C₅×#64159 #13088133

order	1440
isoclinism	isoclinic class of #3753
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₅×#64159

Splittings

Fitting	C₃²×C₅×#41 ⋊ C₂
Hall	C₃²×C₅ ⋊ #135
Sylow	C₃² ⋊ C₅×#135

Subgroups

	subgroup	G/H
center	C₄×C₅	#638
commutator, cosocle	C₃²×C₄	C₂³×C₅
Frattini	C₄	C₂²×C₅×C₃⋊D₃
Fitting	C₃²×C₅×#41	C₂
socle	C₂×C₃²×C₅	C₂×D₄
hypercenter	C₅×#41	C₃⋊D₃
hypocenter	C₃²	C₅×#135
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#41

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

normal subgroups only

This group has 372 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)(7,12)(9,17)(11,18)(13,14)(15,21)(19,20)(25,26,27);(6,8,10,16)(7,11,14,20)(9,15,17,21)(12,18,13,19)(22,23,24);(6,7)(8,11)(9,12)(10,14)(13,17)(15,18)(16,20)(19,21)(23,24)(26,27)

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, 4450);