C₂×#20257172 #27124690

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#20257172

Splittings

Fitting	C₂×D₄×Q₈×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3499

Subgroups

	subgroup	G/H
center	C₂³	C₂⁴×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂×D₄×Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁵
hypercenter	C₂×D₄×Q₈	D₄₇
hypocenter	C₄₇	C₂×#3499
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×D₄×Q₈

11126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 6016

A minimal faithful permutation representation — 6 generators on 69 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)(54,57,58,66)(55,60,61,65)(56,64,63,69)(59,67,62,68);(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)(56,63)(59,62)(64,69)(67,68);(50,51,52,53);(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)(54,56)(55,59)(57,64)(58,63)(60,67)(61,62)(65,68)(66,69);(51,53)(54,55,58,61)(56,62,63,59)(57,65,66,60)(64,67,69,68);(48,49)

Conjugacy classes & character table

1238 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1077980);

C2×#20257172 #27124690