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C₂×D₃×#123 #86004

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

Aliases

C₂×D₃×#123

Splittings

Fitting	C₂×C₃×#123 ⋊ C₂
Sylow	C₃ ⋊ C₂²×#123

Subgroups

	subgroup	G/H
center	C₂³	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁵
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂×C₃×#123	C₂
socle	C₂³×C₃	C₂⁴
prime radical	C₂²×D₃×D₄	C₂
hypercenter	C₂×#123	D₃
hypocenter	C₃	C₂²×#123
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×#123

1968 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:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10710649, D₃×#1468, #10761252, …
maximal normal subgroup of	#10710649, D₃×#1468, #10761252, …
maximal quotient of	C₂×D₃×#381, C₂×D₃×#385, C₂×D₃×#386, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18580);