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C9×D5⋊C8 #10623642

order720
isoclinismisoclinic class of AGL(1,𝔽5)
automorphism?
homologyAbelianization (H1) = C2×C8×C9 · Schur multiplier (H2) = C2

Aliases

direct productC9×D5⋊C8
unambiguous semidirect productD5⋊(C8×C9)

Splittings

derived, SylowC5 ⋊ C2×C8×C9
HallC5×C9 ⋊ C2×C8

Subgroups

subgroupG/H
center, hypercenterC4×C9AGL(1,𝔽5)
commutator, hypocenterC5C2×C8×C9
FrattiniC2×C3C2×C3×AGL(1,𝔽5)
FittingC4×C5×C9C4
socleC2×C3×C5C2×C3×C4
prime radicalC2×C3×D5C3×C4
cosocleC3×Dic5C22×C3
LCSG ▷ C5
UCSC1 ◁ C4×C9

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 360

A minimal faithful permutation representation — 2 generators on 22 points:

(1,2,3,4,5,6,7,8,9)(10,11,12,14,13)(15,17,19,18)(16,20,22,21);(11,12,13,14)(15,16,18,21,19,22,17,20)

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C9⋊AGL(1,𝔽5))⋊C8, #13083919, #13083920, …
maximal normal subgroup of(C9⋊AGL(1,𝔽5))⋊C8, #13083919, #13083920, …
maximal quotient ofC9×D5⋊C16, C9×#3990, C4×C9×#170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 111);