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C₈×C₃⋊D₃ #3742

order	144
isoclinism	isoclinic class of C₃⋊D₃
automorphism	Aut(G) = C₂³×AGL(2,𝔽₃) (order 3456) · Inn(G) = C₃⋊D₃ · Out(G) = C₂³×S₄
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂×C₃

Aliases

C₈×C₃⋊D₃

Splittings

Fitting	C₃²×C₈ ⋊ C₂
derived, Sylow	C₃² ⋊ C₂×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₈	C₃⋊D₃
commutator, hypocenter	C₃²	C₂×C₈
Frattini	C₄	C₂×C₃⋊D₃
Fitting	C₃²×C₈	C₂
socle	C₂×C₃²	C₂×C₄
prime radical	C₂×C₃⋊D₃	C₄
cosocle	C₃²×C₄	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₈

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#11798072, #11798103, #11798148, …
maximal subgroup of	#63542, #63544, C₁₆×C₃⋊D₃, …
maximal normal subgroup of	#63542, #63544, C₁₆×C₃⋊D₃, …
maximal quotient of	C₁₆×C₃⋊D₃, #63625, C₈×C₃⋊Dic₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 85);