C₂×#20254692 #27088513

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

Aliases

direct product

C₂×#20254692

Splittings

Fitting	C₂×C₄₇×#485 ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3244

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#485	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂×#13093998	C₂²
hypercenter	C₂×#485	D₄₇
hypocenter	C₄₇	C₂×#3244
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#485

1742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 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)(51,57)(55,58)(56,65)(63,64)(67,73)(68,76)(72,74)(77,81);(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)(51,56,57,65)(52,59,60,61)(55,63,58,64)(67,72)(68,76)(70,78)(71,79)(73,74)(75,80);(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)(50,52,54,60)(51,55,57,58)(53,61,62,59)(56,64,65,63)(66,68)(67,71)(69,76)(70,77)(72,80)(73,75)(74,79)(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)(52,58)(53,56)(54,57)(55,60)(59,64)(61,63)(62,65)(66,67)(68,75)(69,73)(70,72)(71,76)(74,78)(77,79)(80,81);(48,49)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1041803);

C2×#20254692 #27088513