C₃²×#4016 #13087795

order	1440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂×C₃

supersolvable
metabelian
solvable
monomial

Aliases

direct product

C₃²×#4016

Splittings

Hall	C₃²×C₅ ⋊ #125
Sylow	C₅ ⋊ C₃²×#125

Subgroups

	subgroup	G/H
center	C₂²×C₃²	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂³×C₃²
Frattini	C₂²	C₂²×C₃²×D₅
Fitting	C₃²×C₄²×C₅	C₂
socle, prime radical	C₂²×C₃²×C₅	C₂³
hypercenter	C₃²×C₄²	D₅
hypocenter	C₅	C₃²×#125
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄²

336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 1440

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

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 4112);

C32×#4016 #13087795