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## G = Dic19order 76 = 22·19

### Dicyclic group

Aliases: Dic19, C19⋊C4, C38.C2, C2.D19, SmallGroup(76,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C19 — Dic19
 Chief series C1 — C19 — C38 — Dic19
 Lower central C19 — Dic19
 Upper central C1 — C2

Generators and relations for Dic19
G = < a,b | a38=1, b2=a19, bab-1=a-1 >

Character table of Dic19

 class 1 2 4A 4B 19A 19B 19C 19D 19E 19F 19G 19H 19I 38A 38B 38C 38D 38E 38F 38G 38H 38I size 1 1 19 19 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 i -i 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ4 1 -1 -i i 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ5 2 2 0 0 ζ1910+ζ199 ζ1916+ζ193 ζ1915+ζ194 ζ1911+ζ198 ζ1918+ζ19 ζ1913+ζ196 ζ1912+ζ197 ζ1914+ζ195 ζ1917+ζ192 ζ1917+ζ192 ζ1910+ζ199 ζ1916+ζ193 ζ1915+ζ194 ζ1911+ζ198 ζ1918+ζ19 ζ1913+ζ196 ζ1912+ζ197 ζ1914+ζ195 orthogonal lifted from D19 ρ6 2 2 0 0 ζ1913+ζ196 ζ1917+ζ192 ζ1910+ζ199 ζ1918+ζ19 ζ1912+ζ197 ζ1915+ζ194 ζ1911+ζ198 ζ1916+ζ193 ζ1914+ζ195 ζ1914+ζ195 ζ1913+ζ196 ζ1917+ζ192 ζ1910+ζ199 ζ1918+ζ19 ζ1912+ζ197 ζ1915+ζ194 ζ1911+ζ198 ζ1916+ζ193 orthogonal lifted from D19 ρ7 2 2 0 0 ζ1914+ζ195 ζ1911+ζ198 ζ1917+ζ192 ζ1915+ζ194 ζ1910+ζ199 ζ1916+ζ193 ζ1913+ζ196 ζ1912+ζ197 ζ1918+ζ19 ζ1918+ζ19 ζ1914+ζ195 ζ1911+ζ198 ζ1917+ζ192 ζ1915+ζ194 ζ1910+ζ199 ζ1916+ζ193 ζ1913+ζ196 ζ1912+ζ197 orthogonal lifted from D19 ρ8 2 2 0 0 ζ1911+ζ198 ζ1910+ζ199 ζ1912+ζ197 ζ1914+ζ195 ζ1916+ζ193 ζ1918+ζ19 ζ1917+ζ192 ζ1915+ζ194 ζ1913+ζ196 ζ1913+ζ196 ζ1911+ζ198 ζ1910+ζ199 ζ1912+ζ197 ζ1914+ζ195 ζ1916+ζ193 ζ1918+ζ19 ζ1917+ζ192 ζ1915+ζ194 orthogonal lifted from D19 ρ9 2 2 0 0 ζ1912+ζ197 ζ1915+ζ194 ζ1918+ζ19 ζ1917+ζ192 ζ1914+ζ195 ζ1911+ζ198 ζ1916+ζ193 ζ1913+ζ196 ζ1910+ζ199 ζ1910+ζ199 ζ1912+ζ197 ζ1915+ζ194 ζ1918+ζ19 ζ1917+ζ192 ζ1914+ζ195 ζ1911+ζ198 ζ1916+ζ193 ζ1913+ζ196 orthogonal lifted from D19 ρ10 2 2 0 0 ζ1918+ζ19 ζ1913+ζ196 ζ1911+ζ198 ζ1916+ζ193 ζ1917+ζ192 ζ1912+ζ197 ζ1914+ζ195 ζ1910+ζ199 ζ1915+ζ194 ζ1915+ζ194 ζ1918+ζ19 ζ1913+ζ196 ζ1911+ζ198 ζ1916+ζ193 ζ1917+ζ192 ζ1912+ζ197 ζ1914+ζ195 ζ1910+ζ199 orthogonal lifted from D19 ρ11 2 2 0 0 ζ1916+ζ193 ζ1918+ζ19 ζ1914+ζ195 ζ1910+ζ199 ζ1913+ζ196 ζ1917+ζ192 ζ1915+ζ194 ζ1911+ζ198 ζ1912+ζ197 ζ1912+ζ197 ζ1916+ζ193 ζ1918+ζ19 ζ1914+ζ195 ζ1910+ζ199 ζ1913+ζ196 ζ1917+ζ192 ζ1915+ζ194 ζ1911+ζ198 orthogonal lifted from D19 ρ12 2 2 0 0 ζ1915+ζ194 ζ1914+ζ195 ζ1913+ζ196 ζ1912+ζ197 ζ1911+ζ198 ζ1910+ζ199 ζ1918+ζ19 ζ1917+ζ192 ζ1916+ζ193 ζ1916+ζ193 ζ1915+ζ194 ζ1914+ζ195 ζ1913+ζ196 ζ1912+ζ197 ζ1911+ζ198 ζ1910+ζ199 ζ1918+ζ19 ζ1917+ζ192 orthogonal lifted from D19 ρ13 2 2 0 0 ζ1917+ζ192 ζ1912+ζ197 ζ1916+ζ193 ζ1913+ζ196 ζ1915+ζ194 ζ1914+ζ195 ζ1910+ζ199 ζ1918+ζ19 ζ1911+ζ198 ζ1911+ζ198 ζ1917+ζ192 ζ1912+ζ197 ζ1916+ζ193 ζ1913+ζ196 ζ1915+ζ194 ζ1914+ζ195 ζ1910+ζ199 ζ1918+ζ19 orthogonal lifted from D19 ρ14 2 -2 0 0 ζ1914+ζ195 ζ1911+ζ198 ζ1917+ζ192 ζ1915+ζ194 ζ1910+ζ199 ζ1916+ζ193 ζ1913+ζ196 ζ1912+ζ197 ζ1918+ζ19 -ζ1918-ζ19 -ζ1914-ζ195 -ζ1911-ζ198 -ζ1917-ζ192 -ζ1915-ζ194 -ζ1910-ζ199 -ζ1916-ζ193 -ζ1913-ζ196 -ζ1912-ζ197 symplectic faithful, Schur index 2 ρ15 2 -2 0 0 ζ1913+ζ196 ζ1917+ζ192 ζ1910+ζ199 ζ1918+ζ19 ζ1912+ζ197 ζ1915+ζ194 ζ1911+ζ198 ζ1916+ζ193 ζ1914+ζ195 -ζ1914-ζ195 -ζ1913-ζ196 -ζ1917-ζ192 -ζ1910-ζ199 -ζ1918-ζ19 -ζ1912-ζ197 -ζ1915-ζ194 -ζ1911-ζ198 -ζ1916-ζ193 symplectic faithful, Schur index 2 ρ16 2 -2 0 0 ζ1910+ζ199 ζ1916+ζ193 ζ1915+ζ194 ζ1911+ζ198 ζ1918+ζ19 ζ1913+ζ196 ζ1912+ζ197 ζ1914+ζ195 ζ1917+ζ192 -ζ1917-ζ192 -ζ1910-ζ199 -ζ1916-ζ193 -ζ1915-ζ194 -ζ1911-ζ198 -ζ1918-ζ19 -ζ1913-ζ196 -ζ1912-ζ197 -ζ1914-ζ195 symplectic faithful, Schur index 2 ρ17 2 -2 0 0 ζ1912+ζ197 ζ1915+ζ194 ζ1918+ζ19 ζ1917+ζ192 ζ1914+ζ195 ζ1911+ζ198 ζ1916+ζ193 ζ1913+ζ196 ζ1910+ζ199 -ζ1910-ζ199 -ζ1912-ζ197 -ζ1915-ζ194 -ζ1918-ζ19 -ζ1917-ζ192 -ζ1914-ζ195 -ζ1911-ζ198 -ζ1916-ζ193 -ζ1913-ζ196 symplectic faithful, Schur index 2 ρ18 2 -2 0 0 ζ1911+ζ198 ζ1910+ζ199 ζ1912+ζ197 ζ1914+ζ195 ζ1916+ζ193 ζ1918+ζ19 ζ1917+ζ192 ζ1915+ζ194 ζ1913+ζ196 -ζ1913-ζ196 -ζ1911-ζ198 -ζ1910-ζ199 -ζ1912-ζ197 -ζ1914-ζ195 -ζ1916-ζ193 -ζ1918-ζ19 -ζ1917-ζ192 -ζ1915-ζ194 symplectic faithful, Schur index 2 ρ19 2 -2 0 0 ζ1918+ζ19 ζ1913+ζ196 ζ1911+ζ198 ζ1916+ζ193 ζ1917+ζ192 ζ1912+ζ197 ζ1914+ζ195 ζ1910+ζ199 ζ1915+ζ194 -ζ1915-ζ194 -ζ1918-ζ19 -ζ1913-ζ196 -ζ1911-ζ198 -ζ1916-ζ193 -ζ1917-ζ192 -ζ1912-ζ197 -ζ1914-ζ195 -ζ1910-ζ199 symplectic faithful, Schur index 2 ρ20 2 -2 0 0 ζ1916+ζ193 ζ1918+ζ19 ζ1914+ζ195 ζ1910+ζ199 ζ1913+ζ196 ζ1917+ζ192 ζ1915+ζ194 ζ1911+ζ198 ζ1912+ζ197 -ζ1912-ζ197 -ζ1916-ζ193 -ζ1918-ζ19 -ζ1914-ζ195 -ζ1910-ζ199 -ζ1913-ζ196 -ζ1917-ζ192 -ζ1915-ζ194 -ζ1911-ζ198 symplectic faithful, Schur index 2 ρ21 2 -2 0 0 ζ1915+ζ194 ζ1914+ζ195 ζ1913+ζ196 ζ1912+ζ197 ζ1911+ζ198 ζ1910+ζ199 ζ1918+ζ19 ζ1917+ζ192 ζ1916+ζ193 -ζ1916-ζ193 -ζ1915-ζ194 -ζ1914-ζ195 -ζ1913-ζ196 -ζ1912-ζ197 -ζ1911-ζ198 -ζ1910-ζ199 -ζ1918-ζ19 -ζ1917-ζ192 symplectic faithful, Schur index 2 ρ22 2 -2 0 0 ζ1917+ζ192 ζ1912+ζ197 ζ1916+ζ193 ζ1913+ζ196 ζ1915+ζ194 ζ1914+ζ195 ζ1910+ζ199 ζ1918+ζ19 ζ1911+ζ198 -ζ1911-ζ198 -ζ1917-ζ192 -ζ1912-ζ197 -ζ1916-ζ193 -ζ1913-ζ196 -ζ1915-ζ194 -ζ1914-ζ195 -ζ1910-ζ199 -ζ1918-ζ19 symplectic faithful, Schur index 2

Smallest permutation representation of Dic19
Regular action on 76 points
Generators in S76
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38)(39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76)
(1 57 20 76)(2 56 21 75)(3 55 22 74)(4 54 23 73)(5 53 24 72)(6 52 25 71)(7 51 26 70)(8 50 27 69)(9 49 28 68)(10 48 29 67)(11 47 30 66)(12 46 31 65)(13 45 32 64)(14 44 33 63)(15 43 34 62)(16 42 35 61)(17 41 36 60)(18 40 37 59)(19 39 38 58)

G:=sub<Sym(76)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76), (1,57,20,76)(2,56,21,75)(3,55,22,74)(4,54,23,73)(5,53,24,72)(6,52,25,71)(7,51,26,70)(8,50,27,69)(9,49,28,68)(10,48,29,67)(11,47,30,66)(12,46,31,65)(13,45,32,64)(14,44,33,63)(15,43,34,62)(16,42,35,61)(17,41,36,60)(18,40,37,59)(19,39,38,58)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76), (1,57,20,76)(2,56,21,75)(3,55,22,74)(4,54,23,73)(5,53,24,72)(6,52,25,71)(7,51,26,70)(8,50,27,69)(9,49,28,68)(10,48,29,67)(11,47,30,66)(12,46,31,65)(13,45,32,64)(14,44,33,63)(15,43,34,62)(16,42,35,61)(17,41,36,60)(18,40,37,59)(19,39,38,58) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38),(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76)], [(1,57,20,76),(2,56,21,75),(3,55,22,74),(4,54,23,73),(5,53,24,72),(6,52,25,71),(7,51,26,70),(8,50,27,69),(9,49,28,68),(10,48,29,67),(11,47,30,66),(12,46,31,65),(13,45,32,64),(14,44,33,63),(15,43,34,62),(16,42,35,61),(17,41,36,60),(18,40,37,59),(19,39,38,58)]])

Dic19 is a maximal subgroup of   Dic38  C4×D19  C19⋊D4  C19⋊C12  Dic57  Dic95  C19⋊F5
Dic19 is a maximal quotient of   C19⋊C8  Dic57  Dic95  C19⋊F5

Matrix representation of Dic19 in GL2(𝔽37) generated by

 10 34 34 1
,
 6 18 0 31
G:=sub<GL(2,GF(37))| [10,34,34,1],[6,0,18,31] >;

Dic19 in GAP, Magma, Sage, TeX

{\rm Dic}_{19}
% in TeX

G:=Group("Dic19");
// GroupNames label

G:=SmallGroup(76,1);
// by ID

G=gap.SmallGroup(76,1);
# by ID

G:=PCGroup([3,-2,-2,-19,6,650]);
// Polycyclic

G:=Group<a,b|a^38=1,b^2=a^19,b*a*b^-1=a^-1>;
// generators/relations

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