13749310575
27498621150
27498621150
54997242300
13 A1(q2) 2 2
10606611015
42426444060
14 A1(q2) 3 2
4949751807
19799007228
15 A2(q) 3 3
7576150725
45456904350
60609205800
16 A1(q) 3 2
24748759035
49497518070
17 A1(q) 6 2
33748307775
67496615550
18 A1(q).3 2 6
4583103525
4583103525
4583103525
9166207050
9166207050
9166207050
19 A1(q).3 1 6
13749310575
13749310575
13749310575
27498621150
27498621150
27498621150
20 A0(q) 6 1
33748307775
21 A0(q) 6 1
58615481925
22 A0(q) 3 1
85668781275
23 A0(q) 3 1
53033055075
24 A0(q) 6 1
65511420975
25 A0(q) 6 1
31819833045
26 A0(q).3 2 3
7576150725
7576150725
7576150725
27 A0(q).3 1 3
17677685025
17677685025
17677685025

Last updated: Thu Aug 25 15:11:07 2005 (CET)