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C₂×#5654 #86008

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

Splittings

Fitting	C₂×C₃×#123 ⋊ C₂
Sylow	C₃ ⋊ C₂×#546

Subgroups

	subgroup	G/H
center	C₂³	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁵
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂×C₃×#123	C₂
socle	C₂³×C₃	C₂⁴
hypercenter	C₂×#123	D₃
hypocenter	C₃	C₂×#546
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×#123

1900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 5 generators on 17 points:

(1,2)(3,4,5)(7,9)(10,13)(11,15)(12,16)(14,17);(6,7)(8,9)(10,12,14,15)(11,13,16,17);(10,11)(12,13)(14,16)(15,17);(4,5)(7,9);(1,2)

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10761370, #10810515, C₄×#5654, …
maximal normal subgroup of	#10761370, #10810515, C₄×#5654, …
maximal quotient of	C₂×#74878, C₂×#74121, C₂×#74122, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18584);