Finite Group Databasebeta

← Prev (#1171) Next (#1173) →

Cayley diagram

D19⋊C3 #1172

order114
isoclinismisoclinic class of D19⋊C3
automorphismAut(G) = AGL(1,𝔽19) (order 342) · Inn(G) = G · Out(G) = C3
holomorphG ⋊ Aut(G) = #72228749
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productD19⋊C3
unambiguous semidirect product(C19⋊C3)⋊C2

Splittings

Fitting, socle, derived, SylowC19 ⋊ C2×C3
HallC19⋊C3 ⋊ C2
HallD19 ⋊ C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC19C2×C3
LCSG ▷ C19
UCSC1

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 19

A minimal faithful permutation representation — 2 generators on 19 points:

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

Polycyclic presentation:

2,3,19|c^a=c^18,c^b=c^7

Conjugacy classes & character table

C1C2C3C4C5C6C7C8C9
size16661919191919
order119191923366
χ1111111111
χ211111−1+i32−1−i32−1+i32−1−i32
χ311111−1−i32−1+i32−1−i32−1+i32
χ41111-111-1-1
χ51111-1−1+i32−1−i321−i321+i32
χ61111-1−1−i32−1+i321+i321−i32
χ76−ζ−ζ7−ζ8−ζ11−ζ20−ζ26−ζ31−ζ37−ζ46−ζ49−ζ50−ζ56−ζ4−ζ10−ζ13−ζ23−ζ25−ζ28−ζ29−ζ32−ζ34−ζ44−ζ47−ζ53−ζ2−ζ5−ζ14−ζ16−ζ17−ζ22−ζ35−ζ40−ζ41−ζ43−ζ52−ζ5500000
χ86−ζ4−ζ10−ζ13−ζ23−ζ25−ζ28−ζ29−ζ32−ζ34−ζ44−ζ47−ζ53−ζ2−ζ5−ζ14−ζ16−ζ17−ζ22−ζ35−ζ40−ζ41−ζ43−ζ52−ζ55−ζ−ζ7−ζ8−ζ11−ζ20−ζ26−ζ31−ζ37−ζ46−ζ49−ζ50−ζ5600000
χ96−ζ2−ζ5−ζ14−ζ16−ζ17−ζ22−ζ35−ζ40−ζ41−ζ43−ζ52−ζ55−ζ−ζ7−ζ8−ζ11−ζ20−ζ26−ζ31−ζ37−ζ46−ζ49−ζ50−ζ56−ζ4−ζ10−ζ13−ζ23−ζ25−ζ28−ζ29−ζ32−ζ34−ζ44−ζ47−ζ5300000

Here ζ is a primitive root of unity of order 57 (ζ57 = 1); i = ζ14.

Relationships

maximal subgroup ofC2×D19⋊C3, AGL(1,𝔽19), C3×D19⋊C3, …
maximal normal subgroup ofC2×D19⋊C3, AGL(1,𝔽19), C3×D19⋊C3, …
maximal quotient of(C19⋊C3)⋊C4, C2×D19⋊C3, #4294⋊C2, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(114, 1);