C₅×#63956 #13087226

order	1440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#63956

Splittings

Hall	C₃²×C₅ ⋊ #126
Sylow	C₃² ⋊ C₅×#126

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂×D₃²
commutator, cosocle	C₂²×C₃²	C₂³×C₅
Frattini	C₂²	C₂×C₅×D₃²
Fitting, prime radical	C₂³×C₃²×C₅	C₂²
socle	C₂²×C₃²×C₅	C₂³
hypercenter	C₂³×C₅	D₃²
hypocenter	C₃²	C₅×#126
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅

274 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 88 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)(6,7,8)(9,10,11)(13,17)(16,19)(22,26)(24,27);(7,8)(12,14,15,18)(13,16,17,19)(20,22,23,26)(21,24,25,27);(10,11)(12,13)(14,16)(15,17)(18,19)(20,21,23,25)(22,24,26,27)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 3543);