C₂×#20253880 #27085777

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#20253880

Splittings

Fitting	C₂×C₄₇×#497 ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3156

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#497	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×#13094045	C₂
hypercenter	C₂×#497	D₄₇
hypocenter	C₄₇	C₂×#3156
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#497

3150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6016

A minimal faithful permutation representation — 5 generators on 85 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)(57,69)(61,75)(66,80)(70,79)(73,84)(76,82)(77,85)(81,83);(50,51,52,53)(54,57)(55,61)(56,66)(58,70)(59,69)(60,73)(62,76)(63,75)(64,77)(65,79)(67,81)(68,80)(71,82)(72,83)(74,84)(78,85);(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)(54,56,65,72,59,68,58,67)(55,60,71,78,63,74,62,64)(57,66,79,83,69,80,70,81)(61,73,82,85,75,84,76,77);(51,53)(54,55,59,63)(56,64,68,78)(57,61,69,75)(58,71,65,62)(60,72,74,67)(66,77,80,85)(70,82,79,76)(73,83,84,81);(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, 1039067);

C2×#20253880 #27085777