C₂×#20253679 #27084624

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

Aliases

direct product

C₂×#20253679

Splittings

Fitting	C₂×C₄₇×#519 ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3126

Subgroups

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#519	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂×#15689728	C₂
hypercenter	C₂×#519	D₄₇
hypocenter	C₄₇	C₂×#3126
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#519

2122 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:

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

Conjugacy classes & character table

1628 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1037914);

C2×#20253679 #27084624