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C₂×#5655 #86009

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

Aliases

Splittings

Fitting	C₂×C₃×#123 ⋊ C₂
Sylow	C₃ ⋊ C₂×#547

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×#547
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×#123

1632 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)(10,13,14,12)(11,16,15,17);(1,2)(7,9)(10,12,14,13)(11,16,15,17);(10,11)(12,16)(13,17)(14,15);(4,5)(11,15)(16,17);(1,2)

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10761410, #10810526, C₄×#5655, …
maximal normal subgroup of	#10761410, #10810526, C₄×#5655, …
maximal quotient of	C₂×#74121, C₂×#74672, C₂×#74060, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18585);