C₂×#20255169 #27092483

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#20255169

Splittings

Fitting	C₂×C₄₇×#472 ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3290

Subgroups

	subgroup	G/H
center	C₂²	#15689757
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#472	C₂
socle	C₂²×C₄₇	D₄²
hypercenter	C₂×#472	D₄₇
hypocenter	C₄₇	C₂×#3290
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#472

2650 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)(50,53)(51,58)(52,63)(54,69)(55,70)(56,74)(57,71)(59,78)(60,77)(61,80)(62,64)(65,67)(66,81)(68,72)(73,76)(75,79);(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)(50,53)(51,57)(52,62)(54,68)(55,70)(56,73)(58,71)(59,77)(60,78)(61,80)(63,64)(65,81)(66,67)(69,72)(74,76)(75,79);(50,52)(51,56)(53,67)(54,65)(55,66)(57,74)(58,79)(59,75)(60,76)(61,71)(62,70)(63,68)(64,72)(69,81)(73,78)(77,80);(50,51)(52,61)(53,58)(54,71)(55,60)(56,64)(57,69)(59,72)(62,74)(63,80)(65,76)(66,75)(67,73)(68,78)(70,77)(79,81);(48,49)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1045773);

C2×#20255169 #27092483