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C₅×#3699 #10623701

order	720
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

Fitting	C₅×D₄×C₉ ⋊ C₂
Hall	C₅×C₉ ⋊ #41
Sylow	C₉ ⋊ C₅×#41

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂²×D₉
commutator, cosocle	C₂×C₉	C₂³×C₅
Frattini	C₂×C₃	C₂²×C₅×D₃
Fitting	C₅×D₄×C₉	C₂
socle	C₂×C₃×C₅	C₂²×D₃
hypercenter	C₅×D₄	D₉
hypocenter	C₉	C₅×#41
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₅ ◁ C₅×D₄

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 360

A minimal faithful permutation representation — 3 generators on 22 points:

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#13084245, #13084248, #13084253, …
maximal normal subgroup of	#13084245, #13084248, #13084253, …
maximal quotient of	C₅×#63439, C₅×#63440, C₅×#63447, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 170);