Finite Group Databasebeta

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M₁₁ #72209634

order	7920
isoclinism	isoclinic class of M₁₁
automorphism	Aut(G) = M₁₁ (order 7920) · Inn(G) = G · Out(G) = C₁
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₁

Aliases

M₁₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	G	C₁
LCS	G
UCS	C₁

39 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Permutation generators:

(1,2)(3,5)(4,7)(10,11);(2,3,6,4)(7,8,10,9)

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀
size	1	165	440	720	720	990	990	990	1320	1584
order	1	2	3	11	11	4	8	8	6	5
χ₁	1	1	1	1	1	1	1	1	1	1
χ₂	10	2	1	-1	-1	2	0	0	-1	0
χ₃	10	-2	1	-1	-1	0	$i \sqrt{2}$	$- i \sqrt{2}$	1	0
χ₄	10	-2	1	-1	-1	0	$- i \sqrt{2}$	$i \sqrt{2}$	1	0
χ₅	11	3	2	0	0	-1	-1	-1	0	1
χ₆	16	0	-2	$\frac{- 1 - i \sqrt{11}}{2}$	$\frac{- 1 + i \sqrt{11}}{2}$	0	0	0	0	1
χ₇	16	0	-2	$\frac{- 1 + i \sqrt{11}}{2}$	$\frac{- 1 - i \sqrt{11}}{2}$	0	0	0	0	1
χ₈	44	4	-1	0	0	0	0	0	1	-1
χ₉	45	-3	0	1	1	1	-1	-1	0	0
χ₁₀	55	-1	1	0	0	-1	1	1	-1	0

Here i is the imaginary unit (i² = −1).

Relationships

automorphism group of

M₁₁

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:

gap> G := Group((1,2)(3,5)(4,7)(10,11), (2,3,6,4)(7,8,10,9));