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G = Dic22order 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

C1C22 — Dic22
C1C11C22Dic11 — Dic22
C11C22 — Dic22
C1C2C4

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

11C4
11C4
11Q8

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
ρ522-200ζ117114ζ116115ζ111011ζ119112ζ118113ζ119112ζ118113ζ116115ζ111011ζ117114117114119112118113118113119112117114111011116115116115111011    orthogonal lifted from D22
ρ622-200ζ111011ζ117114ζ118113ζ116115ζ119112ζ116115ζ119112ζ117114ζ118113ζ111011111011116115119112119112116115111011118113117114117114118113    orthogonal lifted from D22
ρ722-200ζ119112ζ118113ζ116115ζ111011ζ117114ζ111011ζ117114ζ118113ζ116115ζ119112119112111011117114117114111011119112116115118113118113116115    orthogonal lifted from D22
ρ822200ζ119112ζ118113ζ116115ζ111011ζ117114ζ111011ζ117114ζ118113ζ116115ζ119112ζ119112ζ111011ζ117114ζ117114ζ111011ζ119112ζ116115ζ118113ζ118113ζ116115    orthogonal lifted from D11
ρ922200ζ117114ζ116115ζ111011ζ119112ζ118113ζ119112ζ118113ζ116115ζ111011ζ117114ζ117114ζ119112ζ118113ζ118113ζ119112ζ117114ζ111011ζ116115ζ116115ζ111011    orthogonal lifted from D11
ρ1022200ζ111011ζ117114ζ118113ζ116115ζ119112ζ116115ζ119112ζ117114ζ118113ζ111011ζ111011ζ116115ζ119112ζ119112ζ116115ζ111011ζ118113ζ117114ζ117114ζ118113    orthogonal lifted from D11
ρ1122-200ζ116115ζ119112ζ117114ζ118113ζ111011ζ118113ζ111011ζ119112ζ117114ζ116115116115118113111011111011118113116115117114119112119112117114    orthogonal lifted from D22
ρ1222-200ζ118113ζ111011ζ119112ζ117114ζ116115ζ117114ζ116115ζ111011ζ119112ζ118113118113117114116115116115117114118113119112111011111011119112    orthogonal lifted from D22
ρ1322200ζ116115ζ119112ζ117114ζ118113ζ111011ζ118113ζ111011ζ119112ζ117114ζ116115ζ116115ζ118113ζ111011ζ111011ζ118113ζ116115ζ117114ζ119112ζ119112ζ117114    orthogonal lifted from D11
ρ1422200ζ118113ζ111011ζ119112ζ117114ζ116115ζ117114ζ116115ζ111011ζ119112ζ118113ζ118113ζ117114ζ116115ζ116115ζ117114ζ118113ζ119112ζ111011ζ111011ζ119112    orthogonal lifted from D11
ρ152-200022222-2-2-2-2-20000000000    symplectic lifted from Q8, Schur index 2
ρ162-2000ζ111011ζ117114ζ118113ζ116115ζ119112116115119112117114118113111011ζ4ζ11104ζ1143ζ11643ζ1154ζ1194ζ112ζ4ζ1194ζ112ζ43ζ11643ζ1154ζ11104ζ1143ζ11843ζ1134ζ1174ζ114ζ4ζ1174ζ114ζ43ζ11843ζ113    symplectic faithful, Schur index 2
ρ172-2000ζ117114ζ116115ζ111011ζ119112ζ1181131191121181131161151110111171144ζ1174ζ114ζ4ζ1194ζ112ζ43ζ11843ζ11343ζ11843ζ1134ζ1194ζ112ζ4ζ1174ζ1144ζ11104ζ1143ζ11643ζ115ζ43ζ11643ζ115ζ4ζ11104ζ11    symplectic faithful, Schur index 2
ρ182-2000ζ118113ζ111011ζ119112ζ117114ζ116115117114116115111011119112118113ζ43ζ11843ζ1134ζ1174ζ114ζ43ζ11643ζ11543ζ11643ζ115ζ4ζ1174ζ11443ζ11843ζ113ζ4ζ1194ζ112ζ4ζ11104ζ114ζ11104ζ114ζ1194ζ112    symplectic faithful, Schur index 2
ρ192-2000ζ116115ζ119112ζ117114ζ118113ζ11101111811311101111911211711411611543ζ11643ζ11543ζ11843ζ113ζ4ζ11104ζ114ζ11104ζ11ζ43ζ11843ζ113ζ43ζ11643ζ115ζ4ζ1174ζ114ζ4ζ1194ζ1124ζ1194ζ1124ζ1174ζ114    symplectic faithful, Schur index 2
ρ202-2000ζ119112ζ118113ζ116115ζ111011ζ1171141110111171141181131161151191124ζ1194ζ112ζ4ζ11104ζ11ζ4ζ1174ζ1144ζ1174ζ1144ζ11104ζ11ζ4ζ1194ζ11243ζ11643ζ115ζ43ζ11843ζ11343ζ11843ζ113ζ43ζ11643ζ115    symplectic faithful, Schur index 2
ρ212-2000ζ119112ζ118113ζ116115ζ111011ζ117114111011117114118113116115119112ζ4ζ1194ζ1124ζ11104ζ114ζ1174ζ114ζ4ζ1174ζ114ζ4ζ11104ζ114ζ1194ζ112ζ43ζ11643ζ11543ζ11843ζ113ζ43ζ11843ζ11343ζ11643ζ115    symplectic faithful, Schur index 2
ρ222-2000ζ116115ζ119112ζ117114ζ118113ζ111011118113111011119112117114116115ζ43ζ11643ζ115ζ43ζ11843ζ1134ζ11104ζ11ζ4ζ11104ζ1143ζ11843ζ11343ζ11643ζ1154ζ1174ζ1144ζ1194ζ112ζ4ζ1194ζ112ζ4ζ1174ζ114    symplectic faithful, Schur index 2
ρ232-2000ζ118113ζ111011ζ119112ζ117114ζ11611511711411611511101111911211811343ζ11843ζ113ζ4ζ1174ζ11443ζ11643ζ115ζ43ζ11643ζ1154ζ1174ζ114ζ43ζ11843ζ1134ζ1194ζ1124ζ11104ζ11ζ4ζ11104ζ11ζ4ζ1194ζ112    symplectic faithful, Schur index 2
ρ242-2000ζ111011ζ117114ζ118113ζ116115ζ1191121161151191121171141181131110114ζ11104ζ11ζ43ζ11643ζ115ζ4ζ1194ζ1124ζ1194ζ11243ζ11643ζ115ζ4ζ11104ζ11ζ43ζ11843ζ113ζ4ζ1174ζ1144ζ1174ζ11443ζ11843ζ113    symplectic faithful, Schur index 2
ρ252-2000ζ117114ζ116115ζ111011ζ119112ζ118113119112118113116115111011117114ζ4ζ1174ζ1144ζ1194ζ11243ζ11843ζ113ζ43ζ11843ζ113ζ4ζ1194ζ1124ζ1174ζ114ζ4ζ11104ζ11ζ43ζ11643ζ11543ζ11643ζ1154ζ11104ζ11    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 69 23 47)(2 68 24 46)(3 67 25 45)(4 66 26 88)(5 65 27 87)(6 64 28 86)(7 63 29 85)(8 62 30 84)(9 61 31 83)(10 60 32 82)(11 59 33 81)(12 58 34 80)(13 57 35 79)(14 56 36 78)(15 55 37 77)(16 54 38 76)(17 53 39 75)(18 52 40 74)(19 51 41 73)(20 50 42 72)(21 49 43 71)(22 48 44 70)

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

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

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

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

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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