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C₃×D₇₂ #88461

order	432
isoclinism	isoclinic class of Dic₃₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

C₃×D₇₂

Splittings

Fitting	C₃×C₈×C₉ ⋊ C₂
Sylow	C₃×C₉ ⋊ D₈

Subgroups

	subgroup	G/H
center	C₂×C₃	D₃₆
commutator, cosocle	C₄×C₉	C₂²×C₃
Frattini	C₃×C₄	C₂×C₃×D₃
Fitting	C₃×C₈×C₉	C₂
socle	C₂×C₃²	D₁₂
hypercenter	C₃×C₈	D₉
hypocenter	C₉	C₃×D₈
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×C₈

74 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 2 generators on 20 points:

(1,2,3)(4,5,8,10,12,11,6,9,7)(13,20,15,14,16,17,18,19);(5,7)(6,10)(8,9)(11,12)(14,17)(15,18)(19,20)

Conjugacy classes & character table

117 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11720232, #11720233, #11720238, …
maximal normal subgroup of	#11720232, #11720233, #11720238, …
maximal quotient of	C₃×D₁₄₄, C₃×#63359, C₃×Dic₇₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 108);