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C5×C9⋊C16 #10623532

order720
isoclinismisoclinic class of D9
automorphism?
homologyAbelianization (H1) = C5×C16 · Schur multiplier (H2) = C1

Aliases

direct productC5×C9⋊C16
unambiguous semidirect productC9⋊(C5×C16)

Splittings

derived, SylowC9 ⋊ C5×C16
HallC5×C9 ⋊ C16

Subgroups

subgroupG/H
center, hypercenterC5×C8D9
commutator, hypocenterC9C5×C16
FrattiniC3×C8C5×D3
FittingC5×C8×C9C2
socle, prime radicalC2×C3×C5C3⋊C8
cosocleC8×C9C2×C5
LCSG ▷ C9
UCSC1 ◁ C5×C8

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 720

A minimal faithful permutation representation — 2 generators on 30 points:

(1,2,3,4,5)(7,9)(8,12)(10,11)(13,14)(15,16,17,20,18,21,23,26,19,22,24,27,25,28,29,30);(6,14,9,12,11,10,8,7,13)

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13083691, #13083693, D5×C9⋊C16, …
maximal normal subgroup of#13083691, #13083693, D5×C9⋊C16, …
maximal quotient ofC5×C9⋊C32, C2×C5×C9⋊C16

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 1);