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C32×ΓL(1,𝔽32) #13063698

order1395
isoclinismisoclinic class of ΓL(1,𝔽32)
automorphism?
homologyAbelianization (H1) = C32×C5 · Schur multiplier (H2) = C3

Aliases

direct productC32×ΓL(1,𝔽32)
unambiguous semidirect product(C32×C31)⋊C5

Splittings

Fitting, socle, HallC32×C31 ⋊ C5
derived, SylowC31 ⋊ C32×C5

Subgroups

subgroupG/H
center, hypercenterC32ΓL(1,𝔽32)
commutator, cosocle, hypocenterC31C32×C5
FrattiniC1G
Fitting, socleC32×C31C5
LCSG ▷ C31
UCSC1 ◁ C32

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 279

A minimal faithful permutation representation — 2 generators on 37 points:

(1,2,3)(8,9,11,14,10)(12,16,21,24,13)(15,19,20,18,17)(22,28,32,30,23)(25,29,27,31,26)(33,37,36,35,34);(4,5,6)(7,8,10,13,14,18,24,30,11,15,20,26,21,27,32,36,9,12,17,23,19,25,31,35,16,22,29,34,28,33,37)

Conjugacy classes & character table

99 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC32×D31⋊C5, C3×C3⋊(D31⋊C5), C3×D3×ΓL(1,𝔽32), …
maximal normal subgroup ofC32×D31⋊C5, C3×C3⋊(D31⋊C5), C3×D3×ΓL(1,𝔽32), …
maximal quotient ofC2×C32×ΓL(1,𝔽32), C32×#13133721, C32×#13133724, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1395, 6);