C₂×#20254544 #27088343

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

Aliases

direct product

C₂×#20254544

Splittings

Fitting	C₂×C₄₇×#487 ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3231

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#487	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂³×D₄₇	C₂×D₄
hypercenter	C₂×#487	D₄₇
hypocenter	C₄₇	C₂×#3231
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#487

1778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 6016

A minimal faithful permutation representation — 5 generators on 81 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)(51,57)(55,58)(56,65)(63,64)(66,69,71,70)(67,73,74,78)(68,76,77,79)(72,80,81,75);(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)(50,53,54,62)(51,56,57,65)(52,59,60,61)(55,63,58,64)(66,68)(67,72)(69,76)(70,79)(71,77)(73,80)(74,81)(75,78);(50,52,54,60)(51,55,57,58)(53,61,62,59)(56,64,65,63);(50,51)(52,58)(53,56)(54,57)(55,60)(59,64)(61,63)(62,65)(66,67,71,74)(68,75,77,80)(69,73,70,78)(72,79,81,76);(48,49)

Conjugacy classes & character table

1076 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1041633);

C2×#20254544 #27088343