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C₂×#5726 #86211

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

Aliases

Splittings

Fitting	C₂×C₃×C₄⋊Q₈ ⋊ 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×C₄⋊Q₈	C₂
socle	C₂³×C₃	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₃
hypocenter	C₃	C₂×#547
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

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

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10686907, #10686937, #10687268, …
maximal normal subgroup of	#10686907, #10686937, #10687268, …
maximal quotient of	C₂×#76613, C₂×#76200, C₂×#74360, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18787);