C₅×#64157 #13088131

order	1440
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²×C₃

supersolvable
metabelian
solvable
monomial

Aliases

direct product

C₅×#64157

Splittings

Hall	C₃²×C₅ ⋊ #131
Sylow	C₃² ⋊ C₅×#131

Subgroups

	subgroup	G/H
center	C₄×C₅	C₂²×C₃⋊D₃
commutator	C₂×C₃²	C₂²×C₄×C₅
Frattini	C₄	C₂²×C₅×C₃⋊D₃
Fitting, prime radical	C₃²×C₅×#41	C₂
socle	C₂×C₃²×C₅	C₂²×C₄
cosocle	C₃²×C₄	C₂³×C₅
hypercenter	C₅×#41	C₃⋊D₃
hypocenter	C₃²	C₅×#131
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄×C₅ ◁ C₅×#41

372 conjugacy classes of subgroups.

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 4448);

C5×#64157 #13088131