C₅×#63809 #13087081

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

Aliases

direct product

C₅×#63809

Splittings

Hall	C₃²×C₅ ⋊ #135
Sylow	C₃² ⋊ C₅×#135

Subgroups

	subgroup	G/H
center	C₂×C₅	#3802
commutator, cosocle	C₃²×C₄	C₂³×C₅
Frattini	C₄	C₂×C₅×D₃²
Fitting	C₃²×C₅×C₈	C₂²
socle	C₂×C₃²×C₅	C₂×D₄
hypercenter	C₅×C₈	D₃²
hypocenter	C₃²	C₅×#135
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×C₈

270 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 240

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

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

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 3398);