C₂×#20252106 #27080159

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#20252106

Splittings

Sylow

C₄₇ ⋊ C₂×#2932

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting, prime radical	C₂×C₄₇×#532	C₂
socle	C₂³×C₄₇	C₂³×C₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂×#532	D₄₇
hypocenter	C₄₇	C₂×#2932
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#532

2806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 6016

A minimal faithful permutation representation — 5 generators on 69 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)(54,57)(55,60)(56,62)(58,66)(59,65)(61,64)(63,69)(67,68);(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)(54,56,58,61,59,63,67,55)(57,62,66,64,65,69,68,60);(51,53)(57,65)(60,64)(62,69)(66,68);(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)(52,53)(54,55,58,56,59,61,67,63)(57,64,66,69,65,60,68,62);(48,49)

Conjugacy classes & character table

1376 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1033449);

C2×#20252106 #27080159