C₅×C₃⋊#564 #11756071

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

Aliases

direct product	C₅×C₃⋊#564
unambiguous semidirect product	C₃⋊(C₅×#564)

Splittings

Hall	C₃×C₅ ⋊ #564
Sylow	C₃ ⋊ C₅×#564

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₃×C₅×#125	C₂
socle, prime radical	C₂²×C₃×C₅	C₂⁴
hypercenter	C₅×#125	D₃
hypocenter	C₃	C₅×#564
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#125

344 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 40
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,3,4,5)(6,7,8)(10,16)(14,22)(15,24)(21,23)(25,28,29,36)(26,32,31,39)(27,34,33,35)(30,37,38,40);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(26,31)(27,33)(32,39)(34,35);(9,11,13,18)(10,14,16,22)(12,19,20,17)(15,23,24,21)(25,27)(26,30)(28,35)(29,33)(31,38)(32,40)(34,36)(37,39);(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,29,31)(27,30,33,38)(28,32,36,39)(34,37,35,40)

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#71250, C₅×#68375, #14290988, …
maximal normal subgroup of	C₅×#71250, C₅×#68375, #14290988, …
maximal quotient of	C₅×#75993, C₅×#74978, C₅×#74979, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 10266);