C₂×#20254839 #27089232

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

Aliases

direct product

C₂×#20254839

Splittings

Sylow

C₄₇ ⋊ C₂×#3261

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#470	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂×#470	D₄₇
hypocenter	C₄₇	C₂×#3261
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#470

1754 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)(60,69)(64,78)(68,83)(70,85)(77,88)(79,86)(82,89)(84,87);(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)(58,61,63,73)(59,65,67,80)(60,68,69,83)(62,75,76,72)(64,77,78,88)(66,81,74,71)(70,87,85,84)(79,89,86,82);(50,51)(52,55)(53,54)(56,57)(58,60,63,69)(59,64,67,78)(61,68,73,83)(62,70,76,85)(65,77,80,88)(66,79,74,86)(71,82,81,89)(72,84,75,87);(51,54)(52,56)(58,59,63,67)(60,64,69,78)(61,71,73,81)(62,74,76,66)(65,75,80,72)(68,82,83,89)(70,86,85,79)(77,87,88,84);(48,49)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1042522);

C2×#20254839 #27089232