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Cayley diagram

C9×C5⋊D5 #90650

order450
isoclinismisoclinic class of C5⋊D5
automorphism?
homologyAbelianization (H1) = C2×C9 · Schur multiplier (H2) = C5

Aliases

direct productC9×C5⋊D5
unambiguous semidirect productC5⋊(C9×D5)

Splittings

Fitting, HallC52×C9 ⋊ C2
derived, SylowC52 ⋊ C2×C9

Subgroups

subgroupG/H
center, hypercenterC9C5⋊D5
commutator, hypocenterC52C2×C9
FrattiniC3C3×C5⋊D5
FittingC52×C9C2
socle, cosocleC3×C52C2×C3
prime radicalC3×C5⋊D5C3
LCSG ▷ C52
UCSC1 ◁ C9

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 225

A minimal faithful permutation representation — 3 generators on 19 points:

(2,3)(4,5)(7,8)(9,10)(11,12,14,13,15,17,16,18,19);(6,7,9,10,8);(1,2,4,5,3)

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

derived subgroup of#14268098, D52⋊D9, C9⋊#6057
maximal subgroup ofC9×#1043, C9×#1044, C9⋊#1043, …
maximal normal subgroup ofC9×#1043, C9×#1044, C9⋊#1043, …
maximal quotient ofC9×#1039, C2×C9×C5⋊D5, C27×C5⋊D5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(450, 15);