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C₅×#3701 #10623703

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

Aliases

Splittings

Fitting	C₅×Q₈×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×Q₈×C₉	C₂
socle	C₂×C₃×C₅	C₂²×D₃
hypercenter	C₅×Q₈	D₉
hypocenter	C₉	C₅×#41
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₅ ◁ C₅×Q₈

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)(15,17,18,21)(16,19,20,22);(15,16,18,20)(17,22,21,19);(7,9)(8,12)(10,11)(13,14)(17,21)(19,22)

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#13084269, #13084272, #13084277, …
maximal normal subgroup of	#13084269, #13084272, #13084277, …
maximal quotient of	C₅×#63452, C₅×#63454, C₅×#63455, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 172);