metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic41, C41⋊2C4, C82.C2, C2.D41, SmallGroup(164,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C41 — Dic41 |
Generators and relations for Dic41
G = < a,b | a82=1, b2=a41, bab-1=a-1 >
Smallest permutation representation of Dic41
►Regular action on 164 points
Generators in S164
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164)
(1 141 42 100)(2 140 43 99)(3 139 44 98)(4 138 45 97)(5 137 46 96)(6 136 47 95)(7 135 48 94)(8 134 49 93)(9 133 50 92)(10 132 51 91)(11 131 52 90)(12 130 53 89)(13 129 54 88)(14 128 55 87)(15 127 56 86)(16 126 57 85)(17 125 58 84)(18 124 59 83)(19 123 60 164)(20 122 61 163)(21 121 62 162)(22 120 63 161)(23 119 64 160)(24 118 65 159)(25 117 66 158)(26 116 67 157)(27 115 68 156)(28 114 69 155)(29 113 70 154)(30 112 71 153)(31 111 72 152)(32 110 73 151)(33 109 74 150)(34 108 75 149)(35 107 76 148)(36 106 77 147)(37 105 78 146)(38 104 79 145)(39 103 80 144)(40 102 81 143)(41 101 82 142)
G:=sub<Sym(164)| (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)(83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164), (1,141,42,100)(2,140,43,99)(3,139,44,98)(4,138,45,97)(5,137,46,96)(6,136,47,95)(7,135,48,94)(8,134,49,93)(9,133,50,92)(10,132,51,91)(11,131,52,90)(12,130,53,89)(13,129,54,88)(14,128,55,87)(15,127,56,86)(16,126,57,85)(17,125,58,84)(18,124,59,83)(19,123,60,164)(20,122,61,163)(21,121,62,162)(22,120,63,161)(23,119,64,160)(24,118,65,159)(25,117,66,158)(26,116,67,157)(27,115,68,156)(28,114,69,155)(29,113,70,154)(30,112,71,153)(31,111,72,152)(32,110,73,151)(33,109,74,150)(34,108,75,149)(35,107,76,148)(36,106,77,147)(37,105,78,146)(38,104,79,145)(39,103,80,144)(40,102,81,143)(41,101,82,142)>;
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)(83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164), (1,141,42,100)(2,140,43,99)(3,139,44,98)(4,138,45,97)(5,137,46,96)(6,136,47,95)(7,135,48,94)(8,134,49,93)(9,133,50,92)(10,132,51,91)(11,131,52,90)(12,130,53,89)(13,129,54,88)(14,128,55,87)(15,127,56,86)(16,126,57,85)(17,125,58,84)(18,124,59,83)(19,123,60,164)(20,122,61,163)(21,121,62,162)(22,120,63,161)(23,119,64,160)(24,118,65,159)(25,117,66,158)(26,116,67,157)(27,115,68,156)(28,114,69,155)(29,113,70,154)(30,112,71,153)(31,111,72,152)(32,110,73,151)(33,109,74,150)(34,108,75,149)(35,107,76,148)(36,106,77,147)(37,105,78,146)(38,104,79,145)(39,103,80,144)(40,102,81,143)(41,101,82,142) );
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),(83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164)], [(1,141,42,100),(2,140,43,99),(3,139,44,98),(4,138,45,97),(5,137,46,96),(6,136,47,95),(7,135,48,94),(8,134,49,93),(9,133,50,92),(10,132,51,91),(11,131,52,90),(12,130,53,89),(13,129,54,88),(14,128,55,87),(15,127,56,86),(16,126,57,85),(17,125,58,84),(18,124,59,83),(19,123,60,164),(20,122,61,163),(21,121,62,162),(22,120,63,161),(23,119,64,160),(24,118,65,159),(25,117,66,158),(26,116,67,157),(27,115,68,156),(28,114,69,155),(29,113,70,154),(30,112,71,153),(31,111,72,152),(32,110,73,151),(33,109,74,150),(34,108,75,149),(35,107,76,148),(36,106,77,147),(37,105,78,146),(38,104,79,145),(39,103,80,144),(40,102,81,143),(41,101,82,142)]])
Dic41 is a maximal subgroup of
C41⋊2C8 Dic82 C4×D41 C41⋊D4 Dic123
Dic41 is a maximal quotient of C41⋊3C8 Dic123
44 conjugacy classes
| class | 1 | 2 | 4A | 4B | 41A | ··· | 41T | 82A | ··· | 82T |
| order | 1 | 2 | 4 | 4 | 41 | ··· | 41 | 82 | ··· | 82 |
| size | 1 | 1 | 41 | 41 | 2 | ··· | 2 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | |
| image | C1 | C2 | C4 | D41 | Dic41 |
| kernel | Dic41 | C82 | C41 | C2 | C1 |
| # reps | 1 | 1 | 2 | 20 | 20 |
Matrix representation of Dic41 ►in GL3(𝔽821) generated by
| 820 | 0 | 0 |
| 0 | 782 | 820 |
| 0 | 1 | 0 |
| 526 | 0 | 0 |
| 0 | 725 | 390 |
| 0 | 29 | 96 |
G:=sub<GL(3,GF(821))| [820,0,0,0,782,1,0,820,0],[526,0,0,0,725,29,0,390,96] >;
Dic41 in GAP, Magma, Sage, TeX
{\rm Dic}_{41} % in TeX
G:=Group("Dic41"); // GroupNames label
G:=SmallGroup(164,1);
// by ID
G=gap.SmallGroup(164,1);
# by ID
G:=PCGroup([3,-2,-2,-41,6,1442]);
// Polycyclic
G:=Group<a,b|a^82=1,b^2=a^41,b*a*b^-1=a^-1>;
// generators/relations
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