C₄×#15689934 #27080507

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

Aliases

direct product

C₄×#15689934

Splittings

Fitting	C₄×H(5,𝔽₂)×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ C₄×#577

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄×H(5,𝔽₂)×C₄₇	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	C₂×#15689934	C₂
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₄×H(5,𝔽₂)	D₄₇
hypocenter	C₄₇	C₄×#577
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₄×H(5,𝔽₂)

2742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 5 generators on 67 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)(54,61)(58,65)(62,67)(64,66);(52,54)(53,58)(55,62)(56,61)(57,64)(59,66)(60,65)(63,67);(53,57)(55,63)(58,64)(59,60)(62,67)(65,66);(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)(52,53,56,60)(54,58,61,65)(55,57,63,59)(62,64,67,66);(48,49,50,51)

Conjugacy classes & character table

1652 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1033797);

C4×#15689934 #27080507