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C9×#713 #10623693

order720
automorphism?
homologyAbelianization (H1) = C23×C9 · Schur multiplier (H2) = C22

Aliases

direct productC9×#713

Splittings

FittingC5×Q8×C9 ⋊ C2
HallC5×C9 ⋊ #41
SylowC5 ⋊ C9×#41

Subgroups

subgroupG/H
centerC2×C9C22×D5
commutatorC2×C5C23×C9
FrattiniC2×C3C22×C3×D5
FittingC5×Q8×C9C2
socle, cosocleC2×C3×C5C23×C3
prime radicalC3×#713C3
hypercenterQ8×C9D5
hypocenterC5C9×#41
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C9 ◁ Q8×C9

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 22 points:

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

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC9×#4010, #13083997, #13084261, …
maximal normal subgroup ofC9×#4010, #13083997, #13084261, …
maximal quotient ofC9×#4036, C9×#4038, C9×#4039, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 162);