C₂×#20254492 #27087668

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₂×#20254492

Splittings

Fitting	C₂×C₄₇×#497 ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3225

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#497	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂×#15689267	C₂
hypercenter	C₂×#497	D₄₇
hypocenter	C₄₇	C₂×#3225
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#497

2086 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)(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)(51,55)(54,56)(58,61,62,63)(59,66,64,70)(60,67,69,71)(65,72,68,73);(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)(52,57)(54,56)(58,60)(59,65)(61,67)(62,69)(63,71)(64,68)(66,72)(70,73);(50,52,53,57)(51,54,55,56)(58,59)(60,68)(61,70)(62,64)(63,66)(65,69)(67,72)(71,73);(50,51)(52,56)(53,55)(54,57)(59,64)(60,67)(61,63)(65,72)(68,73)(69,71);(48,49)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1040958);

C2×#20254492 #27087668