C₅×#5005 #11750504

order	960
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#5005

Splittings

Fitting	C₃×C₅×#117 ⋊ C₂
Hall	C₃×C₅ ⋊ #421
Sylow	C₃ ⋊ C₅×#421

Subgroups

	subgroup	G/H
center	C₄×C₅	C₂×D₁₂
commutator	C₃×C₄	C₂²×C₄×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#117	C₂
socle	C₂×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×C₅×#235	C₂
cosocle	C₂×C₃×C₄	C₂³×C₅
hypercenter	C₅×#117	D₃
hypocenter	C₃	C₅×#421
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#117

308 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 118 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)(10,15,16,17)(11,18)(14,21,22,23)(19,24);(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,21)(20,24);(7,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#70295, C₅×#70054, #14297298, …
maximal normal subgroup of	C₅×#70295, C₅×#70054, #14297298, …
maximal quotient of	C₅×#70755, C₅×#70757, C₅×#70767, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4699);