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C9×#676 #10623565

order720
isoclinismisoclinic class of Dic10
automorphism?
homologyAbelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C1

Aliases

direct productC9×#676

Splittings

FittingC5×C8×C9 ⋊ C2
HallC5×C9 ⋊ M16
SylowC5 ⋊ C9×M16

Subgroups

subgroupG/H
centerC4×C9C2×D5
commutatorC2×C5C2×C4×C9
FrattiniC3×C4C2×C3×D5
FittingC5×C8×C9C2
socleC2×C3×C5C2×C3×C4
prime radicalC2×C3×D5C3×C4
cosocleC3×C4×C5C22×C3
hypercenterC8×C9D5
hypocenterC5C9×M16
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C4×C9 ◁ C8×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)(15,16,17,20,18,19,21,22);(11,12)(13,14)(16,19)(20,22)

Conjugacy classes & character table

234 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13084003, #13084004, D9×#676, …
maximal normal subgroup of#13084003, #13084004, D9×#676, …
maximal quotient ofC9×#3946, C9×#3947, C9×#3952, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 34);