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Cayley diagram

C2×ΓL(1,𝔽32) #64579

order310
isoclinismisoclinic class of ΓL(1,𝔽32)
automorphismAut(G) = AGL(1,𝔽31) (order 930) · Inn(G) = ΓL(1,𝔽32) · Out(G) = C2×C3
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC2×ΓL(1,𝔽32)
unambiguous semidirect product(C2×C31)⋊C5

Splittings

Fitting, socle, HallC2×C31 ⋊ C5
derived, SylowC31 ⋊ C2×C5

Subgroups

subgroupG/H
center, hypercenterC2ΓL(1,𝔽32)
commutator, cosocle, hypocenterC31C2×C5
FrattiniC1G
Fitting, socleC2×C31C5
LCSG ▷ C31
UCSC1 ◁ C2

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 33
transitive degreeμt(G) = 62

A minimal faithful permutation representation — 2 generators on 33 points:

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

Conjugacy classes & character table

22 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic31⋊C5, C4×ΓL(1,𝔽32), C2×D31⋊C5, …
maximal normal subgroup ofDic31⋊C5, C4×ΓL(1,𝔽32), C2×D31⋊C5, …
maximal quotient ofC4×ΓL(1,𝔽32), C22×ΓL(1,𝔽32), D3×ΓL(1,𝔽32), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(310, 2);