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C₄³×D₃ #73312

order	384
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

C₄³×D₃

Splittings

Fitting	C₃×C₄³ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂×C₄³

Subgroups

	subgroup	G/H
center, hypercenter	C₄³	D₃
commutator, hypocenter	C₃	C₂×C₄³
Frattini	C₂³	C₂³×D₃
Fitting	C₃×C₄³	C₂
socle, cosocle	C₂³×C₃	C₂⁴
prime radical	C₂³×D₃	C₂³
LCS	G ▷ C₃
UCS	C₁ ◁ C₄³

1272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 15 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1276, #10684627, C₄²×D₃×C₈, …
maximal normal subgroup of	D₃×#1276, #10684627, C₄²×D₃×C₈, …
maximal quotient of	C₄³×Dic₃, C₄×D₃×#95, C₄×#4648, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5888);