C₅×#64151 #13088127

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

Aliases

direct product

C₅×#64151

Splittings

Fitting	C₂×C₃²×C₅×Q₈ ⋊ C₂
Hall	C₃²×C₅ ⋊ #137
Sylow	C₃² ⋊ C₅×#137

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂×#638
commutator, cosocle	C₃²×C₄	C₂³×C₅
Frattini	C₄	C₂²×C₅×C₃⋊D₃
Fitting	C₂×C₃²×C₅×Q₈	C₂
socle	C₂×C₃²×C₅	C₂×D₄
prime radical	C₅×#3828	C₂
hypercenter	C₂×C₅×Q₈	C₃⋊D₃
hypocenter	C₃²	C₅×#137
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×Q₈

360 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 130 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)(6,9,11,18)(7,13,14,21)(8,15,10,17)(12,20,19,16)(25,26,27);(6,8,11,10)(7,12,14,19)(9,17,18,15)(13,16,21,20)(22,23,24);(6,7)(8,12)(9,16)(10,19)(11,14)(13,15)(17,21)(18,20)(23,24)(26,27)

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 4444);