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## G = Dic22order 88 = 23·11

### Dicyclic group

Aliases: Dic22, C11⋊Q8, C4.D11, C44.1C2, C2.3D22, Dic11.C2, C22.1C22, SmallGroup(88,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — Dic22
 Chief series C1 — C11 — C22 — Dic11 — Dic22
 Lower central C11 — C22 — Dic22
 Upper central C1 — C2 — C4

Generators and relations for Dic22
G = < a,b | a44=1, b2=a22, bab-1=a-1 >

Character table of Dic22

 class 1 2 4A 4B 4C 11A 11B 11C 11D 11E 22A 22B 22C 22D 22E 44A 44B 44C 44D 44E 44F 44G 44H 44I 44J size 1 1 2 22 22 2 2 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 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 -1 -1 -1 linear of order 2 ρ3 1 1 -1 -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 ρ4 1 1 1 -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 ρ5 2 2 -2 0 0 ζ117+ζ114 ζ116+ζ115 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ119+ζ112 ζ118+ζ113 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 -ζ117-ζ114 -ζ119-ζ112 -ζ118-ζ113 -ζ118-ζ113 -ζ119-ζ112 -ζ117-ζ114 -ζ1110-ζ11 -ζ116-ζ115 -ζ116-ζ115 -ζ1110-ζ11 orthogonal lifted from D22 ρ6 2 2 -2 0 0 ζ1110+ζ11 ζ117+ζ114 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ116+ζ115 ζ119+ζ112 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 -ζ1110-ζ11 -ζ116-ζ115 -ζ119-ζ112 -ζ119-ζ112 -ζ116-ζ115 -ζ1110-ζ11 -ζ118-ζ113 -ζ117-ζ114 -ζ117-ζ114 -ζ118-ζ113 orthogonal lifted from D22 ρ7 2 2 -2 0 0 ζ119+ζ112 ζ118+ζ113 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ1110+ζ11 ζ117+ζ114 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 -ζ119-ζ112 -ζ1110-ζ11 -ζ117-ζ114 -ζ117-ζ114 -ζ1110-ζ11 -ζ119-ζ112 -ζ116-ζ115 -ζ118-ζ113 -ζ118-ζ113 -ζ116-ζ115 orthogonal lifted from D22 ρ8 2 2 2 0 0 ζ119+ζ112 ζ118+ζ113 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ1110+ζ11 ζ117+ζ114 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ117+ζ114 ζ1110+ζ11 ζ119+ζ112 ζ116+ζ115 ζ118+ζ113 ζ118+ζ113 ζ116+ζ115 orthogonal lifted from D11 ρ9 2 2 2 0 0 ζ117+ζ114 ζ116+ζ115 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ119+ζ112 ζ118+ζ113 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ118+ζ113 ζ119+ζ112 ζ117+ζ114 ζ1110+ζ11 ζ116+ζ115 ζ116+ζ115 ζ1110+ζ11 orthogonal lifted from D11 ρ10 2 2 2 0 0 ζ1110+ζ11 ζ117+ζ114 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ116+ζ115 ζ119+ζ112 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ119+ζ112 ζ116+ζ115 ζ1110+ζ11 ζ118+ζ113 ζ117+ζ114 ζ117+ζ114 ζ118+ζ113 orthogonal lifted from D11 ρ11 2 2 -2 0 0 ζ116+ζ115 ζ119+ζ112 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ118+ζ113 ζ1110+ζ11 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 -ζ116-ζ115 -ζ118-ζ113 -ζ1110-ζ11 -ζ1110-ζ11 -ζ118-ζ113 -ζ116-ζ115 -ζ117-ζ114 -ζ119-ζ112 -ζ119-ζ112 -ζ117-ζ114 orthogonal lifted from D22 ρ12 2 2 -2 0 0 ζ118+ζ113 ζ1110+ζ11 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ117+ζ114 ζ116+ζ115 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 -ζ118-ζ113 -ζ117-ζ114 -ζ116-ζ115 -ζ116-ζ115 -ζ117-ζ114 -ζ118-ζ113 -ζ119-ζ112 -ζ1110-ζ11 -ζ1110-ζ11 -ζ119-ζ112 orthogonal lifted from D22 ρ13 2 2 2 0 0 ζ116+ζ115 ζ119+ζ112 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ118+ζ113 ζ1110+ζ11 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ1110+ζ11 ζ118+ζ113 ζ116+ζ115 ζ117+ζ114 ζ119+ζ112 ζ119+ζ112 ζ117+ζ114 orthogonal lifted from D11 ρ14 2 2 2 0 0 ζ118+ζ113 ζ1110+ζ11 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ117+ζ114 ζ116+ζ115 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ116+ζ115 ζ117+ζ114 ζ118+ζ113 ζ119+ζ112 ζ1110+ζ11 ζ1110+ζ11 ζ119+ζ112 orthogonal lifted from D11 ρ15 2 -2 0 0 0 2 2 2 2 2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ16 2 -2 0 0 0 ζ1110+ζ11 ζ117+ζ114 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 -ζ116-ζ115 -ζ119-ζ112 -ζ117-ζ114 -ζ118-ζ113 -ζ1110-ζ11 ζ4ζ1110-ζ4ζ11 -ζ43ζ116+ζ43ζ115 -ζ4ζ119+ζ4ζ112 ζ4ζ119-ζ4ζ112 ζ43ζ116-ζ43ζ115 -ζ4ζ1110+ζ4ζ11 -ζ43ζ118+ζ43ζ113 -ζ4ζ117+ζ4ζ114 ζ4ζ117-ζ4ζ114 ζ43ζ118-ζ43ζ113 symplectic faithful, Schur index 2 ρ17 2 -2 0 0 0 ζ117+ζ114 ζ116+ζ115 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 -ζ119-ζ112 -ζ118-ζ113 -ζ116-ζ115 -ζ1110-ζ11 -ζ117-ζ114 -ζ4ζ117+ζ4ζ114 ζ4ζ119-ζ4ζ112 ζ43ζ118-ζ43ζ113 -ζ43ζ118+ζ43ζ113 -ζ4ζ119+ζ4ζ112 ζ4ζ117-ζ4ζ114 -ζ4ζ1110+ζ4ζ11 -ζ43ζ116+ζ43ζ115 ζ43ζ116-ζ43ζ115 ζ4ζ1110-ζ4ζ11 symplectic faithful, Schur index 2 ρ18 2 -2 0 0 0 ζ118+ζ113 ζ1110+ζ11 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 -ζ117-ζ114 -ζ116-ζ115 -ζ1110-ζ11 -ζ119-ζ112 -ζ118-ζ113 ζ43ζ118-ζ43ζ113 -ζ4ζ117+ζ4ζ114 ζ43ζ116-ζ43ζ115 -ζ43ζ116+ζ43ζ115 ζ4ζ117-ζ4ζ114 -ζ43ζ118+ζ43ζ113 ζ4ζ119-ζ4ζ112 ζ4ζ1110-ζ4ζ11 -ζ4ζ1110+ζ4ζ11 -ζ4ζ119+ζ4ζ112 symplectic faithful, Schur index 2 ρ19 2 -2 0 0 0 ζ116+ζ115 ζ119+ζ112 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 -ζ118-ζ113 -ζ1110-ζ11 -ζ119-ζ112 -ζ117-ζ114 -ζ116-ζ115 -ζ43ζ116+ζ43ζ115 -ζ43ζ118+ζ43ζ113 ζ4ζ1110-ζ4ζ11 -ζ4ζ1110+ζ4ζ11 ζ43ζ118-ζ43ζ113 ζ43ζ116-ζ43ζ115 ζ4ζ117-ζ4ζ114 ζ4ζ119-ζ4ζ112 -ζ4ζ119+ζ4ζ112 -ζ4ζ117+ζ4ζ114 symplectic faithful, Schur index 2 ρ20 2 -2 0 0 0 ζ119+ζ112 ζ118+ζ113 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 -ζ1110-ζ11 -ζ117-ζ114 -ζ118-ζ113 -ζ116-ζ115 -ζ119-ζ112 -ζ4ζ119+ζ4ζ112 ζ4ζ1110-ζ4ζ11 ζ4ζ117-ζ4ζ114 -ζ4ζ117+ζ4ζ114 -ζ4ζ1110+ζ4ζ11 ζ4ζ119-ζ4ζ112 -ζ43ζ116+ζ43ζ115 ζ43ζ118-ζ43ζ113 -ζ43ζ118+ζ43ζ113 ζ43ζ116-ζ43ζ115 symplectic faithful, Schur index 2 ρ21 2 -2 0 0 0 ζ119+ζ112 ζ118+ζ113 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 -ζ1110-ζ11 -ζ117-ζ114 -ζ118-ζ113 -ζ116-ζ115 -ζ119-ζ112 ζ4ζ119-ζ4ζ112 -ζ4ζ1110+ζ4ζ11 -ζ4ζ117+ζ4ζ114 ζ4ζ117-ζ4ζ114 ζ4ζ1110-ζ4ζ11 -ζ4ζ119+ζ4ζ112 ζ43ζ116-ζ43ζ115 -ζ43ζ118+ζ43ζ113 ζ43ζ118-ζ43ζ113 -ζ43ζ116+ζ43ζ115 symplectic faithful, Schur index 2 ρ22 2 -2 0 0 0 ζ116+ζ115 ζ119+ζ112 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 -ζ118-ζ113 -ζ1110-ζ11 -ζ119-ζ112 -ζ117-ζ114 -ζ116-ζ115 ζ43ζ116-ζ43ζ115 ζ43ζ118-ζ43ζ113 -ζ4ζ1110+ζ4ζ11 ζ4ζ1110-ζ4ζ11 -ζ43ζ118+ζ43ζ113 -ζ43ζ116+ζ43ζ115 -ζ4ζ117+ζ4ζ114 -ζ4ζ119+ζ4ζ112 ζ4ζ119-ζ4ζ112 ζ4ζ117-ζ4ζ114 symplectic faithful, Schur index 2 ρ23 2 -2 0 0 0 ζ118+ζ113 ζ1110+ζ11 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 -ζ117-ζ114 -ζ116-ζ115 -ζ1110-ζ11 -ζ119-ζ112 -ζ118-ζ113 -ζ43ζ118+ζ43ζ113 ζ4ζ117-ζ4ζ114 -ζ43ζ116+ζ43ζ115 ζ43ζ116-ζ43ζ115 -ζ4ζ117+ζ4ζ114 ζ43ζ118-ζ43ζ113 -ζ4ζ119+ζ4ζ112 -ζ4ζ1110+ζ4ζ11 ζ4ζ1110-ζ4ζ11 ζ4ζ119-ζ4ζ112 symplectic faithful, Schur index 2 ρ24 2 -2 0 0 0 ζ1110+ζ11 ζ117+ζ114 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 -ζ116-ζ115 -ζ119-ζ112 -ζ117-ζ114 -ζ118-ζ113 -ζ1110-ζ11 -ζ4ζ1110+ζ4ζ11 ζ43ζ116-ζ43ζ115 ζ4ζ119-ζ4ζ112 -ζ4ζ119+ζ4ζ112 -ζ43ζ116+ζ43ζ115 ζ4ζ1110-ζ4ζ11 ζ43ζ118-ζ43ζ113 ζ4ζ117-ζ4ζ114 -ζ4ζ117+ζ4ζ114 -ζ43ζ118+ζ43ζ113 symplectic faithful, Schur index 2 ρ25 2 -2 0 0 0 ζ117+ζ114 ζ116+ζ115 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 -ζ119-ζ112 -ζ118-ζ113 -ζ116-ζ115 -ζ1110-ζ11 -ζ117-ζ114 ζ4ζ117-ζ4ζ114 -ζ4ζ119+ζ4ζ112 -ζ43ζ118+ζ43ζ113 ζ43ζ118-ζ43ζ113 ζ4ζ119-ζ4ζ112 -ζ4ζ117+ζ4ζ114 ζ4ζ1110-ζ4ζ11 ζ43ζ116-ζ43ζ115 -ζ43ζ116+ζ43ζ115 -ζ4ζ1110+ζ4ζ11 symplectic faithful, Schur index 2

Smallest permutation representation of Dic22
Regular action on 88 points
Generators in S88
(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 77 78 79 80 81 82 83 84 85 86 87 88)
(1 58 23 80)(2 57 24 79)(3 56 25 78)(4 55 26 77)(5 54 27 76)(6 53 28 75)(7 52 29 74)(8 51 30 73)(9 50 31 72)(10 49 32 71)(11 48 33 70)(12 47 34 69)(13 46 35 68)(14 45 36 67)(15 88 37 66)(16 87 38 65)(17 86 39 64)(18 85 40 63)(19 84 41 62)(20 83 42 61)(21 82 43 60)(22 81 44 59)

G:=sub<Sym(88)| (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)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88), (1,58,23,80)(2,57,24,79)(3,56,25,78)(4,55,26,77)(5,54,27,76)(6,53,28,75)(7,52,29,74)(8,51,30,73)(9,50,31,72)(10,49,32,71)(11,48,33,70)(12,47,34,69)(13,46,35,68)(14,45,36,67)(15,88,37,66)(16,87,38,65)(17,86,39,64)(18,85,40,63)(19,84,41,62)(20,83,42,61)(21,82,43,60)(22,81,44,59)>;

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)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88), (1,58,23,80)(2,57,24,79)(3,56,25,78)(4,55,26,77)(5,54,27,76)(6,53,28,75)(7,52,29,74)(8,51,30,73)(9,50,31,72)(10,49,32,71)(11,48,33,70)(12,47,34,69)(13,46,35,68)(14,45,36,67)(15,88,37,66)(16,87,38,65)(17,86,39,64)(18,85,40,63)(19,84,41,62)(20,83,42,61)(21,82,43,60)(22,81,44,59) );

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),(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88)], [(1,58,23,80),(2,57,24,79),(3,56,25,78),(4,55,26,77),(5,54,27,76),(6,53,28,75),(7,52,29,74),(8,51,30,73),(9,50,31,72),(10,49,32,71),(11,48,33,70),(12,47,34,69),(13,46,35,68),(14,45,36,67),(15,88,37,66),(16,87,38,65),(17,86,39,64),(18,85,40,63),(19,84,41,62),(20,83,42,61),(21,82,43,60),(22,81,44,59)]])

Dic22 is a maximal subgroup of
C8⋊D11  Dic44  D4.D11  C11⋊Q16  D445C2  D42D11  Q8×D11  C33⋊Q8  Dic66  C4.F11  C55⋊Q8  Dic110
Dic22 is a maximal quotient of
Dic11⋊C4  C44⋊C4  C33⋊Q8  Dic66  C55⋊Q8  Dic110

Matrix representation of Dic22 in GL2(𝔽43) generated by

 5 20 20 20
,
 9 41 41 34
G:=sub<GL(2,GF(43))| [5,20,20,20],[9,41,41,34] >;

Dic22 in GAP, Magma, Sage, TeX

{\rm Dic}_{22}
% in TeX

G:=Group("Dic22");
// GroupNames label

G:=SmallGroup(88,3);
// by ID

G=gap.SmallGroup(88,3);
# by ID

G:=PCGroup([4,-2,-2,-2,-11,16,49,21,1283]);
// Polycyclic

G:=Group<a,b|a^44=1,b^2=a^22,b*a*b^-1=a^-1>;
// generators/relations

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