C₅×#5203 #11750702

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

Aliases

direct product

C₅×#5203

Splittings

Fitting	C₃×C₅×#137 ⋊ C₂
Hall	C₃×C₅ ⋊ #471
Sylow	C₃ ⋊ C₅×#471

Subgroups

	subgroup	G/H
center	C₂×C₅	#939
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#137	C₂
socle	C₂×C₃×C₅	#121
prime radical	C₅×#949	C₂
hypercenter	C₅×#137	D₃
hypocenter	C₃	C₅×#471
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#137

208 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 74 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)(9,11,14,21)(10,16,17,24)(12,20,13,18)(15,23,22,19);(9,10)(11,19)(12,15)(13,22)(14,17)(16,20)(18,24)(21,23);(7,8)(10,15)(11,18)(12,13)(17,22)(19,23)(20,21)

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#471, C₂×C₅×#5203, D₅×#5203, …
maximal normal subgroup of	C₅×D₃×#471, C₂×C₅×#5203, D₅×#5203, …
maximal quotient of	C₅×#71100, C₅×#71116, C₅×#71036, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4897);