Frank Lübeck

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

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

Zuletzt geändert: Do 25.08.2005, 13:11:07 (UTC)