C₉×#4061 #13084391

order	1440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#4061

Splittings

Fitting	C₅×C₉×SD₁₆ ⋊ C₂
Hall	C₅×C₉ ⋊ #137
Sylow	C₅ ⋊ C₉×#137

Subgroups

	subgroup	G/H
center	C₂×C₉	D₄×D₅
commutator	C₄×C₅	C₂³×C₉
Frattini	C₃×C₄	C₂²×C₃×D₅
Fitting	C₅×C₉×SD₁₆	C₂
socle	C₂×C₃×C₅	C₂×C₃×D₄
prime radical	C₃×#711	C₂×C₃
cosocle	C₃×C₄×C₅	C₂³×C₃
hypercenter	C₉×SD₁₆	D₅
hypocenter	C₅	C₉×#137
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₉ ◁ C₄×C₉ ◁ C₉×SD₁₆

180 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 720

A minimal faithful permutation representation — 3 generators on 30 points:

(1,2,3,4,5,6,7,8,9)(10,11,13,14,12)(16,21)(17,23)(18,26)(20,27)(24,25)(28,30);(15,17,19,25)(16,20,21,28)(18,23,26,24)(22,30,29,27);(11,12)(13,14)(15,16)(17,20)(18,22)(19,21)(23,30)(24,27)(25,28)(26,29)

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 708);