C₂×#20254046 #27086121

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

Aliases

direct product

C₂×#20254046

Splittings

Fitting	C₂×C₄₇×#491 ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3177

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#491	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂³×D₄₇	C₂×D₄
hypercenter	C₂×#491	D₄₇
hypocenter	C₄₇	C₂×#3177
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#491

1842 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)(64,77)(68,83)(70,82)(76,88)(78,85)(81,89)(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)(58,61,72,79,63,74,62,73)(59,65,75,86,67,80,66,71)(60,68,82,87,69,83,70,84)(64,76,85,89,77,88,78,81);(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)(51,54)(55,57)(58,60,63,69)(59,64,67,77)(61,68,74,83)(62,70,72,82)(65,76,80,88)(66,78,75,85)(71,81,86,89)(73,84,79,87);(50,51)(52,55)(53,54)(56,57)(58,59,63,67)(60,64,69,77)(61,71,74,86)(62,75,72,66)(65,79,80,73)(68,81,83,89)(70,85,82,78)(76,87,88,84);(48,49)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1039411);

C2×#20254046 #27086121