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C9×#674 #10623536

order720
isoclinismisoclinic class of AGL(1,𝔽5)
automorphism?
homologyAbelianization (H1) = C9×C16 · Schur multiplier (H2) = C1

Aliases

direct productC9×#674

Splittings

derived, SylowC5 ⋊ C9×C16
HallC5×C9 ⋊ C16

Subgroups

subgroupG/H
center, hypercenterC4×C9AGL(1,𝔽5)
commutator, hypocenterC5C9×C16
FrattiniC3×C4C3×AGL(1,𝔽5)
FittingC4×C5×C9C4
socle, prime radicalC2×C3×C5C3×C8
cosocleC3×#168C2×C3
LCSG ▷ C5
UCSC1 ◁ C4×C9

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:

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

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD9×#674, #13083907, #13083908, …
maximal normal subgroup ofD9×#674, #13083907, #13083908, …
maximal quotient ofC9×#3928, C2×C9×#674

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 5);