C₉×#4025 #13084356

order	1440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₉×#4025

Splittings

Hall	C₅×C₉ ⋊ #126
Sylow	C₅ ⋊ C₉×#126

Subgroups

	subgroup	G/H
center	C₂²×C₉	C₂²×D₅
commutator	C₂²×C₅	C₂³×C₉
Frattini	C₂²×C₃	C₂²×C₃×D₅
Fitting	C₅×C₉×C₂²⋊C₄	C₂
socle, cosocle	C₂²×C₃×C₅	C₂³×C₃
prime radical	C₂³×C₃×C₅	C₂²×C₃
hypercenter	C₉×C₂²⋊C₄	D₅
hypocenter	C₅	C₉×#126
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×C₂²⋊C₄

180 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 87 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)(16,20)(19,22)(25,29)(27,30);(15,17,18,21)(16,19,20,22)(23,25,26,29)(24,27,28,30);(11,12)(13,14)(15,16)(17,19)(18,20)(21,22)(23,24,26,28)(25,27,29,30)

Conjugacy classes & character table

306 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 673);