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C9×C22⋊AGL(1,𝔽5) #10623648

order720
isoclinismisoclinic class of #700
automorphism?
homologyAbelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C22

Aliases

direct productC9×C22⋊AGL(1,𝔽5)
unambiguous semidirect product(C2×AGL(1,𝔽5))⋊(C2×C9)

Splittings

FittingC22×C5×C9 ⋊ C4
HallC5×C9 ⋊ C22⋊C4
SylowC5 ⋊ C9×C22⋊C4

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×D5C2×C3
cosocleC2×C3×D5C22×C3
hypercenterC22×C9AGL(1,𝔽5)
hypocenterC5C9×C22⋊C4
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C9 ◁ C22×C9

102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 180

A minimal faithful permutation representation — 2 generators on 18 points:

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

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13083926, #13083927, C9×#3999, …
maximal normal subgroup of#13083926, #13083927, C9×#3999, …
maximal quotient ofC9×#3999, C9×#4003, C9×#4005, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 117);