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C₃²×#686 #10623845

order	720
isoclinism	isoclinic class of #686
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

C₃²×#686

Splittings

Fitting	C₃²×C₅×D₄ ⋊ C₂
Hall	C₃²×C₅ ⋊ D₈
Sylow	C₅ ⋊ C₃²×D₈

Subgroups

	subgroup	G/H
center	C₂×C₃²	C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂²×C₃²
Frattini	C₄	C₂×C₃²×D₅
Fitting	C₃²×C₅×D₄	C₂
socle	C₂×C₃²×C₅	D₄
hypercenter	C₃²×D₄	D₅
hypocenter	C₅	C₃²×D₈
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃² ◁ C₃²×C₄ ◁ C₃²×D₄

132 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)(14,16)(15,18)(17,19)

Conjugacy classes & character table

153 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₃×#686, C₃×#91480, C₃×#91481, …
maximal normal subgroup of	C₃×D₃×#686, C₃×#91480, C₃×#91481, …
maximal quotient of	C₃²×#3939, C₃²×#3941, C₃²×#3958, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 314);