C₅×#64121 #13088095

order	1440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₅×#64121

Splittings

Fitting	C₃²×C₅×D₈ ⋊ C₂
Hall	C₃²×C₅ ⋊ #135
Sylow	C₃² ⋊ C₅×#135

Subgroups

	subgroup	G/H
center	C₂×C₅	D₄×C₃⋊D₃
commutator, cosocle	C₃²×C₄	C₂³×C₅
Frattini	C₄	C₂²×C₅×C₃⋊D₃
Fitting	C₃²×C₅×D₈	C₂
socle	C₂×C₃²×C₅	C₂×D₄
hypercenter	C₅×D₈	C₃⋊D₃
hypocenter	C₃²	C₅×#135
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×D₈

372 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 720

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

(1,2,3,4,5)(7,12)(8,15)(9,18)(11,19)(14,21)(16,17)(25,26,27);(6,8)(7,11)(9,15)(10,17)(12,14)(13,21)(16,18)(19,20)(22,23,24);(6,7)(8,14)(9,13)(10,12)(11,17)(15,19)(16,21)(18,20)(23,24)(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, 4412);