Finite Group Databasebeta

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C9×#689 #10623578

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

Aliases

direct productC9×#689

Splittings

HallC5×C9 ⋊ Q16
SylowC5 ⋊ C9×Q16

Subgroups

subgroupG/H
centerC2×C9C22⋊D5
commutatorC4×C5C22×C9
FrattiniC3×C4C2×C3×D5
FittingC5×Q8×C9C2
socle, prime radicalC2×C3×C5C3×D4
cosocleC3×C4×C5C22×C3
hypercenterQ8×C9D5
hypocenterC5C9×Q16
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C9 ◁ C4×C9 ◁ Q8×C9

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 720

A minimal faithful permutation representation — 2 generators on 30 points:

(1,2,3,4,5,6,7,8,9)(10,11,13,14,12)(15,17,19,25)(16,20,22,29)(18,27,28,24)(21,30,26,23);(11,12)(13,14)(15,16,19,22)(17,23,25,30)(18,26,28,21)(20,27,29,24)

Conjugacy classes & character table

153 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD9×#689, #13084277, #13084278, …
maximal normal subgroup ofD9×#689, #13084277, #13084278, …
maximal quotient ofC9×#3939, C9×#3942, C9×#3967, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 47);