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C₂×#5630 #85980

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

Aliases

Splittings

Fitting	C₂×C₃×C₂²⋊Q₈ ⋊ 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×C₂²⋊Q₈	C₂
socle	C₂³×C₃	C₂⁴
hypercenter	C₂×C₂²⋊Q₈	D₃
hypocenter	C₃	C₂×#546
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

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

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10686877, #10810545, #10810602, …
maximal normal subgroup of	#10686877, #10810545, #10810602, …
maximal quotient of	C₂×#74007, C₂×#76613, C₂×#74595, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18556);