Finite Group Databasebeta

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Cayley diagram

C2×C3×Dic20 #91622

order480
isoclinismisoclinic class of #677
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productC2×C3×Dic20

Splittings

HallC3×C5 ⋊ C2×Q16
SylowC5 ⋊ C2×C3×Q16

Subgroups

subgroupG/H
centerC22×C3D20
commutator, cosocleC4×C5C23×C3
FrattiniC4C22×C3×D5
FittingC2×C3×C5×C8C2
socle, prime radicalC22×C3×C5D4
hypercenterC2×C3×C8D5
hypocenterC5C2×C3×Q16
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 480

A minimal faithful permutation representation — 3 generators on 26 points:

(3,4,5)(6,7,9,10,8)(11,13,20,24,15,22,14,21)(12,16,23,26,18,25,17,19);(1,2)(7,8)(9,10)(11,12,15,18)(13,19,22,26)(14,23,20,17)(16,24,25,21);(1,2)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11745897, #11745898, #11745921, …
maximal normal subgroup of#11745897, #11745898, #11745921, …
maximal quotient ofC3×#64970, C3×#64971, C3×C4×Dic20, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 698);