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Cayley diagram

C3×ΓL(1,𝔽32) #90820

order465
isoclinismisoclinic class of ΓL(1,𝔽32)
automorphismAut(G) = C2×AGL(1,𝔽31) (order 1860) · Inn(G) = ΓL(1,𝔽32) · Out(G) = C22×C3
homologyAbelianization (H1) = C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC3×ΓL(1,𝔽32)
unambiguous semidirect product(C3×C31)⋊C5

Splittings

Fitting, socle, HallC3×C31 ⋊ C5
derived, SylowC31 ⋊ C3×C5

Subgroups

subgroupG/H
center, hypercenterC3ΓL(1,𝔽32)
commutator, cosocle, hypocenterC31C3×C5
FrattiniC1G
Fitting, socleC3×C31C5
LCSG ▷ C31
UCSC1 ◁ C3

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 93

A minimal faithful permutation representation — 2 generators on 34 points:

(2,3,5,8,4)(6,10,15,18,7)(9,13,14,12,11)(16,22,26,24,17)(19,23,21,25,20)(27,31,30,29,28)(32,33,34);(1,2,4,7,8,12,18,24,5,9,14,20,15,21,26,30,3,6,11,17,13,19,25,29,10,16,23,28,22,27,31)

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×D31⋊C5, D3×ΓL(1,𝔽32), C3⋊(D31⋊C5), …
maximal normal subgroup ofC3×D31⋊C5, D3×ΓL(1,𝔽32), C3⋊(D31⋊C5), …
maximal quotient ofC2×C3×ΓL(1,𝔽32), C9×ΓL(1,𝔽32), C32×ΓL(1,𝔽32), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(465, 2);