C₂×#20254942 #27091306

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

Aliases

direct product

C₂×#20254942

Splittings

Fitting	C₂×C₄₇×#497 ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3269

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#497	C₂
socle	C₂³×C₄₇	C₂²×D₄
hypercenter	C₂×#497	D₄₇
hypocenter	C₄₇	C₂×#3269
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#497

2774 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)(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)(54,58)(55,57)(56,61)(59,63)(60,62)(64,67)(65,68)(66,69);(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)(51,53)(54,57,60,59)(55,62,63,58)(56,65,66,67)(61,68,69,64);(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)(54,56,60,66)(55,61,63,69)(57,65,59,67)(58,64,62,68);(50,51)(52,53)(54,55,60,63)(56,64,66,68)(57,62,59,58)(61,67,69,65);(48,49)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1044596);

C2×#20254942 #27091306