C₂²×#15689385 #27083650

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

Aliases

direct product

C₂²×#15689385

Splittings

Fitting	C₂²×#100×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ C₂²×#471

Subgroups

	subgroup	G/H
center	C₂³	#13093997
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×#100×C₄₇	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂²×#13094050	C₂
hypercenter	C₂²×#100	D₄₇
hypocenter	C₄₇	C₂²×#471
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#100

2758 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)(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)(52,54)(53,58)(55,61)(56,62)(57,64)(59,67)(60,63)(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)(52,53,55,57,56,59,63,65)(54,58,61,64,62,67,60,66);(53,57)(54,60)(55,63)(59,65)(61,62)(64,66);(50,51);(48,49)

Conjugacy classes & character table

1076 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1036940);

C22×#15689385 #27083650