C₂×#20252121 #27080176

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

Aliases

direct product

C₂×#20252121

Splittings

Fitting	C₂×C₄₇×#435 ⋊ C₂
Sylow	C₄₇ ⋊ C₂×D₄×M₁₆

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#435	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂²×D₄×D₄₇	C₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂×#435	D₄₇
hypocenter	C₄₇	C₂×D₄×M₁₆
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#435

3218 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 3008

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

Conjugacy classes & character table

1388 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1033466);

C2×#20252121 #27080176