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## G = Dic29order 116 = 22·29

### Dicyclic group

Aliases: Dic29, C292C4, C58.C2, C2.D29, SmallGroup(116,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C29 — Dic29
 Chief series C1 — C29 — C58 — Dic29
 Lower central C29 — Dic29
 Upper central C1 — C2

Generators and relations for Dic29
G = < a,b | a58=1, b2=a29, bab-1=a-1 >

Smallest permutation representation of Dic29
Regular action on 116 points
Generators in S116
(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 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116)
(1 109 30 80)(2 108 31 79)(3 107 32 78)(4 106 33 77)(5 105 34 76)(6 104 35 75)(7 103 36 74)(8 102 37 73)(9 101 38 72)(10 100 39 71)(11 99 40 70)(12 98 41 69)(13 97 42 68)(14 96 43 67)(15 95 44 66)(16 94 45 65)(17 93 46 64)(18 92 47 63)(19 91 48 62)(20 90 49 61)(21 89 50 60)(22 88 51 59)(23 87 52 116)(24 86 53 115)(25 85 54 114)(26 84 55 113)(27 83 56 112)(28 82 57 111)(29 81 58 110)

G:=sub<Sym(116)| (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)(59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116), (1,109,30,80)(2,108,31,79)(3,107,32,78)(4,106,33,77)(5,105,34,76)(6,104,35,75)(7,103,36,74)(8,102,37,73)(9,101,38,72)(10,100,39,71)(11,99,40,70)(12,98,41,69)(13,97,42,68)(14,96,43,67)(15,95,44,66)(16,94,45,65)(17,93,46,64)(18,92,47,63)(19,91,48,62)(20,90,49,61)(21,89,50,60)(22,88,51,59)(23,87,52,116)(24,86,53,115)(25,85,54,114)(26,84,55,113)(27,83,56,112)(28,82,57,111)(29,81,58,110)>;

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)(59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116), (1,109,30,80)(2,108,31,79)(3,107,32,78)(4,106,33,77)(5,105,34,76)(6,104,35,75)(7,103,36,74)(8,102,37,73)(9,101,38,72)(10,100,39,71)(11,99,40,70)(12,98,41,69)(13,97,42,68)(14,96,43,67)(15,95,44,66)(16,94,45,65)(17,93,46,64)(18,92,47,63)(19,91,48,62)(20,90,49,61)(21,89,50,60)(22,88,51,59)(23,87,52,116)(24,86,53,115)(25,85,54,114)(26,84,55,113)(27,83,56,112)(28,82,57,111)(29,81,58,110) );

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),(59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116)], [(1,109,30,80),(2,108,31,79),(3,107,32,78),(4,106,33,77),(5,105,34,76),(6,104,35,75),(7,103,36,74),(8,102,37,73),(9,101,38,72),(10,100,39,71),(11,99,40,70),(12,98,41,69),(13,97,42,68),(14,96,43,67),(15,95,44,66),(16,94,45,65),(17,93,46,64),(18,92,47,63),(19,91,48,62),(20,90,49,61),(21,89,50,60),(22,88,51,59),(23,87,52,116),(24,86,53,115),(25,85,54,114),(26,84,55,113),(27,83,56,112),(28,82,57,111),(29,81,58,110)]])

Dic29 is a maximal subgroup of   C29⋊C8  Dic58  C4×D29  C29⋊D4  Dic87
Dic29 is a maximal quotient of   C292C8  Dic87

32 conjugacy classes

 class 1 2 4A 4B 29A ··· 29N 58A ··· 58N order 1 2 4 4 29 ··· 29 58 ··· 58 size 1 1 29 29 2 ··· 2 2 ··· 2

32 irreducible representations

 dim 1 1 1 2 2 type + + + - image C1 C2 C4 D29 Dic29 kernel Dic29 C58 C29 C2 C1 # reps 1 1 2 14 14

Matrix representation of Dic29 in GL3(𝔽233) generated by

 232 0 0 0 28 232 0 1 0
,
 89 0 0 0 225 29 0 38 8
G:=sub<GL(3,GF(233))| [232,0,0,0,28,1,0,232,0],[89,0,0,0,225,38,0,29,8] >;

Dic29 in GAP, Magma, Sage, TeX

{\rm Dic}_{29}
% in TeX

G:=Group("Dic29");
// GroupNames label

G:=SmallGroup(116,1);
// by ID

G=gap.SmallGroup(116,1);
# by ID

G:=PCGroup([3,-2,-2,-29,6,1010]);
// Polycyclic

G:=Group<a,b|a^58=1,b^2=a^29,b*a*b^-1=a^-1>;
// generators/relations

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