C₂×#20253111 #27083677

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#20253111

Splittings

Fitting	C₂×C₄₇×#471 ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3068

Subgroups

	subgroup	G/H
center	C₂²	#121×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#471	C₂
socle	C₂²×C₄₇	C₂×#121
hypercenter	C₂×#471	D₄₇
hypocenter	C₄₇	C₂×#3068
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#471

2746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 1504

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1036967);

C2×#20253111 #27083677