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G = Dic19order 76 = 22·19

Dicyclic group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C19 — Dic19
C1C19C38 — Dic19
C19 — Dic19
C1C2

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

19C4

Character table of Dic19

 class 124A4B19A19B19C19D19E19F19G19H19I38A38B38C38D38E38F38G38H38I
 size 111919222222222222222222
ρ11111111111111111111111    trivial
ρ211-1-1111111111111111111    linear of order 2
ρ31-1i-i111111111-1-1-1-1-1-1-1-1-1    linear of order 4
ρ41-1-ii111111111-1-1-1-1-1-1-1-1-1    linear of order 4
ρ52200ζ1910199ζ1916193ζ1915194ζ1911198ζ191819ζ1913196ζ1912197ζ1914195ζ1917192ζ1917192ζ1910199ζ1916193ζ1915194ζ1911198ζ191819ζ1913196ζ1912197ζ1914195    orthogonal lifted from D19
ρ62200ζ1913196ζ1917192ζ1910199ζ191819ζ1912197ζ1915194ζ1911198ζ1916193ζ1914195ζ1914195ζ1913196ζ1917192ζ1910199ζ191819ζ1912197ζ1915194ζ1911198ζ1916193    orthogonal lifted from D19
ρ72200ζ1914195ζ1911198ζ1917192ζ1915194ζ1910199ζ1916193ζ1913196ζ1912197ζ191819ζ191819ζ1914195ζ1911198ζ1917192ζ1915194ζ1910199ζ1916193ζ1913196ζ1912197    orthogonal lifted from D19
ρ82200ζ1911198ζ1910199ζ1912197ζ1914195ζ1916193ζ191819ζ1917192ζ1915194ζ1913196ζ1913196ζ1911198ζ1910199ζ1912197ζ1914195ζ1916193ζ191819ζ1917192ζ1915194    orthogonal lifted from D19
ρ92200ζ1912197ζ1915194ζ191819ζ1917192ζ1914195ζ1911198ζ1916193ζ1913196ζ1910199ζ1910199ζ1912197ζ1915194ζ191819ζ1917192ζ1914195ζ1911198ζ1916193ζ1913196    orthogonal lifted from D19
ρ102200ζ191819ζ1913196ζ1911198ζ1916193ζ1917192ζ1912197ζ1914195ζ1910199ζ1915194ζ1915194ζ191819ζ1913196ζ1911198ζ1916193ζ1917192ζ1912197ζ1914195ζ1910199    orthogonal lifted from D19
ρ112200ζ1916193ζ191819ζ1914195ζ1910199ζ1913196ζ1917192ζ1915194ζ1911198ζ1912197ζ1912197ζ1916193ζ191819ζ1914195ζ1910199ζ1913196ζ1917192ζ1915194ζ1911198    orthogonal lifted from D19
ρ122200ζ1915194ζ1914195ζ1913196ζ1912197ζ1911198ζ1910199ζ191819ζ1917192ζ1916193ζ1916193ζ1915194ζ1914195ζ1913196ζ1912197ζ1911198ζ1910199ζ191819ζ1917192    orthogonal lifted from D19
ρ132200ζ1917192ζ1912197ζ1916193ζ1913196ζ1915194ζ1914195ζ1910199ζ191819ζ1911198ζ1911198ζ1917192ζ1912197ζ1916193ζ1913196ζ1915194ζ1914195ζ1910199ζ191819    orthogonal lifted from D19
ρ142-200ζ1914195ζ1911198ζ1917192ζ1915194ζ1910199ζ1916193ζ1913196ζ1912197ζ19181919181919141951911198191719219151941910199191619319131961912197    symplectic faithful, Schur index 2
ρ152-200ζ1913196ζ1917192ζ1910199ζ191819ζ1912197ζ1915194ζ1911198ζ1916193ζ191419519141951913196191719219101991918191912197191519419111981916193    symplectic faithful, Schur index 2
ρ162-200ζ1910199ζ1916193ζ1915194ζ1911198ζ191819ζ1913196ζ1912197ζ1914195ζ191719219171921910199191619319151941911198191819191319619121971914195    symplectic faithful, Schur index 2
ρ172-200ζ1912197ζ1915194ζ191819ζ1917192ζ1914195ζ1911198ζ1916193ζ1913196ζ191019919101991912197191519419181919171921914195191119819161931913196    symplectic faithful, Schur index 2
ρ182-200ζ1911198ζ1910199ζ1912197ζ1914195ζ1916193ζ191819ζ1917192ζ1915194ζ191319619131961911198191019919121971914195191619319181919171921915194    symplectic faithful, Schur index 2
ρ192-200ζ191819ζ1913196ζ1911198ζ1916193ζ1917192ζ1912197ζ1914195ζ1910199ζ191519419151941918191913196191119819161931917192191219719141951910199    symplectic faithful, Schur index 2
ρ202-200ζ1916193ζ191819ζ1914195ζ1910199ζ1913196ζ1917192ζ1915194ζ1911198ζ191219719121971916193191819191419519101991913196191719219151941911198    symplectic faithful, Schur index 2
ρ212-200ζ1915194ζ1914195ζ1913196ζ1912197ζ1911198ζ1910199ζ191819ζ1917192ζ191619319161931915194191419519131961912197191119819101991918191917192    symplectic faithful, Schur index 2
ρ222-200ζ1917192ζ1912197ζ1916193ζ1913196ζ1915194ζ1914195ζ1910199ζ191819ζ191119819111981917192191219719161931913196191519419141951910199191819    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 52 20 71)(2 51 21 70)(3 50 22 69)(4 49 23 68)(5 48 24 67)(6 47 25 66)(7 46 26 65)(8 45 27 64)(9 44 28 63)(10 43 29 62)(11 42 30 61)(12 41 31 60)(13 40 32 59)(14 39 33 58)(15 76 34 57)(16 75 35 56)(17 74 36 55)(18 73 37 54)(19 72 38 53)

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

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

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

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

1034
341
,
618
031
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

Export

Subgroup lattice of Dic19 in TeX
Character table of Dic19 in TeX

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