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## G = Dic31order 124 = 22·31

### Dicyclic group

Aliases: Dic31, C31⋊C4, C62.C2, C2.D31, SmallGroup(124,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C31 — Dic31
 Chief series C1 — C31 — C62 — Dic31
 Lower central C31 — Dic31
 Upper central C1 — C2

Generators and relations for Dic31
G = < a,b | a62=1, b2=a31, bab-1=a-1 >

Smallest permutation representation of Dic31
Regular action on 124 points
Generators in S124
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124)
(1 98 32 67)(2 97 33 66)(3 96 34 65)(4 95 35 64)(5 94 36 63)(6 93 37 124)(7 92 38 123)(8 91 39 122)(9 90 40 121)(10 89 41 120)(11 88 42 119)(12 87 43 118)(13 86 44 117)(14 85 45 116)(15 84 46 115)(16 83 47 114)(17 82 48 113)(18 81 49 112)(19 80 50 111)(20 79 51 110)(21 78 52 109)(22 77 53 108)(23 76 54 107)(24 75 55 106)(25 74 56 105)(26 73 57 104)(27 72 58 103)(28 71 59 102)(29 70 60 101)(30 69 61 100)(31 68 62 99)

G:=sub<Sym(124)| (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)(63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124), (1,98,32,67)(2,97,33,66)(3,96,34,65)(4,95,35,64)(5,94,36,63)(6,93,37,124)(7,92,38,123)(8,91,39,122)(9,90,40,121)(10,89,41,120)(11,88,42,119)(12,87,43,118)(13,86,44,117)(14,85,45,116)(15,84,46,115)(16,83,47,114)(17,82,48,113)(18,81,49,112)(19,80,50,111)(20,79,51,110)(21,78,52,109)(22,77,53,108)(23,76,54,107)(24,75,55,106)(25,74,56,105)(26,73,57,104)(27,72,58,103)(28,71,59,102)(29,70,60,101)(30,69,61,100)(31,68,62,99)>;

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)(63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124), (1,98,32,67)(2,97,33,66)(3,96,34,65)(4,95,35,64)(5,94,36,63)(6,93,37,124)(7,92,38,123)(8,91,39,122)(9,90,40,121)(10,89,41,120)(11,88,42,119)(12,87,43,118)(13,86,44,117)(14,85,45,116)(15,84,46,115)(16,83,47,114)(17,82,48,113)(18,81,49,112)(19,80,50,111)(20,79,51,110)(21,78,52,109)(22,77,53,108)(23,76,54,107)(24,75,55,106)(25,74,56,105)(26,73,57,104)(27,72,58,103)(28,71,59,102)(29,70,60,101)(30,69,61,100)(31,68,62,99) );

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),(63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124)], [(1,98,32,67),(2,97,33,66),(3,96,34,65),(4,95,35,64),(5,94,36,63),(6,93,37,124),(7,92,38,123),(8,91,39,122),(9,90,40,121),(10,89,41,120),(11,88,42,119),(12,87,43,118),(13,86,44,117),(14,85,45,116),(15,84,46,115),(16,83,47,114),(17,82,48,113),(18,81,49,112),(19,80,50,111),(20,79,51,110),(21,78,52,109),(22,77,53,108),(23,76,54,107),(24,75,55,106),(25,74,56,105),(26,73,57,104),(27,72,58,103),(28,71,59,102),(29,70,60,101),(30,69,61,100),(31,68,62,99)]])

Dic31 is a maximal subgroup of   Dic62  C4×D31  C31⋊D4  C31⋊C12  Dic93
Dic31 is a maximal quotient of   C31⋊C8  Dic93

34 conjugacy classes

 class 1 2 4A 4B 31A ··· 31O 62A ··· 62O order 1 2 4 4 31 ··· 31 62 ··· 62 size 1 1 31 31 2 ··· 2 2 ··· 2

34 irreducible representations

 dim 1 1 1 2 2 type + + + - image C1 C2 C4 D31 Dic31 kernel Dic31 C62 C31 C2 C1 # reps 1 1 2 15 15

Matrix representation of Dic31 in GL3(𝔽373) generated by

 372 0 0 0 316 372 0 1 0
,
 104 0 0 0 274 241 0 289 99
G:=sub<GL(3,GF(373))| [372,0,0,0,316,1,0,372,0],[104,0,0,0,274,289,0,241,99] >;

Dic31 in GAP, Magma, Sage, TeX

{\rm Dic}_{31}
% in TeX

G:=Group("Dic31");
// GroupNames label

G:=SmallGroup(124,1);
// by ID

G=gap.SmallGroup(124,1);
# by ID

G:=PCGroup([3,-2,-2,-31,6,1082]);
// Polycyclic

G:=Group<a,b|a^62=1,b^2=a^31,b*a*b^-1=a^-1>;
// generators/relations

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