C₂×#20253643 #27084586

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#20253643

Splittings

Sylow

C₄₇ ⋊ C₂×#3124

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting, prime radical	C₂×C₄₇×#449	C₂
socle	C₂³×C₄₇	C₂²×D₄
hypercenter	C₂×#449	D₄₇
hypocenter	C₄₇	C₂×#3124
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#449

2134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 6016

A minimal faithful permutation representation — 5 generators on 73 points:

(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)(50,52,53,56)(51,55,54,57)(60,68)(63,67)(69,73)(71,72);(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)(59,64)(62,70)(63,71)(65,66)(67,72)(69,73);(51,54)(55,57)(58,60,61,68)(59,63,65,67)(62,69,70,73)(64,71,66,72);(50,51)(52,55)(53,54)(56,57)(58,59)(60,67)(61,65)(62,64)(63,68)(66,70)(69,72)(71,73);(48,49)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1037876);

C2×#20253643 #27084586