C₂×#20251998 #27079975

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#20251998

Splittings

Fitting	C₂×C₄₇×#430 ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#2916

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₄×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#430	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂²×#10625350	C₂²
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂×#430	D₄₇
hypocenter	C₄₇	C₂×#2916
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#430

2586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Conjugacy classes & character table

1100 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1033265);

C2×#20251998 #27079975