C₅×#64128 #13088102

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

Aliases

direct product

C₅×#64128

Splittings

Fitting	C₃²×C₅×Q₁₆ ⋊ 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×Q₁₆	C₂
socle	C₂×C₃²×C₅	C₂×D₄
hypercenter	C₅×Q₁₆	C₃⋊D₃
hypocenter	C₃²	C₅×#135
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×Q₁₆

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)(6,8,10,14)(7,12,11,20)(9,18,19,16)(13,21,17,15)(25,26,27);(6,7,10,11)(8,15,14,21)(9,17,19,13)(12,18,20,16)(22,23,24);(7,11)(8,14)(13,17)(16,18)(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, 4419);