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C₄×C₃⋊(C₃⋊D₃) #6309

order	216
isoclinism	isoclinic class of C₃⋊(C₃⋊D₃)
automorphism	Aut(G) order 1213056
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂×C₃³

Aliases

C₄×C₃⋊(C₃⋊D₃)

Splittings

Fitting	C₃³×C₄ ⋊ C₂
derived, Sylow	C₃³ ⋊ C₂×C₄

Subgroups

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

224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 4 generators on 13 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#88785, #88788, C₈×C₃⋊(C₃⋊D₃), …
maximal normal subgroup of	#88785, #88788, C₈×C₃⋊(C₃⋊D₃), …
maximal quotient of	C₈×C₃⋊(C₃⋊D₃), #88850, C₄×C₃⋊(C₃⋊Dic₃), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 146);