C₅×#64298 #13089311

order	1440
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#64298

Splittings

Hall	C₃²×C₅ ⋊ 2₋¹⁺⁴
Sylow	C₃² ⋊ C₅×2₋¹⁺⁴

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂²×D₃²
commutator, cosocle	C₂×C₃²	C₂⁴×C₅
Frattini	C₂	C₂²×C₅×D₃²
Fitting	C₂×C₃²×C₄×C₅	C₂²
socle	C₂×C₃²×C₅	C₂⁴
hypercenter	C₂×C₄×C₅	D₃²
hypocenter	C₃²	C₅×2₋¹⁺⁴
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅

622 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 216 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 — 4 generators on 27 points:

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

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 5628);