C₂×#20257877 #27123997

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

Aliases

direct product

C₂×#20257877

Splittings

Fitting	C₂×Q₈²×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3555

Subgroups

	subgroup	G/H
center	C₂³	C₂⁴×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂×Q₈²×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁵
prime radical	C₂×#13093999	C₂²
hypercenter	C₂×Q₈²	D₄₇
hypocenter	C₄₇	C₂×#3555
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×Q₈²

6446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 6016

A minimal faithful permutation representation — 6 generators on 81 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)(66,69,70,78)(67,72,73,77)(68,76,75,81)(71,79,74,80);(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)(51,57)(55,64)(56,60)(63,65)(68,75)(71,74)(76,81)(79,80);(50,53,54,62)(51,56,57,60)(52,58,59,61)(55,63,64,65);(50,52,54,59)(51,55,57,64)(53,61,62,58)(56,65,60,63)(66,68)(67,71)(69,76)(70,75)(72,79)(73,74)(77,80)(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)(50,51)(52,55)(53,60)(54,57)(56,62)(58,65)(59,64)(61,63)(66,67,70,73)(68,74,75,71)(69,77,78,72)(76,79,81,80);(48,49)

Conjugacy classes & character table

1226 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1077287);

C2×#20257877 #27123997