C₅×#5819 #11756204

order	960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#5819

Splittings

Hall	C₃×C₅ ⋊ #568
Sylow	C₃ ⋊ C₅×#568

Subgroups

	subgroup	G/H
center	C₄×C₅	C₂³×D₃
commutator	C₂×C₃	C₂³×C₄×C₅
Frattini	C₄	C₂³×C₅×D₃
Fitting, prime radical	C₂×C₃×C₅×#41	C₂
socle	C₂×C₃×C₅	C₂³×C₄
cosocle	C₃×C₄	C₂⁴×C₅
hypercenter	C₂×C₅×#41	D₃
hypocenter	C₃	C₅×#568
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×#41

516 conjugacy classes of subgroups; only the 374 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 374 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) = 24
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 4 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×#70001, C₅×#70135, #14299686, …
maximal normal subgroup of	C₅×#70001, C₅×#70135, #14299686, …
maximal quotient of	C₅×#82202, C₅×#82511, C₅×#84074, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 10399);