Finite Group Databasebeta

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C9×#688 #10623577

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

Aliases

direct productC9×#688

Splittings

FittingC5×Q8×C9 ⋊ C2
HallC5×C9 ⋊ SD16
SylowC5 ⋊ C9×SD16

Subgroups

subgroupG/H
centerC2×C9C22⋊D5
commutatorC4×C5C22×C9
FrattiniC3×C4C2×C3×D5
FittingC5×Q8×C9C2
socleC2×C3×C5C3×D4
prime radicalC3×D20C2×C3
cosocleC3×C4×C5C22×C3
hypercenterQ8×C9D5
hypocenterC5C9×SD16
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C9 ◁ C4×C9 ◁ Q8×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)(15,16,18,20)(17,22,21,19);(11,12)(13,14)(16,19)(17,21)(20,22)

Conjugacy classes & character table

153 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD9×#688, #13084269, #13084270, …
maximal normal subgroup ofD9×#688, #13084269, #13084270, …
maximal quotient ofC9×#3940, C9×#3941, C9×#3967, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 46);