C₂×#20257559 #27123536

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#20257559

Splittings

Fitting	C₂²×#123×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3533

Subgroups

	subgroup	G/H
center	C₂³	C₂⁴×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂²×#123×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁵
hypercenter	C₂²×#123	D₄₇
hypocenter	C₄₇	C₂×#3533
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×#123

7430 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 6 generators on 73 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(59,65)(63,66)(64,71)(70,72);(51,55)(54,57)(60,68)(63,66)(67,73)(70,72);(51,55)(52,56)(58,61,62,69)(59,64,65,71)(60,67,68,73)(63,70,66,72);(50,52)(51,54)(53,56)(55,57)(58,60)(59,63)(61,67)(62,68)(64,70)(65,66)(69,73)(71,72);(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)(52,54)(53,55)(56,57)(58,59)(60,66)(61,64)(62,65)(63,68)(67,72)(69,71)(70,73);(48,49)

Conjugacy classes & character table

1364 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1076826);

C2×#20257559 #27123536