Finite Group Databasebeta

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C₂×C₃⋊(C₃⋊(C₃⋊C₈)) #88854

order	432
isoclinism	isoclinic class of C₃⋊(C₃⋊D₃)
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂×C₃³

Aliases

direct product	C₂×C₃⋊(C₃⋊(C₃⋊C₈))
unambiguous semidirect product	(C₂×C₃)⋊(C₃⋊(C₃⋊C₈))

Splittings

C₃³ ⋊ C₂×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	C₃⋊(C₃⋊D₃)
commutator, hypocenter	C₃³	C₂×C₈
Frattini	C₄	C₂×C₃⋊(C₃⋊D₃)
Fitting	C₂×C₃³×C₄	C₂
socle, prime radical	C₂²×C₃³	C₄
cosocle	C₃³×C₄	C₂²
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂×C₄

308 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 432

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₃×C₃⋊(C₃⋊C₈), C₃⋊C₈×C₃⋊Dic₃, #11722019, …
maximal normal subgroup of	Dic₃×C₃⋊(C₃⋊C₈), C₃⋊C₈×C₃⋊Dic₃, #11722019, …
maximal quotient of	C₄×C₃⋊(C₃⋊(C₃⋊C₈)), C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈))), C₂×C₃⋊(C₃⋊(C₃⋊C₁₆)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 501);