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G = Dic22  order 88 = 23·11

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

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 >

C1
C2
C11
C4
11C4
11C4
C22
11Q8
Dic11
Dic11
C44
Dic22

Character table of Dic22

 class 124A4B4C11A11B11C11D11E22A22B22C22D22E44A44B44C44D44E44F44G44H44I44J
 size 112222222222222222222222222
ρ11111111111111111111111111    trivial
ρ211-11-11111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ311-1-111111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ4111-1-111111111111111111111    linear of order 2
ρ522-200ζ117+ζ114ζ116+ζ115ζ1110+ζ11ζ119+ζ112ζ118+ζ113ζ119+ζ112ζ118+ζ113ζ116+ζ115ζ1110+ζ11ζ117+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
ρ622-200ζ1110+ζ11ζ117+ζ114ζ118+ζ113ζ116+ζ115ζ119+ζ112ζ116+ζ115ζ119+ζ112ζ117+ζ114ζ118+ζ113ζ1110+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
ρ722-200ζ119+ζ112ζ118+ζ113ζ116+ζ115ζ1110+ζ11ζ117+ζ114ζ1110+ζ11ζ117+ζ114ζ118+ζ113ζ116+ζ115ζ119+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
ρ822200ζ119+ζ112ζ118+ζ113ζ116+ζ115ζ1110+ζ11ζ117+ζ114ζ1110+ζ11ζ117+ζ114ζ118+ζ113ζ116+ζ115ζ119+ζ112ζ119+ζ112ζ1110+ζ11ζ117+ζ114ζ117+ζ114ζ1110+ζ11ζ119+ζ112ζ116+ζ115ζ118+ζ113ζ118+ζ113ζ116+ζ115    orthogonal lifted from D11
ρ922200ζ117+ζ114ζ116+ζ115ζ1110+ζ11ζ119+ζ112ζ118+ζ113ζ119+ζ112ζ118+ζ113ζ116+ζ115ζ1110+ζ11ζ117+ζ114ζ117+ζ114ζ119+ζ112ζ118+ζ113ζ118+ζ113ζ119+ζ112ζ117+ζ114ζ1110+ζ11ζ116+ζ115ζ116+ζ115ζ1110+ζ11    orthogonal lifted from D11
ρ1022200ζ1110+ζ11ζ117+ζ114ζ118+ζ113ζ116+ζ115ζ119+ζ112ζ116+ζ115ζ119+ζ112ζ117+ζ114ζ118+ζ113ζ1110+ζ11ζ1110+ζ11ζ116+ζ115ζ119+ζ112ζ119+ζ112ζ116+ζ115ζ1110+ζ11ζ118+ζ113ζ117+ζ114ζ117+ζ114ζ118+ζ113    orthogonal lifted from D11
ρ1122-200ζ116+ζ115ζ119+ζ112ζ117+ζ114ζ118+ζ113ζ1110+ζ11ζ118+ζ113ζ1110+ζ11ζ119+ζ112ζ117+ζ114ζ116+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
ρ1222-200ζ118+ζ113ζ1110+ζ11ζ119+ζ112ζ117+ζ114ζ116+ζ115ζ117+ζ114ζ116+ζ115ζ1110+ζ11ζ119+ζ112ζ118+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
ρ1322200ζ116+ζ115ζ119+ζ112ζ117+ζ114ζ118+ζ113ζ1110+ζ11ζ118+ζ113ζ1110+ζ11ζ119+ζ112ζ117+ζ114ζ116+ζ115ζ116+ζ115ζ118+ζ113ζ1110+ζ11ζ1110+ζ11ζ118+ζ113ζ116+ζ115ζ117+ζ114ζ119+ζ112ζ119+ζ112ζ117+ζ114    orthogonal lifted from D11
ρ1422200ζ118+ζ113ζ1110+ζ11ζ119+ζ112ζ117+ζ114ζ116+ζ115ζ117+ζ114ζ116+ζ115ζ1110+ζ11ζ119+ζ112ζ118+ζ113ζ118+ζ113ζ117+ζ114ζ116+ζ115ζ116+ζ115ζ117+ζ114ζ118+ζ113ζ119+ζ112ζ1110+ζ11ζ1110+ζ11ζ119+ζ112    orthogonal lifted from D11
ρ152-200022222-2-2-2-2-20000000000    symplectic lifted from Q8, Schur index 2
ρ162-2000ζ1110+ζ11ζ117+ζ114ζ118+ζ113ζ116+ζ115ζ119+ζ112-ζ116-ζ115-ζ119-ζ112-ζ117-ζ114-ζ118-ζ113-ζ1110-ζ11ζ4ζ1110-ζ4ζ11-ζ43ζ116+ζ43ζ115-ζ4ζ119+ζ4ζ112ζ4ζ119-ζ4ζ112ζ43ζ116-ζ43ζ115-ζ4ζ1110+ζ4ζ11-ζ43ζ118+ζ43ζ113-ζ4ζ117+ζ4ζ114ζ4ζ117-ζ4ζ114ζ43ζ118-ζ43ζ113    symplectic faithful, Schur index 2
ρ172-2000ζ117+ζ114ζ116+ζ115ζ1110+ζ11ζ119+ζ112ζ118+ζ113-ζ119-ζ112-ζ118-ζ113-ζ116-ζ115-ζ1110-ζ11-ζ117-ζ114-ζ4ζ117+ζ4ζ114ζ4ζ119-ζ4ζ112ζ43ζ118-ζ43ζ113-ζ43ζ118+ζ43ζ113-ζ4ζ119+ζ4ζ112ζ4ζ117-ζ4ζ114-ζ4ζ1110+ζ4ζ11-ζ43ζ116+ζ43ζ115ζ43ζ116-ζ43ζ115ζ4ζ1110-ζ4ζ11    symplectic faithful, Schur index 2
ρ182-2000ζ118+ζ113ζ1110+ζ11ζ119+ζ112ζ117+ζ114ζ116+ζ115-ζ117-ζ114-ζ116-ζ115-ζ1110-ζ11-ζ119-ζ112-ζ118-ζ113ζ43ζ118-ζ43ζ113-ζ4ζ117+ζ4ζ114ζ43ζ116-ζ43ζ115-ζ43ζ116+ζ43ζ115ζ4ζ117-ζ4ζ114-ζ43ζ118+ζ43ζ113ζ4ζ119-ζ4ζ112ζ4ζ1110-ζ4ζ11-ζ4ζ1110+ζ4ζ11-ζ4ζ119+ζ4ζ112    symplectic faithful, Schur index 2
ρ192-2000ζ116+ζ115ζ119+ζ112ζ117+ζ114ζ118+ζ113ζ1110+ζ11-ζ118-ζ113-ζ1110-ζ11-ζ119-ζ112-ζ117-ζ114-ζ116-ζ115-ζ43ζ116+ζ43ζ115-ζ43ζ118+ζ43ζ113ζ4ζ1110-ζ4ζ11-ζ4ζ1110+ζ4ζ11ζ43ζ118-ζ43ζ113ζ43ζ116-ζ43ζ115ζ4ζ117-ζ4ζ114ζ4ζ119-ζ4ζ112-ζ4ζ119+ζ4ζ112-ζ4ζ117+ζ4ζ114    symplectic faithful, Schur index 2
ρ202-2000ζ119+ζ112ζ118+ζ113ζ116+ζ115ζ1110+ζ11ζ117+ζ114-ζ1110-ζ11-ζ117-ζ114-ζ118-ζ113-ζ116-ζ115-ζ119-ζ112-ζ4ζ119+ζ4ζ112ζ4ζ1110-ζ4ζ11ζ4ζ117-ζ4ζ114-ζ4ζ117+ζ4ζ114-ζ4ζ1110+ζ4ζ11ζ4ζ119-ζ4ζ112-ζ43ζ116+ζ43ζ115ζ43ζ118-ζ43ζ113-ζ43ζ118+ζ43ζ113ζ43ζ116-ζ43ζ115    symplectic faithful, Schur index 2
ρ212-2000ζ119+ζ112ζ118+ζ113ζ116+ζ115ζ1110+ζ11ζ117+ζ114-ζ1110-ζ11-ζ117-ζ114-ζ118-ζ113-ζ116-ζ115-ζ119-ζ112ζ4ζ119-ζ4ζ112-ζ4ζ1110+ζ4ζ11-ζ4ζ117+ζ4ζ114ζ4ζ117-ζ4ζ114ζ4ζ1110-ζ4ζ11-ζ4ζ119+ζ4ζ112ζ43ζ116-ζ43ζ115-ζ43ζ118+ζ43ζ113ζ43ζ118-ζ43ζ113-ζ43ζ116+ζ43ζ115    symplectic faithful, Schur index 2
ρ222-2000ζ116+ζ115ζ119+ζ112ζ117+ζ114ζ118+ζ113ζ1110+ζ11-ζ118-ζ113-ζ1110-ζ11-ζ119-ζ112-ζ117-ζ114-ζ116-ζ115ζ43ζ116-ζ43ζ115ζ43ζ118-ζ43ζ113-ζ4ζ1110+ζ4ζ11ζ4ζ1110-ζ4ζ11-ζ43ζ118+ζ43ζ113-ζ43ζ116+ζ43ζ115-ζ4ζ117+ζ4ζ114-ζ4ζ119+ζ4ζ112ζ4ζ119-ζ4ζ112ζ4ζ117-ζ4ζ114    symplectic faithful, Schur index 2
ρ232-2000ζ118+ζ113ζ1110+ζ11ζ119+ζ112ζ117+ζ114ζ116+ζ115-ζ117-ζ114-ζ116-ζ115-ζ1110-ζ11-ζ119-ζ112-ζ118-ζ113-ζ43ζ118+ζ43ζ113ζ4ζ117-ζ4ζ114-ζ43ζ116+ζ43ζ115ζ43ζ116-ζ43ζ115-ζ4ζ117+ζ4ζ114ζ43ζ118-ζ43ζ113-ζ4ζ119+ζ4ζ112-ζ4ζ1110+ζ4ζ11ζ4ζ1110-ζ4ζ11ζ4ζ119-ζ4ζ112    symplectic faithful, Schur index 2
ρ242-2000ζ1110+ζ11ζ117+ζ114ζ118+ζ113ζ116+ζ115ζ119+ζ112-ζ116-ζ115-ζ119-ζ112-ζ117-ζ114-ζ118-ζ113-ζ1110-ζ11-ζ4ζ1110+ζ4ζ11ζ43ζ116-ζ43ζ115ζ4ζ119-ζ4ζ112-ζ4ζ119+ζ4ζ112-ζ43ζ116+ζ43ζ115ζ4ζ1110-ζ4ζ11ζ43ζ118-ζ43ζ113ζ4ζ117-ζ4ζ114-ζ4ζ117+ζ4ζ114-ζ43ζ118+ζ43ζ113    symplectic faithful, Schur index 2
ρ252-2000ζ117+ζ114ζ116+ζ115ζ1110+ζ11ζ119+ζ112ζ118+ζ113-ζ119-ζ112-ζ118-ζ113-ζ116-ζ115-ζ1110-ζ11-ζ117-ζ114ζ4ζ117-ζ4ζ114-ζ4ζ119+ζ4ζ112-ζ43ζ118+ζ43ζ113ζ43ζ118-ζ43ζ113ζ4ζ119-ζ4ζ112-ζ4ζ117+ζ4ζ114ζ4ζ1110-ζ4ζ11ζ43ζ116-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  D44⋊5C2  D4⋊2D11  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

520
2020
,
941
4134
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
 

Export

Subgroup lattice of Dic22 in TeX
Character table of Dic22 in TeX

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