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Cayley diagram

C3×#4005 #91209

order480
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×#4005

Splittings

HallC3×C5 ⋊ #101
SylowC5 ⋊ C3×#101

Subgroups

subgroupG/H
centerC2×C3C22⋊AGL(1,𝔽5)
commutatorC22×C5C2×C3×C4
FrattiniC22C2×C3×AGL(1,𝔽5)
FittingC2×C3×C4×C5C4
socleC2×C3×C5C22⋊C4
prime radicalC22×C3×C5C2×C4
cosocleC2×Dic5C22×C3
hypercenterC2×C3×C4AGL(1,𝔽5)
hypocenterC5C3×#101
LCSG ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×C4

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 24 points:

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

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11746558, #11746559, #11746560, …
maximal normal subgroup of#11746558, #11746559, #11746560, …
maximal quotient ofC3×#64870, C3×#64873, C3×#64913, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 285);