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Cayley diagram

C2×D5×D9 #67115

order360
isoclinismisoclinic class of D5×D9
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

direct productC2×D5×D9
unambiguous semidirect product(C5×D9)⋊C22
unambiguous semidirect product(C9×D5)⋊C22
unambiguous semidirect productD45⋊C22

Splittings

FittingC2×C5×C9 ⋊ C22
derived, HallC5×C9 ⋊ C23
SylowC9 ⋊ C22×D5
SylowC5 ⋊ C22×D9

Subgroups

subgroupG/H
center, hypercenterC2D5×D9
commutator, cosocle, hypocenterC5×C9C23
FrattiniC3C2×D3×D5
FittingC2×C5×C9C22
socleC2×C3×C5C2×D3
LCSG ▷ C5×C9
UCSC1 ◁ C2

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 90

A minimal faithful permutation representation — 3 generators on 16 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×D9)⋊AGL(1,𝔽5), C4×D5×D9, D5×D36, …
maximal normal subgroup of(C2×D9)⋊AGL(1,𝔽5), C4×D5×D9, D5×D36, …
maximal quotient ofD9×Dic10, #10623657, #10623658, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 45);