C₅×#64364 #13089497

order	1440
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵×C₃

Aliases

direct product

C₅×#64364

Splittings

Fitting	C₂×C₃²×C₅×Q₈ ⋊ C₂
Hall	C₃²×C₅ ⋊ 2₋¹⁺⁴
Sylow	C₃² ⋊ C₅×2₋¹⁺⁴

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂³×C₃⋊D₃
commutator, cosocle	C₂×C₃²	C₂⁴×C₅
Frattini	C₂	C₂³×C₅×C₃⋊D₃
Fitting	C₂×C₃²×C₅×Q₈	C₂
socle	C₂×C₃²×C₅	C₂⁴
hypercenter	C₂×C₅×Q₈	C₃⋊D₃
hypocenter	C₃²	C₅×2₋¹⁺⁴
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₅×Q₈

876 conjugacy classes of subgroups; only the 306 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 306 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 — 4 generators on 27 points:

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

Conjugacy classes & character table

285 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 5814);