C₂×#20253690 #27084636

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#20253690

Splittings

Fitting	C₂×C₄₇×#479 ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3127

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#479	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂³×D₄₇	C₂×D₄
hypercenter	C₂×#479	D₄₇
hypocenter	C₄₇	C₂×#3127
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#479

1994 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6016

A minimal faithful permutation representation — 5 generators on 89 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)(50,52,53,56)(51,55,54,57)(60,70)(64,68)(69,85)(71,87)(78,82)(79,83)(84,89)(86,88);(58,61,62,73)(59,65,66,80)(60,69,70,85)(63,77,75,74)(64,78,68,82)(67,81,76,72)(71,88,87,86)(79,84,83,89);(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)(51,54)(55,57)(58,60)(59,64)(61,69)(62,70)(63,71)(65,78)(66,68)(67,79)(72,89)(73,85)(74,86)(75,87)(76,83)(77,88)(80,82)(81,84);(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)(50,51)(52,55)(53,54)(56,57)(58,59,62,66)(60,68,70,64)(61,72,73,81)(63,76,75,67)(65,77,80,74)(69,84,85,89)(71,79,87,83)(78,86,82,88);(48,49)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1037926);

C2×#20253690 #27084636