Finite Group Databasebeta

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Cayley diagram

C2×C3×#683 #91640

order480
isoclinismisoclinic class of Dic10
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×C3×#683

Splittings

HallC3×C5 ⋊ C2×C4⋊C4
SylowC5 ⋊ C2×C3×C4⋊C4

Subgroups

subgroupG/H
centerC23×C3C2×D5
commutatorC2×C5C22×C3×C4
FrattiniC22C22×C3×D5
FittingC22×C3×C4×C5C2
socle, prime radicalC23×C3×C5C22
cosocleC22×C5C23×C3
hypercenterC22×C3×C4D5
hypocenterC5C2×C3×C4⋊C4
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 480

A minimal faithful permutation representation — 3 generators on 18 points:

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11745951, C3×#64689, C3×#64914, …
maximal normal subgroup of#11745951, C3×#64689, C3×#64914, …
maximal quotient ofC3×#65217, C3×C4×#683, C3×#65231, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 716);