C₄×#15689924 #27080491

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

Aliases

direct product

C₄×#15689924

Splittings

Fitting	C₂×C₄×D₈×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ C₄×#576

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₄×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄×D₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	C₂×#15689924	C₂
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂×C₄×D₈	D₄₇
hypocenter	C₄₇	C₄×#576
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

2826 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:

(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)(54,61)(58,65)(62,67)(64,66);(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)(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);(52,53)(54,58)(55,57)(56,60)(59,63)(61,65)(62,64)(66,67);(48,49,50,51)

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, 1033781);

C4×#15689924 #27080491