C₂²×#15689446 #27084014

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₂²×#15689446

Splittings

Fitting	C₂²×#102×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ C₂²×#482

Subgroups

	subgroup	G/H
center	C₂⁴	D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×#102×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂×D₄
prime radical	C₂²×#13093995	C₂
hypercenter	C₂²×#102	D₄₇
hypocenter	C₄₇	C₂²×#482
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#102

2702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 6016

A minimal faithful permutation representation — 5 generators on 67 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)(52,54,55,58)(53,57,56,59)(60,62,63,66)(61,65,64,67);(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)(53,56)(57,59)(60,61,63,64)(62,65,66,67);(52,53)(54,57)(55,56)(58,59)(61,64)(62,65)(66,67);(50,51);(48,49)

Conjugacy classes & character table

1364 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1037304);

C22×#15689446 #27084014