metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.D29, C58.8D4, C4.2D58, C29⋊2SD16, Dic58⋊2C2, C116.2C22, C29⋊2C8⋊2C2, (D4×C29).1C2, C2.5(C29⋊D4), SmallGroup(464,16)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D29
G = < a,b,c,d | a4=b2=c29=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >
Smallest permutation representation of D4.D29
►On 232 points
Generators in S232
(1 111 35 75)(2 112 36 76)(3 113 37 77)(4 114 38 78)(5 115 39 79)(6 116 40 80)(7 88 41 81)(8 89 42 82)(9 90 43 83)(10 91 44 84)(11 92 45 85)(12 93 46 86)(13 94 47 87)(14 95 48 59)(15 96 49 60)(16 97 50 61)(17 98 51 62)(18 99 52 63)(19 100 53 64)(20 101 54 65)(21 102 55 66)(22 103 56 67)(23 104 57 68)(24 105 58 69)(25 106 30 70)(26 107 31 71)(27 108 32 72)(28 109 33 73)(29 110 34 74)(117 175 146 204)(118 176 147 205)(119 177 148 206)(120 178 149 207)(121 179 150 208)(122 180 151 209)(123 181 152 210)(124 182 153 211)(125 183 154 212)(126 184 155 213)(127 185 156 214)(128 186 157 215)(129 187 158 216)(130 188 159 217)(131 189 160 218)(132 190 161 219)(133 191 162 220)(134 192 163 221)(135 193 164 222)(136 194 165 223)(137 195 166 224)(138 196 167 225)(139 197 168 226)(140 198 169 227)(141 199 170 228)(142 200 171 229)(143 201 172 230)(144 202 173 231)(145 203 174 232)
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 81)(8 82)(9 83)(10 84)(11 85)(12 86)(13 87)(14 59)(15 60)(16 61)(17 62)(18 63)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 73)(29 74)(30 106)(31 107)(32 108)(33 109)(34 110)(35 111)(36 112)(37 113)(38 114)(39 115)(40 116)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 96)(50 97)(51 98)(52 99)(53 100)(54 101)(55 102)(56 103)(57 104)(58 105)(117 146)(118 147)(119 148)(120 149)(121 150)(122 151)(123 152)(124 153)(125 154)(126 155)(127 156)(128 157)(129 158)(130 159)(131 160)(132 161)(133 162)(134 163)(135 164)(136 165)(137 166)(138 167)(139 168)(140 169)(141 170)(142 171)(143 172)(144 173)(145 174)
(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)
(1 174 35 145)(2 173 36 144)(3 172 37 143)(4 171 38 142)(5 170 39 141)(6 169 40 140)(7 168 41 139)(8 167 42 138)(9 166 43 137)(10 165 44 136)(11 164 45 135)(12 163 46 134)(13 162 47 133)(14 161 48 132)(15 160 49 131)(16 159 50 130)(17 158 51 129)(18 157 52 128)(19 156 53 127)(20 155 54 126)(21 154 55 125)(22 153 56 124)(23 152 57 123)(24 151 58 122)(25 150 30 121)(26 149 31 120)(27 148 32 119)(28 147 33 118)(29 146 34 117)(59 219 95 190)(60 218 96 189)(61 217 97 188)(62 216 98 187)(63 215 99 186)(64 214 100 185)(65 213 101 184)(66 212 102 183)(67 211 103 182)(68 210 104 181)(69 209 105 180)(70 208 106 179)(71 207 107 178)(72 206 108 177)(73 205 109 176)(74 204 110 175)(75 232 111 203)(76 231 112 202)(77 230 113 201)(78 229 114 200)(79 228 115 199)(80 227 116 198)(81 226 88 197)(82 225 89 196)(83 224 90 195)(84 223 91 194)(85 222 92 193)(86 221 93 192)(87 220 94 191)
G:=sub<Sym(232)| (1,111,35,75)(2,112,36,76)(3,113,37,77)(4,114,38,78)(5,115,39,79)(6,116,40,80)(7,88,41,81)(8,89,42,82)(9,90,43,83)(10,91,44,84)(11,92,45,85)(12,93,46,86)(13,94,47,87)(14,95,48,59)(15,96,49,60)(16,97,50,61)(17,98,51,62)(18,99,52,63)(19,100,53,64)(20,101,54,65)(21,102,55,66)(22,103,56,67)(23,104,57,68)(24,105,58,69)(25,106,30,70)(26,107,31,71)(27,108,32,72)(28,109,33,73)(29,110,34,74)(117,175,146,204)(118,176,147,205)(119,177,148,206)(120,178,149,207)(121,179,150,208)(122,180,151,209)(123,181,152,210)(124,182,153,211)(125,183,154,212)(126,184,155,213)(127,185,156,214)(128,186,157,215)(129,187,158,216)(130,188,159,217)(131,189,160,218)(132,190,161,219)(133,191,162,220)(134,192,163,221)(135,193,164,222)(136,194,165,223)(137,195,166,224)(138,196,167,225)(139,197,168,226)(140,198,169,227)(141,199,170,228)(142,200,171,229)(143,201,172,230)(144,202,173,231)(145,203,174,232), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,59)(15,60)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,106)(31,107)(32,108)(33,109)(34,110)(35,111)(36,112)(37,113)(38,114)(39,115)(40,116)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,100)(54,101)(55,102)(56,103)(57,104)(58,105)(117,146)(118,147)(119,148)(120,149)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,157)(129,158)(130,159)(131,160)(132,161)(133,162)(134,163)(135,164)(136,165)(137,166)(138,167)(139,168)(140,169)(141,170)(142,171)(143,172)(144,173)(145,174), (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), (1,174,35,145)(2,173,36,144)(3,172,37,143)(4,171,38,142)(5,170,39,141)(6,169,40,140)(7,168,41,139)(8,167,42,138)(9,166,43,137)(10,165,44,136)(11,164,45,135)(12,163,46,134)(13,162,47,133)(14,161,48,132)(15,160,49,131)(16,159,50,130)(17,158,51,129)(18,157,52,128)(19,156,53,127)(20,155,54,126)(21,154,55,125)(22,153,56,124)(23,152,57,123)(24,151,58,122)(25,150,30,121)(26,149,31,120)(27,148,32,119)(28,147,33,118)(29,146,34,117)(59,219,95,190)(60,218,96,189)(61,217,97,188)(62,216,98,187)(63,215,99,186)(64,214,100,185)(65,213,101,184)(66,212,102,183)(67,211,103,182)(68,210,104,181)(69,209,105,180)(70,208,106,179)(71,207,107,178)(72,206,108,177)(73,205,109,176)(74,204,110,175)(75,232,111,203)(76,231,112,202)(77,230,113,201)(78,229,114,200)(79,228,115,199)(80,227,116,198)(81,226,88,197)(82,225,89,196)(83,224,90,195)(84,223,91,194)(85,222,92,193)(86,221,93,192)(87,220,94,191)>;
G:=Group( (1,111,35,75)(2,112,36,76)(3,113,37,77)(4,114,38,78)(5,115,39,79)(6,116,40,80)(7,88,41,81)(8,89,42,82)(9,90,43,83)(10,91,44,84)(11,92,45,85)(12,93,46,86)(13,94,47,87)(14,95,48,59)(15,96,49,60)(16,97,50,61)(17,98,51,62)(18,99,52,63)(19,100,53,64)(20,101,54,65)(21,102,55,66)(22,103,56,67)(23,104,57,68)(24,105,58,69)(25,106,30,70)(26,107,31,71)(27,108,32,72)(28,109,33,73)(29,110,34,74)(117,175,146,204)(118,176,147,205)(119,177,148,206)(120,178,149,207)(121,179,150,208)(122,180,151,209)(123,181,152,210)(124,182,153,211)(125,183,154,212)(126,184,155,213)(127,185,156,214)(128,186,157,215)(129,187,158,216)(130,188,159,217)(131,189,160,218)(132,190,161,219)(133,191,162,220)(134,192,163,221)(135,193,164,222)(136,194,165,223)(137,195,166,224)(138,196,167,225)(139,197,168,226)(140,198,169,227)(141,199,170,228)(142,200,171,229)(143,201,172,230)(144,202,173,231)(145,203,174,232), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,59)(15,60)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,106)(31,107)(32,108)(33,109)(34,110)(35,111)(36,112)(37,113)(38,114)(39,115)(40,116)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,100)(54,101)(55,102)(56,103)(57,104)(58,105)(117,146)(118,147)(119,148)(120,149)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,157)(129,158)(130,159)(131,160)(132,161)(133,162)(134,163)(135,164)(136,165)(137,166)(138,167)(139,168)(140,169)(141,170)(142,171)(143,172)(144,173)(145,174), (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), (1,174,35,145)(2,173,36,144)(3,172,37,143)(4,171,38,142)(5,170,39,141)(6,169,40,140)(7,168,41,139)(8,167,42,138)(9,166,43,137)(10,165,44,136)(11,164,45,135)(12,163,46,134)(13,162,47,133)(14,161,48,132)(15,160,49,131)(16,159,50,130)(17,158,51,129)(18,157,52,128)(19,156,53,127)(20,155,54,126)(21,154,55,125)(22,153,56,124)(23,152,57,123)(24,151,58,122)(25,150,30,121)(26,149,31,120)(27,148,32,119)(28,147,33,118)(29,146,34,117)(59,219,95,190)(60,218,96,189)(61,217,97,188)(62,216,98,187)(63,215,99,186)(64,214,100,185)(65,213,101,184)(66,212,102,183)(67,211,103,182)(68,210,104,181)(69,209,105,180)(70,208,106,179)(71,207,107,178)(72,206,108,177)(73,205,109,176)(74,204,110,175)(75,232,111,203)(76,231,112,202)(77,230,113,201)(78,229,114,200)(79,228,115,199)(80,227,116,198)(81,226,88,197)(82,225,89,196)(83,224,90,195)(84,223,91,194)(85,222,92,193)(86,221,93,192)(87,220,94,191) );
G=PermutationGroup([[(1,111,35,75),(2,112,36,76),(3,113,37,77),(4,114,38,78),(5,115,39,79),(6,116,40,80),(7,88,41,81),(8,89,42,82),(9,90,43,83),(10,91,44,84),(11,92,45,85),(12,93,46,86),(13,94,47,87),(14,95,48,59),(15,96,49,60),(16,97,50,61),(17,98,51,62),(18,99,52,63),(19,100,53,64),(20,101,54,65),(21,102,55,66),(22,103,56,67),(23,104,57,68),(24,105,58,69),(25,106,30,70),(26,107,31,71),(27,108,32,72),(28,109,33,73),(29,110,34,74),(117,175,146,204),(118,176,147,205),(119,177,148,206),(120,178,149,207),(121,179,150,208),(122,180,151,209),(123,181,152,210),(124,182,153,211),(125,183,154,212),(126,184,155,213),(127,185,156,214),(128,186,157,215),(129,187,158,216),(130,188,159,217),(131,189,160,218),(132,190,161,219),(133,191,162,220),(134,192,163,221),(135,193,164,222),(136,194,165,223),(137,195,166,224),(138,196,167,225),(139,197,168,226),(140,198,169,227),(141,199,170,228),(142,200,171,229),(143,201,172,230),(144,202,173,231),(145,203,174,232)], [(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,81),(8,82),(9,83),(10,84),(11,85),(12,86),(13,87),(14,59),(15,60),(16,61),(17,62),(18,63),(19,64),(20,65),(21,66),(22,67),(23,68),(24,69),(25,70),(26,71),(27,72),(28,73),(29,74),(30,106),(31,107),(32,108),(33,109),(34,110),(35,111),(36,112),(37,113),(38,114),(39,115),(40,116),(41,88),(42,89),(43,90),(44,91),(45,92),(46,93),(47,94),(48,95),(49,96),(50,97),(51,98),(52,99),(53,100),(54,101),(55,102),(56,103),(57,104),(58,105),(117,146),(118,147),(119,148),(120,149),(121,150),(122,151),(123,152),(124,153),(125,154),(126,155),(127,156),(128,157),(129,158),(130,159),(131,160),(132,161),(133,162),(134,163),(135,164),(136,165),(137,166),(138,167),(139,168),(140,169),(141,170),(142,171),(143,172),(144,173),(145,174)], [(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)], [(1,174,35,145),(2,173,36,144),(3,172,37,143),(4,171,38,142),(5,170,39,141),(6,169,40,140),(7,168,41,139),(8,167,42,138),(9,166,43,137),(10,165,44,136),(11,164,45,135),(12,163,46,134),(13,162,47,133),(14,161,48,132),(15,160,49,131),(16,159,50,130),(17,158,51,129),(18,157,52,128),(19,156,53,127),(20,155,54,126),(21,154,55,125),(22,153,56,124),(23,152,57,123),(24,151,58,122),(25,150,30,121),(26,149,31,120),(27,148,32,119),(28,147,33,118),(29,146,34,117),(59,219,95,190),(60,218,96,189),(61,217,97,188),(62,216,98,187),(63,215,99,186),(64,214,100,185),(65,213,101,184),(66,212,102,183),(67,211,103,182),(68,210,104,181),(69,209,105,180),(70,208,106,179),(71,207,107,178),(72,206,108,177),(73,205,109,176),(74,204,110,175),(75,232,111,203),(76,231,112,202),(77,230,113,201),(78,229,114,200),(79,228,115,199),(80,227,116,198),(81,226,88,197),(82,225,89,196),(83,224,90,195),(84,223,91,194),(85,222,92,193),(86,221,93,192),(87,220,94,191)]])
77 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 29A | ··· | 29N | 58A | ··· | 58N | 58O | ··· | 58AP | 116A | ··· | 116N |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 29 | ··· | 29 | 58 | ··· | 58 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 4 | 2 | 116 | 58 | 58 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | D4 | SD16 | D29 | D58 | C29⋊D4 | D4.D29 |
| kernel | D4.D29 | C29⋊2C8 | Dic58 | D4×C29 | C58 | C29 | D4 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 14 | 14 | 28 | 14 |
Matrix representation of D4.D29 ►in GL4(𝔽233) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 91 | 180 |
| 0 | 0 | 231 | 142 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 91 | 180 |
| 0 | 0 | 187 | 142 |
| 27 | 1 | 0 | 0 |
| 101 | 185 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 104 | 139 | 0 | 0 |
| 63 | 129 | 0 | 0 |
| 0 | 0 | 221 | 209 |
| 0 | 0 | 74 | 12 |
G:=sub<GL(4,GF(233))| [1,0,0,0,0,1,0,0,0,0,91,231,0,0,180,142],[1,0,0,0,0,1,0,0,0,0,91,187,0,0,180,142],[27,101,0,0,1,185,0,0,0,0,1,0,0,0,0,1],[104,63,0,0,139,129,0,0,0,0,221,74,0,0,209,12] >;
D4.D29 in GAP, Magma, Sage, TeX
D_4.D_{29} % in TeX
G:=Group("D4.D29"); // GroupNames label
G:=SmallGroup(464,16);
// by ID
G=gap.SmallGroup(464,16);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-29,40,61,182,97,42,11204]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^29=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations
Export