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C₂×#5791 #87223

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

Aliases

Splittings

Fitting	C₂⁴×C₃×C₄ ⋊ C₂
Sylow	C₃ ⋊ C₂×#525

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁵
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂⁴×C₃×C₄	C₂
socle	C₂³×C₃	C₂⁴
hypercenter	C₂⁴×C₄	D₃
hypocenter	C₃	C₂×#525
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄

1880 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:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10686640, #10686643, #10701703, …
maximal normal subgroup of	#10686640, #10686643, #10701703, …
maximal quotient of	C₂×#84027, C₂×#82065, C₂×#82069, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 19799);