C₂×#20263480 #27162452

order	12642
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₂×#20263480

Splittings

Fitting, socle, Hall	C₂×C₇²×C₄₃ ⋊ C₃
derived, Hall	C₇²×C₄₃ ⋊ C₂×C₃
Hall	C₂×C₇² ⋊ C₄₃⋊C₃
Hall	C₂×C₄₃ ⋊ #3861
Sylow	C₇² ⋊ C₂×C₄₃⋊C₃
Sylow	C₄₃ ⋊ C₂×#3861

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#20263480
commutator, cosocle, hypocenter	C₇²×C₄₃	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₄₃	C₃
LCS	G ▷ C₇²×C₄₃
UCS	C₁ ◁ C₂

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 59
transitive degree	μ_t(G) = 4214

A minimal faithful permutation representation — 3 generators on 59 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,30)(18,26,31)(21,29,28)(23,32,25)(27,36,38)(33,39,37)(34,40,35)(41,43,42)(45,46,47)(48,50,49)(52,53,54)(55,57,56)(58,59);(1,2,4,8,15,17,3,6,11,18,27,26,7,13,21,30,38,36,14,23,33,40,22,31,24,34,41,43,32,12,19,28,37,42,39,20,5,9,16,25,35,29,10)(51,52,53,55,54,56,57);(44,45,46,48,47,49,50)

Conjugacy classes & character table

1410 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#38760312, #38760314, #38760322, …
maximal normal subgroup of	#38760312, #38760314, #38760322, …
maximal quotient of	C₄×#20263480, C₂²×#20263480, C₂×#34038921, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12642, 117);

C2×#20263480 #27162452