C₂×#20258082 #27123707

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#20258082

Splittings

Fitting	C₂×C₄₇×H(3,𝔽₄) ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3568

Subgroups

	subgroup	G/H
center	C₂³	C₂⁴×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂×C₄₇×H(3,𝔽₄)	C₂
socle	C₂³×C₄₇	C₂⁵
hypercenter	C₂×H(3,𝔽₄)	D₄₇
hypocenter	C₄₇	C₂×#3568
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×H(3,𝔽₄)

6726 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 6 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)(56,61)(64,65)(66,70,71,69)(67,74,73,78)(68,77,75,76)(72,79,80,81);(51,57)(52,60)(56,61)(59,62)(66,69,71,70)(67,74,73,78)(68,76,75,77)(72,79,80,81);(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)(50,53,54,63)(51,56,57,61)(52,59,60,62)(55,64,58,65)(67,73)(68,75)(74,78)(76,77);(50,52)(51,55)(53,62)(54,60)(56,65)(57,58)(59,63)(61,64)(66,68)(67,72)(69,76)(70,77)(71,75)(73,80)(74,79)(78,81);(50,51,54,57)(52,58,60,55)(53,61,63,56)(59,64,62,65)(66,67)(68,72)(69,78)(70,74)(71,73)(75,80)(76,81)(77,79);(48,49)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1076997);

C2×#20258082 #27123707