C₂×#20254909 #27091271

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#20254909

Splittings

Fitting	C₂×C₄₇×#546 ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3267

Subgroups

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#546	C₂
socle	C₂³×C₄₇	C₂²×D₄
hypercenter	C₂×#546	D₄₇
hypocenter	C₄₇	C₂×#3267
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×#546

2914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 5 generators on 69 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(55,61)(57,65)(59,62)(60,67)(63,69)(66,68);(51,53)(54,56,58,64)(55,60,62,68)(57,63,65,69)(59,66,61,67);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(54,55)(56,60)(57,61)(58,62)(59,65)(63,67)(64,68)(66,69);(50,51)(52,53)(55,59)(57,65)(60,66)(61,62)(63,69)(67,68);(48,49)

Conjugacy classes & character table

1208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1044561);

C2×#20254909 #27091271