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C₂×#5599 #85932

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

Aliases

Splittings

Fitting	C₂×C₃×#121 ⋊ C₂
Sylow	C₃ ⋊ C₂×#550

Subgroups

	subgroup	G/H
center	C₂³	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁵
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂×C₃×#121	C₂
socle	C₂³×C₃	C₂⁴
hypercenter	C₂×#121	D₃
hypocenter	C₃	C₂×#550
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×#121

1736 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 192

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

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

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10686777, #10761436, #10810580, …
maximal normal subgroup of	#10686777, #10761436, #10810580, …
maximal quotient of	C₂×#73877, C₂×#76301, C₂×#75981, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18508);