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Cayley diagram

D31⋊C3 #4413

order186
isoclinismisoclinic class of D31⋊C3
automorphismAut(G) = AGL(1,𝔽31) (order 930) · Inn(G) = G · Out(G) = C5
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productD31⋊C3
unambiguous semidirect product(C31⋊C3)⋊C2

Splittings

Fitting, socle, derived, SylowC31 ⋊ C2×C3
HallC31⋊C3 ⋊ C2
HallD31 ⋊ C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC31C2×C3
LCSG ▷ C31
UCSC1

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 31

A minimal faithful permutation representation — 2 generators on 31 points:

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

Polycyclic presentation:

2,3,31|c^a=c^30,c^b=c^25

Conjugacy classes & character table

C1C2C3C4C5C6C7C8C9C10C11
size1666663131313131
order1313131313123366
χ111111111111
χ21111111−1+i32−1−i32−1+i32−1−i32
χ31111111−1−i32−1+i32−1−i32−1+i32
χ4111111-111-1-1
χ5111111-1−1+i32−1−i321−i321+i32
χ6111111-1−1−i32−1+i321+i321−i32
χ76−ζ8−ζ14−ζ17−ζ22−ζ23−ζ40−ζ53−ζ70−ζ71−ζ76−ζ79−ζ85−ζ−ζ5−ζ25−ζ26−ζ32−ζ37−ζ56−ζ61−ζ67−ζ68−ζ88−ζ92−ζ4−ζ7−ζ11−ζ20−ζ35−ζ38−ζ55−ζ58−ζ73−ζ82−ζ86−ζ89−ζ13−ζ16−ζ28−ζ34−ζ44−ζ46−ζ47−ζ49−ζ59−ζ65−ζ77−ζ80−ζ2−ζ10−ζ19−ζ29−ζ41−ζ43−ζ50−ζ52−ζ64−ζ74−ζ83−ζ9100000
χ86−ζ−ζ5−ζ25−ζ26−ζ32−ζ37−ζ56−ζ61−ζ67−ζ68−ζ88−ζ92−ζ4−ζ7−ζ11−ζ20−ζ35−ζ38−ζ55−ζ58−ζ73−ζ82−ζ86−ζ89−ζ13−ζ16−ζ28−ζ34−ζ44−ζ46−ζ47−ζ49−ζ59−ζ65−ζ77−ζ80−ζ2−ζ10−ζ19−ζ29−ζ41−ζ43−ζ50−ζ52−ζ64−ζ74−ζ83−ζ91−ζ8−ζ14−ζ17−ζ22−ζ23−ζ40−ζ53−ζ70−ζ71−ζ76−ζ79−ζ8500000
χ96−ζ4−ζ7−ζ11−ζ20−ζ35−ζ38−ζ55−ζ58−ζ73−ζ82−ζ86−ζ89−ζ13−ζ16−ζ28−ζ34−ζ44−ζ46−ζ47−ζ49−ζ59−ζ65−ζ77−ζ80−ζ2−ζ10−ζ19−ζ29−ζ41−ζ43−ζ50−ζ52−ζ64−ζ74−ζ83−ζ91−ζ8−ζ14−ζ17−ζ22−ζ23−ζ40−ζ53−ζ70−ζ71−ζ76−ζ79−ζ85−ζ−ζ5−ζ25−ζ26−ζ32−ζ37−ζ56−ζ61−ζ67−ζ68−ζ88−ζ9200000
χ106−ζ13−ζ16−ζ28−ζ34−ζ44−ζ46−ζ47−ζ49−ζ59−ζ65−ζ77−ζ80−ζ2−ζ10−ζ19−ζ29−ζ41−ζ43−ζ50−ζ52−ζ64−ζ74−ζ83−ζ91−ζ8−ζ14−ζ17−ζ22−ζ23−ζ40−ζ53−ζ70−ζ71−ζ76−ζ79−ζ85−ζ−ζ5−ζ25−ζ26−ζ32−ζ37−ζ56−ζ61−ζ67−ζ68−ζ88−ζ92−ζ4−ζ7−ζ11−ζ20−ζ35−ζ38−ζ55−ζ58−ζ73−ζ82−ζ86−ζ8900000
χ116−ζ2−ζ10−ζ19−ζ29−ζ41−ζ43−ζ50−ζ52−ζ64−ζ74−ζ83−ζ91−ζ8−ζ14−ζ17−ζ22−ζ23−ζ40−ζ53−ζ70−ζ71−ζ76−ζ79−ζ85−ζ−ζ5−ζ25−ζ26−ζ32−ζ37−ζ56−ζ61−ζ67−ζ68−ζ88−ζ92−ζ4−ζ7−ζ11−ζ20−ζ35−ζ38−ζ55−ζ58−ζ73−ζ82−ζ86−ζ89−ζ13−ζ16−ζ28−ζ34−ζ44−ζ46−ζ47−ζ49−ζ59−ζ65−ζ77−ζ8000000

Here ζ is a primitive root of unity of order 93 (ζ93 = 1); i = ζ23.

Relationships

maximal subgroup ofC2×D31⋊C3, C3×D31⋊C3, C3⋊(D31⋊C3), …
maximal normal subgroup ofC2×D31⋊C3, C3×D31⋊C3, AGL(1,𝔽31), …
maximal quotient of(C31⋊C3)⋊C4, C2×D31⋊C3, D31⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(186, 1);