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C₄²×C₃⋊C₈ #69386

order	384
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₄³

Aliases

C₄²×C₃⋊C₈

Splittings

C₃ ⋊ C₄²×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₄³	D₃
commutator, hypocenter	C₃	C₄²×C₈
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×C₄³	C₂
socle, prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³
LCS	G ▷ C₃
UCS	C₁ ◁ C₄³

408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10681714, C₄×C₈×C₃⋊C₈, C₄×#4455, …
maximal normal subgroup of	#10681714, C₄×C₈×C₃⋊C₈, C₄×#4455, …
maximal quotient of	C₄×C₈×C₃⋊C₈, C₄×#4455, C₃⋊C₈×#97, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 1962);