direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: D8×C29, D4⋊C58, C8⋊1C58, C232⋊5C2, C58.14D4, C116.17C22, (D4×C29)⋊4C2, C4.1(C2×C58), C2.3(D4×C29), SmallGroup(464,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8×C29
G = < a,b,c | a29=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D8×C29
►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)
(1 183 141 229 62 104 36 150)(2 184 142 230 63 105 37 151)(3 185 143 231 64 106 38 152)(4 186 144 232 65 107 39 153)(5 187 145 204 66 108 40 154)(6 188 117 205 67 109 41 155)(7 189 118 206 68 110 42 156)(8 190 119 207 69 111 43 157)(9 191 120 208 70 112 44 158)(10 192 121 209 71 113 45 159)(11 193 122 210 72 114 46 160)(12 194 123 211 73 115 47 161)(13 195 124 212 74 116 48 162)(14 196 125 213 75 88 49 163)(15 197 126 214 76 89 50 164)(16 198 127 215 77 90 51 165)(17 199 128 216 78 91 52 166)(18 200 129 217 79 92 53 167)(19 201 130 218 80 93 54 168)(20 202 131 219 81 94 55 169)(21 203 132 220 82 95 56 170)(22 175 133 221 83 96 57 171)(23 176 134 222 84 97 58 172)(24 177 135 223 85 98 30 173)(25 178 136 224 86 99 31 174)(26 179 137 225 87 100 32 146)(27 180 138 226 59 101 33 147)(28 181 139 227 60 102 34 148)(29 182 140 228 61 103 35 149)
(30 135)(31 136)(32 137)(33 138)(34 139)(35 140)(36 141)(37 142)(38 143)(39 144)(40 145)(41 117)(42 118)(43 119)(44 120)(45 121)(46 122)(47 123)(48 124)(49 125)(50 126)(51 127)(52 128)(53 129)(54 130)(55 131)(56 132)(57 133)(58 134)(88 213)(89 214)(90 215)(91 216)(92 217)(93 218)(94 219)(95 220)(96 221)(97 222)(98 223)(99 224)(100 225)(101 226)(102 227)(103 228)(104 229)(105 230)(106 231)(107 232)(108 204)(109 205)(110 206)(111 207)(112 208)(113 209)(114 210)(115 211)(116 212)(146 179)(147 180)(148 181)(149 182)(150 183)(151 184)(152 185)(153 186)(154 187)(155 188)(156 189)(157 190)(158 191)(159 192)(160 193)(161 194)(162 195)(163 196)(164 197)(165 198)(166 199)(167 200)(168 201)(169 202)(170 203)(171 175)(172 176)(173 177)(174 178)
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), (1,183,141,229,62,104,36,150)(2,184,142,230,63,105,37,151)(3,185,143,231,64,106,38,152)(4,186,144,232,65,107,39,153)(5,187,145,204,66,108,40,154)(6,188,117,205,67,109,41,155)(7,189,118,206,68,110,42,156)(8,190,119,207,69,111,43,157)(9,191,120,208,70,112,44,158)(10,192,121,209,71,113,45,159)(11,193,122,210,72,114,46,160)(12,194,123,211,73,115,47,161)(13,195,124,212,74,116,48,162)(14,196,125,213,75,88,49,163)(15,197,126,214,76,89,50,164)(16,198,127,215,77,90,51,165)(17,199,128,216,78,91,52,166)(18,200,129,217,79,92,53,167)(19,201,130,218,80,93,54,168)(20,202,131,219,81,94,55,169)(21,203,132,220,82,95,56,170)(22,175,133,221,83,96,57,171)(23,176,134,222,84,97,58,172)(24,177,135,223,85,98,30,173)(25,178,136,224,86,99,31,174)(26,179,137,225,87,100,32,146)(27,180,138,226,59,101,33,147)(28,181,139,227,60,102,34,148)(29,182,140,228,61,103,35,149), (30,135)(31,136)(32,137)(33,138)(34,139)(35,140)(36,141)(37,142)(38,143)(39,144)(40,145)(41,117)(42,118)(43,119)(44,120)(45,121)(46,122)(47,123)(48,124)(49,125)(50,126)(51,127)(52,128)(53,129)(54,130)(55,131)(56,132)(57,133)(58,134)(88,213)(89,214)(90,215)(91,216)(92,217)(93,218)(94,219)(95,220)(96,221)(97,222)(98,223)(99,224)(100,225)(101,226)(102,227)(103,228)(104,229)(105,230)(106,231)(107,232)(108,204)(109,205)(110,206)(111,207)(112,208)(113,209)(114,210)(115,211)(116,212)(146,179)(147,180)(148,181)(149,182)(150,183)(151,184)(152,185)(153,186)(154,187)(155,188)(156,189)(157,190)(158,191)(159,192)(160,193)(161,194)(162,195)(163,196)(164,197)(165,198)(166,199)(167,200)(168,201)(169,202)(170,203)(171,175)(172,176)(173,177)(174,178)>;
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), (1,183,141,229,62,104,36,150)(2,184,142,230,63,105,37,151)(3,185,143,231,64,106,38,152)(4,186,144,232,65,107,39,153)(5,187,145,204,66,108,40,154)(6,188,117,205,67,109,41,155)(7,189,118,206,68,110,42,156)(8,190,119,207,69,111,43,157)(9,191,120,208,70,112,44,158)(10,192,121,209,71,113,45,159)(11,193,122,210,72,114,46,160)(12,194,123,211,73,115,47,161)(13,195,124,212,74,116,48,162)(14,196,125,213,75,88,49,163)(15,197,126,214,76,89,50,164)(16,198,127,215,77,90,51,165)(17,199,128,216,78,91,52,166)(18,200,129,217,79,92,53,167)(19,201,130,218,80,93,54,168)(20,202,131,219,81,94,55,169)(21,203,132,220,82,95,56,170)(22,175,133,221,83,96,57,171)(23,176,134,222,84,97,58,172)(24,177,135,223,85,98,30,173)(25,178,136,224,86,99,31,174)(26,179,137,225,87,100,32,146)(27,180,138,226,59,101,33,147)(28,181,139,227,60,102,34,148)(29,182,140,228,61,103,35,149), (30,135)(31,136)(32,137)(33,138)(34,139)(35,140)(36,141)(37,142)(38,143)(39,144)(40,145)(41,117)(42,118)(43,119)(44,120)(45,121)(46,122)(47,123)(48,124)(49,125)(50,126)(51,127)(52,128)(53,129)(54,130)(55,131)(56,132)(57,133)(58,134)(88,213)(89,214)(90,215)(91,216)(92,217)(93,218)(94,219)(95,220)(96,221)(97,222)(98,223)(99,224)(100,225)(101,226)(102,227)(103,228)(104,229)(105,230)(106,231)(107,232)(108,204)(109,205)(110,206)(111,207)(112,208)(113,209)(114,210)(115,211)(116,212)(146,179)(147,180)(148,181)(149,182)(150,183)(151,184)(152,185)(153,186)(154,187)(155,188)(156,189)(157,190)(158,191)(159,192)(160,193)(161,194)(162,195)(163,196)(164,197)(165,198)(166,199)(167,200)(168,201)(169,202)(170,203)(171,175)(172,176)(173,177)(174,178) );
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)], [(1,183,141,229,62,104,36,150),(2,184,142,230,63,105,37,151),(3,185,143,231,64,106,38,152),(4,186,144,232,65,107,39,153),(5,187,145,204,66,108,40,154),(6,188,117,205,67,109,41,155),(7,189,118,206,68,110,42,156),(8,190,119,207,69,111,43,157),(9,191,120,208,70,112,44,158),(10,192,121,209,71,113,45,159),(11,193,122,210,72,114,46,160),(12,194,123,211,73,115,47,161),(13,195,124,212,74,116,48,162),(14,196,125,213,75,88,49,163),(15,197,126,214,76,89,50,164),(16,198,127,215,77,90,51,165),(17,199,128,216,78,91,52,166),(18,200,129,217,79,92,53,167),(19,201,130,218,80,93,54,168),(20,202,131,219,81,94,55,169),(21,203,132,220,82,95,56,170),(22,175,133,221,83,96,57,171),(23,176,134,222,84,97,58,172),(24,177,135,223,85,98,30,173),(25,178,136,224,86,99,31,174),(26,179,137,225,87,100,32,146),(27,180,138,226,59,101,33,147),(28,181,139,227,60,102,34,148),(29,182,140,228,61,103,35,149)], [(30,135),(31,136),(32,137),(33,138),(34,139),(35,140),(36,141),(37,142),(38,143),(39,144),(40,145),(41,117),(42,118),(43,119),(44,120),(45,121),(46,122),(47,123),(48,124),(49,125),(50,126),(51,127),(52,128),(53,129),(54,130),(55,131),(56,132),(57,133),(58,134),(88,213),(89,214),(90,215),(91,216),(92,217),(93,218),(94,219),(95,220),(96,221),(97,222),(98,223),(99,224),(100,225),(101,226),(102,227),(103,228),(104,229),(105,230),(106,231),(107,232),(108,204),(109,205),(110,206),(111,207),(112,208),(113,209),(114,210),(115,211),(116,212),(146,179),(147,180),(148,181),(149,182),(150,183),(151,184),(152,185),(153,186),(154,187),(155,188),(156,189),(157,190),(158,191),(159,192),(160,193),(161,194),(162,195),(163,196),(164,197),(165,198),(166,199),(167,200),(168,201),(169,202),(170,203),(171,175),(172,176),(173,177),(174,178)]])
203 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 8A | 8B | 29A | ··· | 29AB | 58A | ··· | 58AB | 58AC | ··· | 58CF | 116A | ··· | 116AB | 232A | ··· | 232BD |
| order | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 29 | ··· | 29 | 58 | ··· | 58 | 58 | ··· | 58 | 116 | ··· | 116 | 232 | ··· | 232 |
| size | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 |
203 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C29 | C58 | C58 | D4 | D8 | D4×C29 | D8×C29 |
| kernel | D8×C29 | C232 | D4×C29 | D8 | C8 | D4 | C58 | C29 | C2 | C1 |
| # reps | 1 | 1 | 2 | 28 | 28 | 56 | 1 | 2 | 28 | 56 |
Matrix representation of D8×C29 ►in GL2(𝔽233) generated by
| 102 | 0 |
| 0 | 102 |
| 159 | 74 |
| 159 | 159 |
| 1 | 0 |
| 0 | 232 |
G:=sub<GL(2,GF(233))| [102,0,0,102],[159,159,74,159],[1,0,0,232] >;
D8×C29 in GAP, Magma, Sage, TeX
D_8\times C_{29} % in TeX
G:=Group("D8xC29"); // GroupNames label
G:=SmallGroup(464,25);
// by ID
G=gap.SmallGroup(464,25);
# by ID
G:=PCGroup([5,-2,-2,-29,-2,-2,1181,6963,3488,58]);
// Polycyclic
G:=Group<a,b,c|a^29=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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