Ordinary Representations Database

Allan Steel (January 2012)

This page has a small selection from the database of over 1000 absolutely irreducible ordinary representations of finite groups which I constructed in 2010-2011.

The database includes the following absolutely irreducible ordinary representations (always realized over a minimal field):

All 669 non-modular entries of the main table in the Hiss/Malle classification to degree 250 of representations of quasi-simple groups.
The minimal-degree representation of every sporadic group and its covers excepting only the Monster (degree 196883 over Q) and the double cover 2.B of the Baby Monster (degree 96256 over Q).
All representations of every sporadic group to degree 10000 at least.
All representations of every cover of every sporadic group to degree 1000 at least.
All representations of every Mathieu group and its covers.

The database was built by my implementation in Magma of the algorithms for constructing representations which are in my PhD thesis Construction of Ordinary Irreducible Representations of Finite Groups (2012).

The following selection picks out some examples which are discussed in my thesis, plus a few others of interest. Most of these representations have never been constructed before (at least over a minimal field and with small entries), and so this fills very many gaps in characteristic zero at the Online ATLAS of Finite Group Representations.

The selection of representations given here will be expanded greatly!

Information on the table:

For each representation, clicking on the degree gives Magma-readable input (using the standard G-module type).
The group is usually defined via the online ATLAS (within Magma), but for some groups which are not in the ATLAS, definitions have been used from the database built by Derek Holt; the input code temporarily uses a special intrinsic which can be found here (simply save that file and attach it in Magma first).
An asterisk (*) after the name of the group indicates that this is a minimal-degree faithful ordinary representation of the group.
CFD = Character Field Degree (the degree of a minimal-degree field which contains all the character values).
SI = Schur Index. Note that all representations are realized over a number field of minimal degree D, so D always equals CFD*SI (and the field is Q iff D=1).
The total time for the construction of each representation is also given. Times are in seconds except for "Xh", which indicates X hours.

Degree Group CFD SI Time Notes

12 6.Suz * 2 1 66
18 3.J3 * 4 1 7.5
35 Sz(8) 3 1 0.8
36 6.A7 2 1 2.9
36 2.L2(71) 2 1 11
40 2.Sz(8) * 3 1 3.6
51 U4(4) * 4 1 26
51 He * 2 1 18
56 J1 * 2 1 0.7
60 12_2.L3(4) 8 1 139
66 6.M22 * 4 1 6.2
66 3.Suz * 2 1 23
70 J2 2 1 1.7
75 U3(4) 4 1 4.9
80 4_1.L3(4) 4 1 6.8
80 4_2.L3(4) 4 1 17.6
84 L2(83) 20 1 50
85 J3 * 2 1 50
96 L3(5) 10 1 12
120 12.M22 * 8 1 309
121 S10(3) * 2 1 1111
122 2.S10(3) * 2 1 2334
126 3.McL 4 1 72
128 2.A16 * 1 1 228
128 2.A17 * 2 1 3165
133 HN * 2 1 16
143 Suz * 1 1 13
157 U3(13) 6 1 104
170 U9(2) * 1 2 135
171 3.U9(2) * 2 1 110 Irreducible extension of J3 (NF deg 2) representation
171 S6(7) * 2 1 334
171 3.J3 2 1 41
171 3.J3 4 1 860
172 2.S6(7) * 2 1 139
189 3.G2(3) 4 1 97
204 U5(4) * 1 2 57
205 5.U5(4) * 4 1 467
208 2.S4(5) * 2 2 33
216 12_1.U4(3) 4 1 293
216 12_2.U4(3) 4 1 179
220 2.Suz * 1 2 79
221 U4(4) 2 1 38
240 U3(16) * 1 2 108 0.9MB
241 U3(16) 16 1 1.8h 6.6MB
248 Th * 1 1 110
273 3.U3(17) * 6 1 1333
276 Co1 * 1 1 18
342 3.O'N * 4 1 16.2h 3.5MB
351 3.Fi22 * 2 1 2745 3.4MB
352 2.Fi22 * 1 1 85 0.9MB
378 Ru * 2 1 31 1.5MB
760 HN 1 1 359 1.6MB gzipped file (6.4MB uncompressed)
782 Fi23 * 1 1 596 1.0MB gzipped file (4.2MB uncompressed)
783 3.Fi24' * 2 1 2.2h 2.1MB gzipped file (21MB uncompressed)
1333 J4 * 2 1 1354 2.3MB gzipped file (28MB uncompressed)
1728 6.Fi22 * 2 1 47.8h 41MB gzipped file (130MB uncompressed)
2480 Ly * 2 1 20.7h 21MB gzipped file (135MB uncompressed)
4371 B * 1 1 39.6h 39MB gzipped file (230MB uncompressed)

Degree	Group	CFD	SI	Time	Notes
12	6.Suz *	2	1	66
18	3.J3 *	4	1	7.5
35	Sz(8)	3	1	0.8
36	6.A7	2	1	2.9
36	2.L2(71)	2	1	11
40	2.Sz(8) *	3	1	3.6
51	U4(4) *	4	1	26
51	He *	2	1	18
56	J1 *	2	1	0.7
60	12_2.L3(4)	8	1	139
66	6.M22 *	4	1	6.2
66	3.Suz *	2	1	23
70	J2	2	1	1.7
75	U3(4)	4	1	4.9
80	4_1.L3(4)	4	1	6.8
80	4_2.L3(4)	4	1	17.6
84	L2(83)	20	1	50
85	J3 *	2	1	50
96	L3(5)	10	1	12
120	12.M22 *	8	1	309
121	S10(3) *	2	1	1111
122	2.S10(3) *	2	1	2334
126	3.McL	4	1	72
128	2.A16 *	1	1	228
128	2.A17 *	2	1	3165
133	HN *	2	1	16
143	Suz *	1	1	13
157	U3(13)	6	1	104
170	U9(2) *	1	2	135
171	3.U9(2) *	2	1	110	Irreducible extension of J3 (NF deg 2) representation
171	S6(7) *	2	1	334
171	3.J3	2	1	41
171	3.J3	4	1	860
172	2.S6(7) *	2	1	139
189	3.G2(3)	4	1	97
204	U5(4) *	1	2	57
205	5.U5(4) *	4	1	467
208	2.S4(5) *	2	2	33
216	12_1.U4(3)	4	1	293
216	12_2.U4(3)	4	1	179
220	2.Suz *	1	2	79
221	U4(4)	2	1	38
240	U3(16) *	1	2	108	0.9MB
241	U3(16)	16	1	1.8h	6.6MB
248	Th *	1	1	110
273	3.U3(17) *	6	1	1333
276	Co1 *	1	1	18
342	3.O'N *	4	1	16.2h	3.5MB
351	3.Fi22 *	2	1	2745	3.4MB
352	2.Fi22 *	1	1	85	0.9MB
378	Ru *	2	1	31	1.5MB
760	HN	1	1	359	1.6MB gzipped file (6.4MB uncompressed)
782	Fi23 *	1	1	596	1.0MB gzipped file (4.2MB uncompressed)
783	3.Fi24' *	2	1	2.2h	2.1MB gzipped file (21MB uncompressed)
1333	J4 *	2	1	1354	2.3MB gzipped file (28MB uncompressed)
1728	6.Fi22 *	2	1	47.8h	41MB gzipped file (130MB uncompressed)
2480	Ly *	2	1	20.7h	21MB gzipped file (135MB uncompressed)
4371	B *	1	1	39.6h	39MB gzipped file (230MB uncompressed)

Minimal-degree representations of the Fischer group Fi24 (degree 8671 over Q) and the O'Nan group O'N (degree 10944 over Q) have also been constructed, but take a lot of space! I will eventually make them available, when I have developed a suitable compressed format.