Finite Group Databasebeta

← Prev (#86187) Next (#86189) →

C₂×#5718 #86188

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

Aliases

Splittings

Fitting	C₂×C₃×#127 ⋊ C₂
Sylow	C₃ ⋊ C₂×#550

Subgroups

	subgroup	G/H
center	C₂³	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁵
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂×C₃×#127	C₂
socle	C₂³×C₃	C₂⁴
hypercenter	C₂×#127	D₃
hypocenter	C₃	C₂×#550
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×#127

1736 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)(7,9)(10,12)(11,14)(13,16)(15,17);(10,11,13,15)(12,17,16,14);(6,7,8,9);(4,5)(7,9)(12,16)(14,17);(1,2)

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10686892, #10686897, #10687256, …
maximal normal subgroup of	#10686892, #10686897, #10687256, …
maximal quotient of	C₂×#76301, C₂×#74323, C₂×#76302, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18764);