C₂×#20252497 #27081323

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#20252497

Splittings

Fitting	C₂²×C₄⋊C₈×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#2988

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₇
commutator	C₂×C₄₇	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×C₄⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂²×#10625350	C₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂²×C₄⋊C₈	D₄₇
hypocenter	C₄₇	C₂×#2988
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈

2434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 6016

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

Conjugacy classes & character table

1976 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1034613);

C2×#20252497 #27081323