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C₃²×#687 #10623846

order	720
isoclinism	isoclinic class of #686
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₃

Aliases

C₃²×#687

Splittings

Hall	C₃²×C₅ ⋊ SD₁₆
Sylow	C₅ ⋊ C₃²×SD₁₆

Subgroups

	subgroup	G/H
center	C₂×C₃²	C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂²×C₃²
Frattini	C₄	C₂×C₃²×D₅
Fitting, prime radical	C₃²×C₅×D₄	C₂
socle	C₂×C₃²×C₅	D₄
hypercenter	C₃²×D₄	D₅
hypocenter	C₅	C₃²×SD₁₆
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃² ◁ C₃²×C₄ ◁ C₃²×D₄

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

Conjugacy classes & character table

153 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₃×#687, C₃×#91486, C₃×#91487, …
maximal normal subgroup of	C₃×D₃×#687, C₃×#91486, C₃×#91487, …
maximal quotient of	C₃²×#3940, C₃²×#3942, C₃²×#3964, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 315);