C₂×#20258045 #27123801

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

Aliases

direct product

C₂×#20258045

Splittings

Fitting	C₂×C₄₇×#560 ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3566

Subgroups

	subgroup	G/H
center	C₂³	C₂⁴×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂×C₄₇×#560	C₂
socle	C₂³×C₄₇	C₂⁵
hypercenter	C₂×#560	D₄₇
hypocenter	C₄₇	C₂×#3566
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#560

6694 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)(56,63)(62,64)(66,69,70,76)(67,73,72,79)(68,75,74,80)(71,78,77,81);(52,60)(55,58)(59,65)(62,64)(67,72)(71,77)(73,79)(78,81);(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)(50,53,54,61)(51,56,57,63)(52,59,60,65)(55,62,58,64)(67,72)(68,74)(73,79)(75,80);(50,52)(51,55)(53,59)(54,60)(56,62)(57,58)(61,65)(63,64)(66,68)(67,71)(69,75)(70,74)(72,77)(73,78)(76,80)(79,81);(50,51)(52,58)(53,56)(54,57)(55,60)(59,64)(61,63)(62,65)(66,67)(68,71)(69,73)(70,72)(74,77)(75,78)(76,79)(80,81);(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, 1077091);

C2×#20258045 #27123801