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C9×#704 #10623647

order720
isoclinismisoclinic class of #700
automorphism?
homologyAbelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C1

Aliases

direct productC9×#704

Splittings

HallC5×C9 ⋊ M16
SylowC5 ⋊ C9×M16

Subgroups

subgroupG/H
centerC2×C9C2×AGL(1,𝔽5)
commutatorC2×C5C2×C4×C9
FrattiniC2×C3C2×C3×AGL(1,𝔽5)
FittingC22×C5×C9C4
socleC2×C3×C5C2×C3×C4
prime radicalC22×C3×C5C3×C4
cosocleC3×Dic5C22×C3
hypercenterC22×C9AGL(1,𝔽5)
hypocenterC5C9×M16
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C9 ◁ C22×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)(10,11,12,14,13)(16,19)(20,22);(11,12,13,14)(15,16,17,20,18,19,21,22)

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13083934, #13083935, C9×#4005, …
maximal normal subgroup of#13083934, #13083935, C9×#4005, …
maximal quotient ofC9×#4002, C9×#4004, C9×C22⋊#170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 116);