C₂×#20257534 #27123178

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

Aliases

direct product

C₂×#20257534

Splittings

Fitting	C₂×C₄₇×#563 ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3531

Subgroups

	subgroup	G/H
center	C₂³	C₂⁴×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂×C₄₇×#563	C₂
socle	C₂³×C₄₇	C₂⁵
hypercenter	C₂×#563	D₄₇
hypocenter	C₄₇	C₂×#3531
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#563

6554 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)(66,69,70,77)(67,73,72,79)(68,76,75,81)(71,78,74,80);(51,57)(55,64)(56,60)(63,65)(67,72)(68,75)(73,79)(76,81);(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)(50,53,54,62)(51,56,57,60)(52,58,59,61)(55,63,64,65)(68,75)(71,74)(76,81)(78,80);(50,52,54,59)(51,55,57,64)(53,61,62,58)(56,65,60,63)(66,68)(67,71)(69,76)(70,75)(72,74)(73,78)(77,81)(79,80);(50,51)(52,55)(53,60)(54,57)(56,62)(58,65)(59,64)(61,63)(66,67)(68,74)(69,73)(70,72)(71,75)(76,80)(77,79)(78,81);(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, 1076468);

C2×#20257534 #27123178