Finite Group Databasebeta

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#11757163

order	960
isoclinism	isoclinic class of #11757163
automorphism	Aut(G) = #11757163⋊C₂ (order 1920) · Inn(G) = G · Out(G) = C₂
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

C₂⁴ ⋊ A₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	G	C₁
Fitting, socle, cosocle	C₂⁴	A₅
LCS	G
UCS	C₁

63 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 10

A minimal faithful permutation representation — 2 generators on 10 points:

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

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
size	1	5	10	60	60	80	80	80	80	120	192	192
order	1	2	2	2	4	3	6	6	6	4	5	5
χ₁	1	1	1	1	1	1	1	1	1	1	1	1
χ₂	3	3	3	-1	-1	0	0	0	0	-1	$\frac{1 - \sqrt{5}}{2}$	$\frac{1 + \sqrt{5}}{2}$
χ₃	3	3	3	-1	-1	0	0	0	0	-1	$\frac{1 + \sqrt{5}}{2}$	$\frac{1 - \sqrt{5}}{2}$
χ₄	4	4	4	0	0	1	1	1	1	0	-1	-1
χ₅	5	5	5	1	1	-1	-1	-1	-1	1	0	0
χ₆	5	-3	1	1	1	2	0	0	-2	-1	0	0
χ₇	5	-3	1	1	1	-1	$i \sqrt{3}$	$- i \sqrt{3}$	1	-1	0	0
χ₈	5	-3	1	1	1	-1	$- i \sqrt{3}$	$i \sqrt{3}$	1	-1	0	0
χ₉	10	2	-2	2	-2	1	-1	-1	1	0	0	0
χ₁₀	10	2	-2	-2	2	1	-1	-1	1	0	0	0
χ₁₁	15	-9	3	-1	-1	0	0	0	0	1	0	0
χ₁₂	20	4	-4	0	0	-1	1	1	-1	0	0	0

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

Relationships

derived subgroup of	#11757163⋊C₂, C₂×#11757163, C₂×C₃×#11757163, …
maximal subgroup of	#11757163⋊C₂, C₂×#11757163, #72209614, …
maximal normal subgroup of	#11757163⋊C₂, C₂×#11757163, C₃×#11757163, …
maximal quotient of	C₂×#11757163, #14515304, #14515305, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 11358);