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Cayley diagram

C3×#4133 #91979

order480
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC3×#4133

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×Q8C4
socleC2×C3×C5C22×C4
prime radicalC3×#713C2
cosocleDic5C23×C3
hypercenterC3×Q8AGL(1,𝔽5)
hypocenterC5C3×#131
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C3×Q8

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

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11751975, #11751976, #11751977, …
maximal normal subgroup of#11751975, #11751976, #11751977, …
maximal quotient ofC3×#65701, C3×#65702, C3×#65704, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1055);