metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic58, C29⋊Q8, C4.D29, C2.3D58, C116.1C2, C58.1C22, Dic29.1C2, SmallGroup(232,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic58
G = < a,b | a116=1, b2=a58, bab-1=a-1 >
Smallest permutation representation of Dic58
►Regular action on 232 points
Generators in S232
(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)(117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232)
(1 134 59 192)(2 133 60 191)(3 132 61 190)(4 131 62 189)(5 130 63 188)(6 129 64 187)(7 128 65 186)(8 127 66 185)(9 126 67 184)(10 125 68 183)(11 124 69 182)(12 123 70 181)(13 122 71 180)(14 121 72 179)(15 120 73 178)(16 119 74 177)(17 118 75 176)(18 117 76 175)(19 232 77 174)(20 231 78 173)(21 230 79 172)(22 229 80 171)(23 228 81 170)(24 227 82 169)(25 226 83 168)(26 225 84 167)(27 224 85 166)(28 223 86 165)(29 222 87 164)(30 221 88 163)(31 220 89 162)(32 219 90 161)(33 218 91 160)(34 217 92 159)(35 216 93 158)(36 215 94 157)(37 214 95 156)(38 213 96 155)(39 212 97 154)(40 211 98 153)(41 210 99 152)(42 209 100 151)(43 208 101 150)(44 207 102 149)(45 206 103 148)(46 205 104 147)(47 204 105 146)(48 203 106 145)(49 202 107 144)(50 201 108 143)(51 200 109 142)(52 199 110 141)(53 198 111 140)(54 197 112 139)(55 196 113 138)(56 195 114 137)(57 194 115 136)(58 193 116 135)
G:=sub<Sym(232)| (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)(117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232), (1,134,59,192)(2,133,60,191)(3,132,61,190)(4,131,62,189)(5,130,63,188)(6,129,64,187)(7,128,65,186)(8,127,66,185)(9,126,67,184)(10,125,68,183)(11,124,69,182)(12,123,70,181)(13,122,71,180)(14,121,72,179)(15,120,73,178)(16,119,74,177)(17,118,75,176)(18,117,76,175)(19,232,77,174)(20,231,78,173)(21,230,79,172)(22,229,80,171)(23,228,81,170)(24,227,82,169)(25,226,83,168)(26,225,84,167)(27,224,85,166)(28,223,86,165)(29,222,87,164)(30,221,88,163)(31,220,89,162)(32,219,90,161)(33,218,91,160)(34,217,92,159)(35,216,93,158)(36,215,94,157)(37,214,95,156)(38,213,96,155)(39,212,97,154)(40,211,98,153)(41,210,99,152)(42,209,100,151)(43,208,101,150)(44,207,102,149)(45,206,103,148)(46,205,104,147)(47,204,105,146)(48,203,106,145)(49,202,107,144)(50,201,108,143)(51,200,109,142)(52,199,110,141)(53,198,111,140)(54,197,112,139)(55,196,113,138)(56,195,114,137)(57,194,115,136)(58,193,116,135)>;
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)(117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232), (1,134,59,192)(2,133,60,191)(3,132,61,190)(4,131,62,189)(5,130,63,188)(6,129,64,187)(7,128,65,186)(8,127,66,185)(9,126,67,184)(10,125,68,183)(11,124,69,182)(12,123,70,181)(13,122,71,180)(14,121,72,179)(15,120,73,178)(16,119,74,177)(17,118,75,176)(18,117,76,175)(19,232,77,174)(20,231,78,173)(21,230,79,172)(22,229,80,171)(23,228,81,170)(24,227,82,169)(25,226,83,168)(26,225,84,167)(27,224,85,166)(28,223,86,165)(29,222,87,164)(30,221,88,163)(31,220,89,162)(32,219,90,161)(33,218,91,160)(34,217,92,159)(35,216,93,158)(36,215,94,157)(37,214,95,156)(38,213,96,155)(39,212,97,154)(40,211,98,153)(41,210,99,152)(42,209,100,151)(43,208,101,150)(44,207,102,149)(45,206,103,148)(46,205,104,147)(47,204,105,146)(48,203,106,145)(49,202,107,144)(50,201,108,143)(51,200,109,142)(52,199,110,141)(53,198,111,140)(54,197,112,139)(55,196,113,138)(56,195,114,137)(57,194,115,136)(58,193,116,135) );
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),(117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232)], [(1,134,59,192),(2,133,60,191),(3,132,61,190),(4,131,62,189),(5,130,63,188),(6,129,64,187),(7,128,65,186),(8,127,66,185),(9,126,67,184),(10,125,68,183),(11,124,69,182),(12,123,70,181),(13,122,71,180),(14,121,72,179),(15,120,73,178),(16,119,74,177),(17,118,75,176),(18,117,76,175),(19,232,77,174),(20,231,78,173),(21,230,79,172),(22,229,80,171),(23,228,81,170),(24,227,82,169),(25,226,83,168),(26,225,84,167),(27,224,85,166),(28,223,86,165),(29,222,87,164),(30,221,88,163),(31,220,89,162),(32,219,90,161),(33,218,91,160),(34,217,92,159),(35,216,93,158),(36,215,94,157),(37,214,95,156),(38,213,96,155),(39,212,97,154),(40,211,98,153),(41,210,99,152),(42,209,100,151),(43,208,101,150),(44,207,102,149),(45,206,103,148),(46,205,104,147),(47,204,105,146),(48,203,106,145),(49,202,107,144),(50,201,108,143),(51,200,109,142),(52,199,110,141),(53,198,111,140),(54,197,112,139),(55,196,113,138),(56,195,114,137),(57,194,115,136),(58,193,116,135)]])
Dic58 is a maximal subgroup of
C232⋊C2 Dic116 D4.D29 C29⋊Q16 D116⋊5C2 D4⋊2D29 Q8×D29
Dic58 is a maximal quotient of C58.D4 C4⋊Dic29
61 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 29A | ··· | 29N | 58A | ··· | 58N | 116A | ··· | 116AB |
| order | 1 | 2 | 4 | 4 | 4 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 2 | 58 | 58 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
61 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | Q8 | D29 | D58 | Dic58 |
| kernel | Dic58 | Dic29 | C116 | C29 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 14 | 14 | 28 |
Matrix representation of Dic58 ►in GL2(𝔽233) generated by
| 224 | 49 |
| 184 | 189 |
| 33 | 148 |
| 191 | 200 |
G:=sub<GL(2,GF(233))| [224,184,49,189],[33,191,148,200] >;
Dic58 in GAP, Magma, Sage, TeX
{\rm Dic}_{58} % in TeX
G:=Group("Dic58"); // GroupNames label
G:=SmallGroup(232,4);
// by ID
G=gap.SmallGroup(232,4);
# by ID
G:=PCGroup([4,-2,-2,-2,-29,16,49,21,3587]);
// Polycyclic
G:=Group<a,b|a^116=1,b^2=a^58,b*a*b^-1=a^-1>;
// generators/relations
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