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C₉×#700 #10623643

order	720
isoclinism	isoclinic class of #700
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₉ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Hall	C₅×C₉ ⋊ M₁₆
Sylow	C₅ ⋊ C₉×M₁₆

Subgroups

	subgroup	G/H
center	C₂×C₉	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₄×C₉
Frattini	C₂×C₃	C₂×C₃×AGL(1,𝔽₅)
Fitting	C₄×C₅×C₉	C₄
socle	C₂×C₃×C₅	C₂×C₃×C₄
prime radical	C₂×C₃×D₅	C₃×C₄
cosocle	C₃×Dic₅	C₂²×C₃
hypercenter	C₄×C₉	AGL(1,𝔽₅)
hypocenter	C₅	C₉×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₉ ◁ C₄×C₉

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 22 points:

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

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#13083922, #13083923, #13083924, …
maximal normal subgroup of	#13083922, #13083923, #13083924, …
maximal quotient of	C₉×#4002, C₉×#4003, C₉×C₄⋊#170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 112);