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Cayley diagram

C22×C9×D5 #67117

order360
isoclinismisoclinic class of D5
automorphism?
homologyAbelianization (H1) = C23×C9 · Schur multiplier (H2) = C23

Aliases

direct productC22×C9×D5
unambiguous semidirect productC5⋊(C23×C9)
unambiguous semidirect product(C2×C5)⋊(C22×C9)

Splittings

FittingC22×C5×C9 ⋊ C2
derived, SylowC5 ⋊ C23×C9
HallC5×C9 ⋊ C23

Subgroups

subgroupG/H
center, hypercenterC22×C9D5
commutator, hypocenterC5C23×C9
FrattiniC3C22×C3×D5
FittingC22×C5×C9C2
socleC22×C3×C5C2×C3
prime radicalC22×C3×D5C3
cosocleC3×C5C23×C3
LCSG ▷ C5
UCSC1 ◁ C22×C9

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 18 points:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10623562, C9×#685, C9×C22⋊AGL(1,𝔽5), …
maximal normal subgroup of#10623562, C9×#685, C9×C22⋊AGL(1,𝔽5), …
maximal quotient ofC2×C9×Dic10, C2×C4×C9×D5, C2×C9×D20, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 47);