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C₂×D₃×C₃₂ #69364

order	384
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂³

Aliases

C₂×D₃×C₃₂

Splittings

Fitting	C₂×C₃×C₃₂ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂²×C₃₂

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃₂	D₃
commutator, hypocenter	C₃	C₂²×C₃₂
Frattini	C₁₆	C₂²×D₃
Fitting	C₂×C₃×C₃₂	C₂
socle	C₂²×C₃	C₂×C₁₆
prime radical	C₂²×D₃	C₁₆
cosocle	C₃×C₁₆	C₂³
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₃₂

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	(C₂×D₃)⋊C₆₄, C₄×D₃×C₃₂, D₃×#1397, …
maximal normal subgroup of	(C₂×D₃)⋊C₆₄, C₄×D₃×C₃₂, D₃×#1397, …
maximal quotient of	C₄×D₃×C₃₂, C₂×Dic₃×C₃₂, D₃×C₂²⋊C₃₂, …
automorphism group of	C₃⋊C₁₂₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 1940);