Finite Group Databasebeta

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C9×#687 #10623576

order720
isoclinismisoclinic class of #686
automorphism?
homologyAbelianization (H1) = C22×C9 · Schur multiplier (H2) = C1

Aliases

direct productC9×#687

Splittings

HallC5×C9 ⋊ SD16
SylowC5 ⋊ C9×SD16

Subgroups

subgroupG/H
centerC2×C9C22⋊D5
commutatorC4×C5C22×C9
FrattiniC3×C4C2×C3×D5
FittingC5×D4×C9C2
socleC2×C3×C5C3×D4
prime radicalC3×C5×D4C2×C3
cosocleC3×C4×C5C22×C3
hypercenterD4×C9D5
hypocenterC5C9×SD16
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C9 ◁ C4×C9 ◁ D4×C9

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 360

A minimal faithful permutation representation — 2 generators on 22 points:

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

Conjugacy classes & character table

153 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD9×#687, #13084253, #13084254, …
maximal normal subgroup ofD9×#687, #13084253, #13084254, …
maximal quotient ofC9×#3940, C9×#3942, C9×#3964, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 45);