C₅×#5196 #11750695

order	960
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#5196

Splittings

Hall	C₃×C₅ ⋊ #443
Sylow	C₃ ⋊ C₅×#443

Subgroups

	subgroup	G/H
center	C₄×C₅	D₃×D₄
commutator	C₃×C₄	C₂²×C₄×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting, prime radical	C₃×C₅×#135	C₂
socle	C₂×C₃×C₅	C₄×D₄
cosocle	C₂×C₃×C₄	C₂³×C₅
hypercenter	C₅×#135	D₃
hypocenter	C₃	C₅×#443
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#135

212 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 106 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 — 3 generators on 24 points:

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

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#69238, C₅×#71511, #14315649, …
maximal normal subgroup of	C₅×#69238, C₅×#71511, #14315649, …
maximal quotient of	C₅×D₈×C₃⋊C₈, C₅×SD₁₆×C₃⋊C₈, C₅×Q₁₆×C₃⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4890);