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C₃×C₄×D₃ #630

order	72
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂³×D₃ (order 48) · Inn(G) = D₃ · Out(G) = C₂³
holomorph	G ⋊ Aut(G) = D₃×#10596675
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

C₃×C₄×D₃

Splittings

Fitting	C₃²×C₄ ⋊ C₂
derived	C₃ ⋊ C₂×C₃×C₄
Sylow	C₃² ⋊ C₂×C₄

Subgroups

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

35 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 10 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×C₃⋊C₈, #3711, C₃×D₃×C₈, …
maximal normal subgroup of	D₃×C₃⋊C₈, #3711, C₃×D₃×C₈, …
maximal quotient of	C₃×D₃×C₈, C₃×ΓL(1,𝔽₂₅), C₃×C₄×Dic₃, …
automorphism group of	C₂²×C₁₃, D₃×C₁₃
outer automorphism group of	C₂²×C₁₃, Q₈×C₁₃, C₁₃×3₋¹⁺², …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 27);