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C₃²×#674 #10623720

order	720
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₁₆ · Schur multiplier (H₂) = C₃

Aliases

C₃²×#674

Splittings

derived, Sylow	C₅ ⋊ C₃²×C₁₆
Hall	C₃²×C₅ ⋊ C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₃²×C₄	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₃²×C₁₆
Frattini	C₄	C₃²×AGL(1,𝔽₅)
Fitting	C₃²×C₄×C₅	C₄
socle, prime radical	C₂×C₃²×C₅	C₈
cosocle	#168	C₂×C₃²
LCS	G ▷ C₅
UCS	C₁ ◁ C₃²×C₄

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 720

A minimal faithful permutation representation — 2 generators on 27 points:

(1,2,3)(8,9,10,11)(12,13,14,17,15,18,20,23,16,19,21,24,22,25,26,27);(4,5,6)(7,8,9,11,10)

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₃×#674, C₃×#91164, C₃×#91165, …
maximal normal subgroup of	C₃×D₃×#674, C₃×#91164, C₃×#91165, …
maximal quotient of	C₃²×#3928, C₂×C₃²×#674

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 189);