C₅×#63716 #13085341

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

Aliases

direct product

C₅×#63716

Splittings

Fitting	C₅×C₉×#41 ⋊ C₂
Hall	C₅×C₉ ⋊ 2₋¹⁺⁴
Sylow	C₉ ⋊ C₅×2₋¹⁺⁴

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂³×D₉
commutator, cosocle	C₂×C₉	C₂⁴×C₅
Frattini	C₂×C₃	C₂³×C₅×D₃
Fitting	C₅×C₉×#41	C₂
socle	C₂×C₃×C₅	C₂³×D₃
hypercenter	C₅×#41	D₉
hypocenter	C₉	C₅×2₋¹⁺⁴
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₅ ◁ C₅×#41

438 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 204 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) = 30
transitive degree	μ_t(G) = 720

A minimal faithful permutation representation — 4 generators on 30 points:

(1,2,3,4,5)(6,11,13,12,7,9,8,14,10)(15,19,20,18)(16,22,23,27)(17,24,25,26)(21,29,30,28);(15,18,20,19)(16,22,23,27)(17,24,25,26)(21,28,30,29);(15,17,20,25)(16,21,23,30)(18,26,19,24)(22,29,27,28);(7,9)(8,12)(10,11)(13,14)(15,16)(17,21)(18,22)(19,27)(20,23)(24,28)(25,30)(26,29)

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, 1658);