Finite Group Databasebeta

← Prev (#91976) Next (#91978) →

Cayley diagram

C3×#4131 #91977

order480
automorphismAut(G) = C22×D4×AGL(1,𝔽5) (order 640) · Inn(G) = C22×AGL(1,𝔽5) · Out(G) = C23
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC3×#4131

Splittings

HallC3×C5 ⋊ #131
SylowC5 ⋊ C3×#131

Subgroups

subgroupG/H
centerC2×C3C22×AGL(1,𝔽5)
commutatorC2×C5C22×C3×C4
FrattiniC2C22×C3×AGL(1,𝔽5)
FittingC3×C5×D4C4
socleC2×C3×C5C22×C4
prime radicalC3×#711C2
cosocleDic5C23×C3
hypercenterC3×D4AGL(1,𝔽5)
hypocenterC5C3×#131
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C3×D4

124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 240

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,3,5,4)(8,14)(11,17)(15,20)(18,21)(22,23,24);(6,8)(7,11)(9,15)(10,14)(12,18)(13,17)(16,20)(19,21);(2,5,4,3)(6,12,16,7,10,19,9,13)(8,18,20,11,14,21,15,17)

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11751951, #11751952, #11751953, …
maximal normal subgroup of#11751951, #11751952, #11751953, …
maximal quotient ofC3×#65691, C3×#65692, C3×#65693, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1053);