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C₄×C₃⋊C₃₂ #67507

order	384
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₄×C₃₂ · Schur multiplier (H₂) = C₄

Aliases

C₄×C₃⋊C₃₂

Splittings

C₃ ⋊ C₄×C₃₂

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₁₆	D₃
commutator, hypocenter	C₃	C₄×C₃₂
Frattini	C₂×C₁₆	C₂×D₃
Fitting	C₃×C₄×C₁₆	C₂
socle, prime radical	C₂²×C₃	C₂×C₁₆
cosocle	C₂×C₃×C₁₆	C₂²
LCS	G ▷ C₃
UCS	C₁ ◁ C₄×C₁₆

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 2 generators on 39 points:

(2,3)(8,9,10,14,11,15,18,24,12,16,19,25,21,27,30,34,13,17,20,26,22,28,31,35,23,29,32,36,33,37,38,39);(1,2,3)(4,5,6,7)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃⋊C₈×C₃₂, C₈×C₃⋊C₃₂, #10682514, …
maximal normal subgroup of	C₃⋊C₈×C₃₂, C₈×C₃⋊C₃₂, #10682514, …
maximal quotient of	C₈×C₃⋊C₃₂, #10682518, #10682874, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 83);