C₅×#63810 #13087083

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

Aliases

direct product

C₅×#63810

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)(6,8,16,19,10,17,9,14)(7,11,18,21,13,20,12,15)(22,23,24)(25,26,27);(6,7,10,13)(8,15,17,21)(9,18,16,12)(11,19,20,14)(23,24);(8,14)(9,16)(11,15)(12,18)(17,19)(20,21)(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, 3400);