C₂×#20255179 #27092494

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

Aliases

direct product

C₂×#20255179

Splittings

Fitting	C₂×C₄₇×#577 ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3291

Subgroups

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#577	C₂
socle	C₂²×C₄₇	D₄²
hypercenter	C₂×#577	D₄₇
hypocenter	C₄₇	C₂×#3291
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#577

2682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 1504

A minimal faithful permutation representation — 5 generators on 65 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)(50,53)(51,58)(52,59)(54,62)(55,64)(56,57)(60,63)(61,65);(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)(51,52)(54,64)(55,62)(56,57)(58,59)(60,65)(61,63);(50,52)(51,57)(53,61)(54,65)(55,58)(56,63)(59,64)(60,62);(50,51,56,61)(52,62,63,64)(53,58,57,65)(54,60,55,59);(48,49)

Conjugacy classes & character table

1070 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1045784);

C2×#20255179 #27092494