C₂×#20257881 #27124112

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

Aliases

direct product

C₂×#20257881

Splittings

Sylow

C₄₇ ⋊ C₂×#3556

Subgroups

	subgroup	G/H
center	C₂³	C₂⁴×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂×C₄₇×#541	C₂
socle	C₂³×C₄₇	C₂⁵
prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×#541	D₄₇
hypocenter	C₄₇	C₂×#3556
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#541

6338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Conjugacy classes & character table

1088 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1077402);

C2×#20257881 #27124112