C₂×#20255979 #27094990

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

Aliases

direct product

C₂×#20255979

Splittings

Fitting	C₂×C₄₇×#428 ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3354

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#428	C₂
socle	C₂³×C₄₇	C₂²×D₄
hypercenter	C₂×#428	D₄₇
hypocenter	C₄₇	C₂×#3354
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#428

2146 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)(52,57)(55,56)(59,65)(63,64);(50,53,54,51)(52,56,57,55)(59,64)(60,61)(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)(50,52)(51,55)(53,56)(54,57)(58,60,62,61)(59,63,65,64);(50,51,54,53)(52,56,57,55)(58,59,62,65)(60,63,61,64);(48,49)

Conjugacy classes & character table

1070 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1048280);

C2×#20255979 #27094990