C₅×#5053 #11750552

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

Aliases

direct product

C₅×#5053

Splittings

Fitting	C₃×C₅×#123 ⋊ C₂
Hall	C₃×C₅ ⋊ #458
Sylow	C₃ ⋊ C₅×#458

Subgroups

	subgroup	G/H
center	C₂×C₅	#937
commutator, cosocle	C₂³×C₃	C₂³×C₅
Frattini	C₂³	C₂²×C₅×D₃
Fitting	C₃×C₅×#123	C₂
socle	C₂×C₃×C₅	#120
prime radical	C₅×#1004	C₂
hypercenter	C₅×#123	D₃
hypocenter	C₃	C₅×#458
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ C₅×#123

320 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 78 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)(9,12,13,19)(10,15,16,23)(11,17,18,24)(14,20,21,22);(9,11)(10,14)(12,17)(13,18)(15,22)(16,21)(19,24)(20,23);(7,8)(9,10)(11,14)(12,15)(13,16)(17,22)(18,21)(19,23)(20,24)

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#71922, C₅×#69138, #14316694, …
maximal normal subgroup of	C₅×#71922, C₅×#69138, #14316694, …
maximal quotient of	C₅×#71164, C₅×#69708, C₅×#71172, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4747);