C₅×#64160 #13088134

order	1440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₅×#64160

Splittings

Hall	C₃²×C₅ ⋊ #137
Sylow	C₃² ⋊ C₅×#137

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂×#638
commutator, cosocle	C₃²×C₄	C₂³×C₅
Frattini	C₄	C₂²×C₅×C₃⋊D₃
Fitting, prime radical	C₃²×C₅×#41	C₂
socle	C₂×C₃²×C₅	C₂×D₄
hypercenter	C₅×#41	C₃⋊D₃
hypocenter	C₃²	C₅×#137
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#41

360 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 27
transitive degree	μ_t(G) = 720

A minimal faithful permutation representation — 3 generators on 27 points:

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

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 4451);