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C₈×C₃⋊(C₃⋊D₃) #88849

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

Aliases

C₈×C₃⋊(C₃⋊D₃)

Splittings

Fitting	C₃³×C₈ ⋊ C₂
derived, Sylow	C₃³ ⋊ C₂×C₈

Subgroups

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

308 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 4 generators on 17 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11722000, #11722003, C₁₆×C₃⋊(C₃⋊D₃), …
maximal normal subgroup of	#11722000, #11722003, C₁₆×C₃⋊(C₃⋊D₃), …
maximal quotient of	C₁₆×C₃⋊(C₃⋊D₃), #11722168, C₈×C₃⋊(C₃⋊Dic₃), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 496);