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C₄×C₃⋊D₃ #635

order	72
isoclinism	isoclinic class of C₃⋊D₃
automorphism	Aut(G) = C₂²×AGL(2,𝔽₃) (order 1728) · Inn(G) = C₃⋊D₃ · Out(G) = C₂²×S₄
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

C₄×C₃⋊D₃

Splittings

Fitting	C₃²×C₄ ⋊ C₂
derived, Sylow	C₃² ⋊ C₂×C₄

Subgroups

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

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 3 generators on 10 points:

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

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#10593284, #10593285, #10593286, …
maximal subgroup of	#3710, #3712, C₈×C₃⋊D₃, …
maximal normal subgroup of	#3710, #3712, C₈×C₃⋊D₃, …
maximal quotient of	C₈×C₃⋊D₃, #3743, C₄×C₃⋊Dic₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 32);