Generators and Representations for G = M₂₂

Standard generators

• Standard generators of the Mathieu group M₂₂ are a and b where a has order 2, b is in class 4A, ab has order 11 and ababb has order 11. (Source: ATLAS)
• Standard generators of the double cover 2.M₂₂ are preimages A and B where A is in +2A, B is in -4A and AB has order 11 (any two of these conditions imply the third). An equivalent set of conditions is that AB has order 11 and ABABB has order 11. (Source: ATLAS)
• Standard generators of the triple cover 3.M₂₂ are preimages A and B where A has order 2 and B has order 4. (Source: ATLAS)
• Standard generators of the fourfold cover 4.M₂₂ are preimages A and B where [A has order 2,] AB has order 11 and ABABB has order 11. (Source: ATLAS)
• Standard generators of the sixfold cover 6.M₂₂ are preimages A and B where A is in class +2A, and B is in class -4A. (Equivalently, A, B, AB and ABABB have orders 2, 4, 33 and 33 respectively.) (Source: ATLAS)
• Standard generators of the twelvefold cover 12.M₂₂ are preimages A and B where A has order 2, B has order 4, AB has order 33 and ABABB has order 33. (Source: ATLAS)
• Standard generators of the automorphism group M₂₂:2 are c and d where c is in class 2B, d is in class 4C and cd has order 11. (Source: ATLAS)

BBOX programs

• Finder for M₂₂
• Finder for M₂₂.2
• Checker for M₂₂
• Checker for M₂₂.2
• CCL rep tester for M₂₂

Representations

Representation genus			Representations available			Test results		Miscellaneous information
Group	Dim	Ind	File	Char	Ring	Checker	CCL	Sparsity	Max	Size (Kb)	Comments
M22	21	+	21	χ2	Z			92%	1	2.3	perm rep on 23 pts
M22	45	o	45(a)	χ3	C			82%		14.0	exterior square of 10-dim rep of 2.M22
			45(b)	χ4	C			82%		14.0	exterior square of 10-dim rep of 2.M22
M22	55	+	55	χ5	Z			94%	3	13.0	perm rep on 77 pts
M22	99	+	99	χ6	Z			81%	39	43.3	perm rep on 330 pts, LLL'ed
M22	154	+	154	χ7	Z			96%	3	98.6	perm rep on 176 pts, LLL'ed
M22	210	+	210	χ8	Z			79%	66	194.5	perm rep on 462(b) pts, LLL'ed.
M22	231	+	231	χ9	Z			82%	42	231.1	perm rep on 462 pts, LLL'ed
2.M22	10	o	10(a)	χ13	C			64%		1.3	Catalogue of lattices, Nebe & Sloane, split and reseeded
			10(b)	χ14	C			66%		1.2	Catalogue of lattices, Nebe & Sloane, split and reseeded
			20	χ13+χ14	Z			56%	15	2.4	Catalogue of lattices, Nebe & Sloane, reseeded
2.M22	56	+	56	χ15	Z			54%	18	14.9	perm rep on 352 pts
2.M22	120	+	120	χ16	Z			74%	29	63.6	perm rep on 352 pts
2.M22	126	o	No representations available
2.M22	154	o	No representations available
2.M22	210	+	210	χ21	Z			89%	10	185.7	perm rep on 660 pts
3.M22	21	o	21(a)	χ32	Q(ω)			72%		5.3	perm rep on 693 pts
3.M22	45	o	No representations available
3.M22	99	o	No representations available
3.M22	105	o	No representations available
3.M22	210	o	No representations available
3.M22	231	o	No representations available
4.M22	56	o	No representations available
4.M22	144	o	No representations available
4.M22	160	o	No representations available
4.M22	176	o	No representations available
6.M22	66	o	No representations available
6.M22	120	o	No representations available
6.M22	126	o	No representations available
6.M22	210	o	No representations available
12.M22	120	o	No representations available
12.M22	144	o	No representations available

Last generated on Sat Sep 24 18:52:37 2005