Finite Group Databasebeta

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Cayley diagram

C22×C3×#168 #91637

order480
isoclinismisoclinic class of D5
automorphism?
homologyAbelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C23

Aliases

direct productC22×C3×#168

Splittings

derived, SylowC5 ⋊ C22×C3×C8
HallC3×C5 ⋊ C22×C8

Subgroups

subgroupG/H
center, hypercenterC22×C3×C4D5
commutator, hypocenterC5C22×C3×C8
FrattiniC4C22×C3×D5
FittingC22×C3×C4×C5C2
socle, prime radicalC23×C3×C5C4
cosocleC4×C5C23×C3
LCSG ▷ C5
UCSC1 ◁ C22×C3×C4

152 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)(3,4)(5,6,7)(9,10)(11,12)(13,14,15,17,16,18,19,20);(3,4)(8,9,11,12,10);(1,2)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11745967, #11745968, #11745981, …
maximal normal subgroup of#11745967, #11745968, #11745981, …
maximal quotient ofC2×C3×C4×#168, C2×C3×C4⋊#168, C3×#65214, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 713);