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C₂×#5633 #85982

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

Aliases

Splittings

Fitting	C₂×C₃×C₂²⋊Q₈ ⋊ C₂
Sylow	C₃ ⋊ C₂×#548

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×#548
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

1620 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:

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

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10686802, #10761396, #10810561, …
maximal normal subgroup of	#10686802, #10761396, #10810561, …
maximal quotient of	C₂×#76614, C₂×#73877, C₂×#75982, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18558);