C₅×#5752 #11756137

order	960
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#5752

Splittings

Fitting	C₃×C₅×#131 ⋊ C₂
Hall	C₃×C₅ ⋊ #576
Sylow	C₃ ⋊ C₅×#576

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂²×D₁₂
commutator, cosocle	C₃×C₄	C₂⁴×C₅
Frattini	C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#131	C₂
socle	C₂×C₃×C₅	C₂²×D₄
hypercenter	C₅×#131	D₃
hypocenter	C₃	C₅×#576
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₅×#41 ◁ C₅×#131

536 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 214 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)(9,11,18,23,13,19,12,17)(10,15,21,24,14,20,16,22);(10,14)(15,20)(16,21)(22,24);(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24);(7,8)(11,17)(12,18)(15,22)(16,21)(19,23)(20,24)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#71094, C₅×#69126, #14316411, …
maximal normal subgroup of	C₅×#71094, C₅×#69126, #14316411, …
maximal quotient of	C₅×#83797, C₅×#83785, C₅×#78560, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 10332);