C₂²×#15689382 #27083406

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

Aliases

direct product

C₂²×#15689382

Splittings

Fitting	C₂²×#106×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ C₂²×#470

Subgroups

	subgroup	G/H
center	C₂⁴	D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×#106×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂×D₄
prime radical	C₂⁴×D₄₇	D₄
hypercenter	C₂²×#106	D₄₇
hypocenter	C₄₇	C₂²×#470
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#106

2878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 6016

A minimal faithful permutation representation — 5 generators on 71 points:

(1,2)(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)(53,55)(58,66)(61,70)(64,71)(67,69);(56,58,65,69,60,66,59,67)(57,61,68,71,63,70,62,64);(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)(52,53)(54,55)(56,57,60,63)(58,64,66,71)(59,68,65,62)(61,69,70,67);(50,51);(48,49)

Conjugacy classes & character table

1352 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1036696);

C22×#15689382 #27083406