Finite Group Databasebeta

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Cayley diagram

C3×#4033 #91604

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

Aliases

direct productC3×#4033

Splittings

HallC3×C5 ⋊ C4×Q8
SylowC5 ⋊ C3×C4×Q8

Subgroups

subgroupG/H
centerC22×C3C22×D5
commutatorC2×C5C22×C3×C4
FrattiniC22C22×C3×D5
FittingC3×C5×C4⋊C4C2
socle, prime radicalC22×C3×C5C23
cosocleC22×C5C23×C3
hypercenterC3×C4⋊C4D5
hypocenterC5C3×C4×Q8
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 480

A minimal faithful permutation representation — 3 generators on 20 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#64872, #11746743, #11747217, …
maximal normal subgroup ofC3×#64872, #11746743, #11747217, …
maximal quotient ofC3×#64935, C3×#64938, C3×#64941, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 680);