C₅×#64099 #13088074

order	1440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₅×#64099

Splittings

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

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂²×C₃⋊D₃
commutator, cosocle	C₂²×C₃²	C₂³×C₅
Frattini	C₂²	C₂²×C₅×C₃⋊D₃
Fitting	C₃²×C₅×C₄⋊C₄	C₂
socle, prime radical	C₂²×C₃²×C₅	C₂³
hypercenter	C₅×C₄⋊C₄	C₃⋊D₃
hypocenter	C₃²	C₅×#125
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₄⋊C₄

336 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 1440

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

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 4391);