C₂²×#15689160 #27079927

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂²×#15689160

Splittings

Fitting	C₂²×#107×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ C₂²×#427

Subgroups

	subgroup	G/H
center	C₂⁴	D₄×D₄₇
commutator	C₄×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×#107×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂×D₄
prime radical	C₂⁴×D₄₇	D₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂²×#107	D₄₇
hypocenter	C₄₇	C₂²×#427
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#107

3006 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)(56,59,61,60)(57,63,64,69)(58,66,68,67)(62,65,70,71);(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)(56,58,60,67,61,68,59,66)(57,62,63,65,64,70,69,71);(52,53,54,55)(56,57)(58,65)(59,63)(60,69)(61,64)(62,67)(66,70)(68,71);(50,51);(48,49)

Conjugacy classes & character table

1400 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1033217);

C22×#15689160 #27079927