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C9×#685 #10623574

order720
isoclinismisoclinic class of Dic10
automorphism?
homologyAbelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C22

Aliases

direct productC9×#685

Splittings

FittingC2×C4×C5×C9 ⋊ C2
HallC5×C9 ⋊ C22⋊C4
SylowC5 ⋊ C9×C22⋊C4

Subgroups

subgroupG/H
centerC22×C9C2×D5
commutatorC2×C5C2×C4×C9
FrattiniC22×C3C2×C3×D5
FittingC2×C4×C5×C9C2
socle, cosocleC22×C3×C5C22×C3
prime radicalC22×C3×D5C2×C3
hypercenterC2×C4×C9D5
hypocenterC5C9×C22⋊C4
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C9 ◁ C2×C4×C9

102 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)(15,16,17,18)(20,22);(11,12)(13,14)(19,20)(21,22)

Conjugacy classes & character table

234 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13084127, #13084128, #13084129, …
maximal normal subgroup of#13084127, #13084128, #13084129, …
maximal quotient ofC9×#3937, C9×#3938, C9×#3941, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 43);