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C₅×#3753 #10623893

order	720
isoclinism	isoclinic class of #3753
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂×C₃

Aliases

Splittings

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

Subgroups

	subgroup	G/H
center	C₂×C₅	#638
commutator, cosocle	C₃²×C₄	C₂²×C₅
Frattini	C₄	C₂×C₅×C₃⋊D₃
Fitting	C₃²×C₅×D₄	C₂
socle	C₂×C₃²×C₅	D₄
hypercenter	C₅×D₄	C₃⋊D₃
hypocenter	C₃²	C₅×D₈
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×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 — 3 generators on 19 points:

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

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#3753, #13086961, #13086964, …
maximal normal subgroup of	D₅×#3753, #13086961, #13086964, …
maximal quotient of	C₅×#63634, C₅×#63636, C₅×#63653, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 362);