Finite Group Databasebeta

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Cayley diagram

C5×#809 #91052

order480
isoclinismisoclinic class of #213
automorphism?
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C22

Aliases

direct productC5×#809

Splittings

FittingC3×C5×C4⋊C4 ⋊ C2
HallC3×C5 ⋊ #102
SylowC3 ⋊ C5×#102

Subgroups

subgroupG/H
centerC22×C5#67
commutatorC3×C4C2×C4×C5
FrattiniC2×C4C2×C5×D3
FittingC3×C5×C4⋊C4C2
socleC22×C3×C5D4
prime radicalC2×C5×D12C2
cosocleC2×C3×C4C22×C5
hypercenterC5×C4⋊C4D3
hypocenterC3C5×#102
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C2×C4×C5 ◁ C5×C4⋊C4

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 240

A minimal faithful permutation representation — 2 generators on 20 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11747138, #11747142, #11747216, …
maximal normal subgroup of#11747138, #11747142, #11747216, …
maximal quotient ofC5×#4451, C5×#4482, C5×#4483, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 128);