C₅×#63713 #13085338

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

Aliases

direct product

C₅×#63713

Splittings

Fitting	C₂×C₅×Q₈×C₉ ⋊ 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	C₂×C₅×Q₈×C₉	C₂
socle	C₂×C₃×C₅	C₂³×D₃
hypercenter	C₂×C₅×Q₈	D₉
hypocenter	C₉	C₅×2₋¹⁺⁴
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₅×Q₈

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

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, 1655);