C₂×#20257893 #27123721

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

Aliases

direct product

C₂×#20257893

Splittings

Sylow

C₄₇ ⋊ C₂×#3557

Subgroups

	subgroup	G/H
center	C₂³	C₂⁴×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting, prime radical	C₂×C₄₇×#540	C₂
socle	C₂³×C₄₇	C₂⁵
hypercenter	C₂×#540	D₄₇
hypocenter	C₄₇	C₂×#3557
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#540

6554 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)(66,70,71,69)(67,75,73,74)(68,78,77,79)(72,80,76,81);(51,57)(55,64)(56,60)(63,65)(66,69,71,70)(67,74,73,75)(68,78,77,79)(72,80,76,81);(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)(50,53,54,62)(51,56,57,60)(52,58,59,61)(55,63,64,65)(67,73)(68,77)(74,75)(78,79);(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)(50,52,54,59)(51,55,57,64)(53,61,62,58)(56,65,60,63)(66,68,71,77)(67,72,73,76)(69,79,70,78)(74,81,75,80);(50,51)(52,55)(53,60)(54,57)(56,62)(58,65)(59,64)(61,63)(66,67)(68,76)(69,75)(70,74)(71,73)(72,77)(78,80)(79,81);(48,49)

Conjugacy classes & character table

1088 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1077011);

C2×#20257893 #27123721