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C₃²×#704 #10624083

order	720
isoclinism	isoclinic class of #700
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₄ · Schur multiplier (H₂) = C₃

Aliases

C₃²×#704

Splittings

Hall	C₃²×C₅ ⋊ M₁₆
Sylow	C₅ ⋊ C₃²×M₁₆

Subgroups

	subgroup	G/H
center	C₂×C₃²	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₃²×C₄
Frattini	C₂	C₂×C₃²×AGL(1,𝔽₅)
Fitting, prime radical	C₂²×C₃²×C₅	C₄
socle	C₂×C₃²×C₅	C₂×C₄
cosocle	Dic₅	C₂²×C₃²
hypercenter	C₂²×C₃²	AGL(1,𝔽₅)
hypocenter	C₅	C₃²×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃² ◁ C₂²×C₃²

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 360

A minimal faithful permutation representation — 2 generators on 19 points:

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

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#91174, C₃×#91175, #13085438, …
maximal normal subgroup of	C₃×#91174, C₃×#91175, #13085438, …
maximal quotient of	C₃²×#4002, C₃²×#4004, C₃²×C₂²⋊#170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 552);