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D₃×#367 #69344

order	384
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

Fitting	C₃×#367 ⋊ C₂
Sylow	C₃ ⋊ C₂×#367

Subgroups

	subgroup	G/H
center	C₂²	D₃×D₈
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#367	C₂
socle	C₂²×C₃	C₂×D₈
prime radical	C₂²×D₃	D₈
cosocle	C₂×C₃×C₈	C₂³
hypercenter	#367	D₃
hypocenter	C₃	C₂×#367
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 23 points:

(5,6,11,20,8,12,7,10,9,13,14,18,15,17,16,19)(21,22,23);(1,2,3,4)(6,10)(7,14)(8,15)(11,16)(12,20)(13,19)(17,18);(22,23)

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1418, D₃×#1426, C₂×D₃×#367, …
maximal normal subgroup of	D₃×#1418, D₃×#1426, C₂×D₃×#367, …
maximal quotient of	D₃×#1372, #10706830, #10706870, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 1920);