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G = Dic31order 124 = 22·31

Dicyclic group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C31 — Dic31
C1C31C62 — Dic31
C31 — Dic31
C1C2

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

31C4

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

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

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

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

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 4A4B31A···31O62A···62O
order124431···3162···62
size1131312···22···2

34 irreducible representations

dim11122
type+++-
imageC1C2C4D31Dic31
kernelDic31C62C31C2C1
# reps1121515

Matrix representation of Dic31 in GL3(𝔽373) generated by

37200
0316372
010
,
10400
0274241
028999
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

Export

Subgroup lattice of Dic31 in TeX

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