metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊D29, C29⋊2D8, C4.1D58, C58.7D4, D116⋊2C2, C116.1C22, C29⋊2C8⋊1C2, (D4×C29)⋊1C2, C2.4(C29⋊D4), SmallGroup(464,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊D29
G = < a,b,c,d | a4=b2=c29=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=c-1 >
Smallest permutation representation of D4⋊D29
►On 232 points
Generators in S232
(1 109 51 76)(2 110 52 77)(3 111 53 78)(4 112 54 79)(5 113 55 80)(6 114 56 81)(7 115 57 82)(8 116 58 83)(9 88 30 84)(10 89 31 85)(11 90 32 86)(12 91 33 87)(13 92 34 59)(14 93 35 60)(15 94 36 61)(16 95 37 62)(17 96 38 63)(18 97 39 64)(19 98 40 65)(20 99 41 66)(21 100 42 67)(22 101 43 68)(23 102 44 69)(24 103 45 70)(25 104 46 71)(26 105 47 72)(27 106 48 73)(28 107 49 74)(29 108 50 75)(117 177 154 228)(118 178 155 229)(119 179 156 230)(120 180 157 231)(121 181 158 232)(122 182 159 204)(123 183 160 205)(124 184 161 206)(125 185 162 207)(126 186 163 208)(127 187 164 209)(128 188 165 210)(129 189 166 211)(130 190 167 212)(131 191 168 213)(132 192 169 214)(133 193 170 215)(134 194 171 216)(135 195 172 217)(136 196 173 218)(137 197 174 219)(138 198 146 220)(139 199 147 221)(140 200 148 222)(141 201 149 223)(142 202 150 224)(143 203 151 225)(144 175 152 226)(145 176 153 227)
(1 220)(2 221)(3 222)(4 223)(5 224)(6 225)(7 226)(8 227)(9 228)(10 229)(11 230)(12 231)(13 232)(14 204)(15 205)(16 206)(17 207)(18 208)(19 209)(20 210)(21 211)(22 212)(23 213)(24 214)(25 215)(26 216)(27 217)(28 218)(29 219)(30 177)(31 178)(32 179)(33 180)(34 181)(35 182)(36 183)(37 184)(38 185)(39 186)(40 187)(41 188)(42 189)(43 190)(44 191)(45 192)(46 193)(47 194)(48 195)(49 196)(50 197)(51 198)(52 199)(53 200)(54 201)(55 202)(56 203)(57 175)(58 176)(59 121)(60 122)(61 123)(62 124)(63 125)(64 126)(65 127)(66 128)(67 129)(68 130)(69 131)(70 132)(71 133)(72 134)(73 135)(74 136)(75 137)(76 138)(77 139)(78 140)(79 141)(80 142)(81 143)(82 144)(83 145)(84 117)(85 118)(86 119)(87 120)(88 154)(89 155)(90 156)(91 157)(92 158)(93 159)(94 160)(95 161)(96 162)(97 163)(98 164)(99 165)(100 166)(101 167)(102 168)(103 169)(104 170)(105 171)(106 172)(107 173)(108 174)(109 146)(110 147)(111 148)(112 149)(113 150)(114 151)(115 152)(116 153)
(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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(59 96)(60 95)(61 94)(62 93)(63 92)(64 91)(65 90)(66 89)(67 88)(68 116)(69 115)(70 114)(71 113)(72 112)(73 111)(74 110)(75 109)(76 108)(77 107)(78 106)(79 105)(80 104)(81 103)(82 102)(83 101)(84 100)(85 99)(86 98)(87 97)(117 189)(118 188)(119 187)(120 186)(121 185)(122 184)(123 183)(124 182)(125 181)(126 180)(127 179)(128 178)(129 177)(130 176)(131 175)(132 203)(133 202)(134 201)(135 200)(136 199)(137 198)(138 197)(139 196)(140 195)(141 194)(142 193)(143 192)(144 191)(145 190)(146 219)(147 218)(148 217)(149 216)(150 215)(151 214)(152 213)(153 212)(154 211)(155 210)(156 209)(157 208)(158 207)(159 206)(160 205)(161 204)(162 232)(163 231)(164 230)(165 229)(166 228)(167 227)(168 226)(169 225)(170 224)(171 223)(172 222)(173 221)(174 220)
G:=sub<Sym(232)| (1,109,51,76)(2,110,52,77)(3,111,53,78)(4,112,54,79)(5,113,55,80)(6,114,56,81)(7,115,57,82)(8,116,58,83)(9,88,30,84)(10,89,31,85)(11,90,32,86)(12,91,33,87)(13,92,34,59)(14,93,35,60)(15,94,36,61)(16,95,37,62)(17,96,38,63)(18,97,39,64)(19,98,40,65)(20,99,41,66)(21,100,42,67)(22,101,43,68)(23,102,44,69)(24,103,45,70)(25,104,46,71)(26,105,47,72)(27,106,48,73)(28,107,49,74)(29,108,50,75)(117,177,154,228)(118,178,155,229)(119,179,156,230)(120,180,157,231)(121,181,158,232)(122,182,159,204)(123,183,160,205)(124,184,161,206)(125,185,162,207)(126,186,163,208)(127,187,164,209)(128,188,165,210)(129,189,166,211)(130,190,167,212)(131,191,168,213)(132,192,169,214)(133,193,170,215)(134,194,171,216)(135,195,172,217)(136,196,173,218)(137,197,174,219)(138,198,146,220)(139,199,147,221)(140,200,148,222)(141,201,149,223)(142,202,150,224)(143,203,151,225)(144,175,152,226)(145,176,153,227), (1,220)(2,221)(3,222)(4,223)(5,224)(6,225)(7,226)(8,227)(9,228)(10,229)(11,230)(12,231)(13,232)(14,204)(15,205)(16,206)(17,207)(18,208)(19,209)(20,210)(21,211)(22,212)(23,213)(24,214)(25,215)(26,216)(27,217)(28,218)(29,219)(30,177)(31,178)(32,179)(33,180)(34,181)(35,182)(36,183)(37,184)(38,185)(39,186)(40,187)(41,188)(42,189)(43,190)(44,191)(45,192)(46,193)(47,194)(48,195)(49,196)(50,197)(51,198)(52,199)(53,200)(54,201)(55,202)(56,203)(57,175)(58,176)(59,121)(60,122)(61,123)(62,124)(63,125)(64,126)(65,127)(66,128)(67,129)(68,130)(69,131)(70,132)(71,133)(72,134)(73,135)(74,136)(75,137)(76,138)(77,139)(78,140)(79,141)(80,142)(81,143)(82,144)(83,145)(84,117)(85,118)(86,119)(87,120)(88,154)(89,155)(90,156)(91,157)(92,158)(93,159)(94,160)(95,161)(96,162)(97,163)(98,164)(99,165)(100,166)(101,167)(102,168)(103,169)(104,170)(105,171)(106,172)(107,173)(108,174)(109,146)(110,147)(111,148)(112,149)(113,150)(114,151)(115,152)(116,153), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,116)(69,115)(70,114)(71,113)(72,112)(73,111)(74,110)(75,109)(76,108)(77,107)(78,106)(79,105)(80,104)(81,103)(82,102)(83,101)(84,100)(85,99)(86,98)(87,97)(117,189)(118,188)(119,187)(120,186)(121,185)(122,184)(123,183)(124,182)(125,181)(126,180)(127,179)(128,178)(129,177)(130,176)(131,175)(132,203)(133,202)(134,201)(135,200)(136,199)(137,198)(138,197)(139,196)(140,195)(141,194)(142,193)(143,192)(144,191)(145,190)(146,219)(147,218)(148,217)(149,216)(150,215)(151,214)(152,213)(153,212)(154,211)(155,210)(156,209)(157,208)(158,207)(159,206)(160,205)(161,204)(162,232)(163,231)(164,230)(165,229)(166,228)(167,227)(168,226)(169,225)(170,224)(171,223)(172,222)(173,221)(174,220)>;
G:=Group( (1,109,51,76)(2,110,52,77)(3,111,53,78)(4,112,54,79)(5,113,55,80)(6,114,56,81)(7,115,57,82)(8,116,58,83)(9,88,30,84)(10,89,31,85)(11,90,32,86)(12,91,33,87)(13,92,34,59)(14,93,35,60)(15,94,36,61)(16,95,37,62)(17,96,38,63)(18,97,39,64)(19,98,40,65)(20,99,41,66)(21,100,42,67)(22,101,43,68)(23,102,44,69)(24,103,45,70)(25,104,46,71)(26,105,47,72)(27,106,48,73)(28,107,49,74)(29,108,50,75)(117,177,154,228)(118,178,155,229)(119,179,156,230)(120,180,157,231)(121,181,158,232)(122,182,159,204)(123,183,160,205)(124,184,161,206)(125,185,162,207)(126,186,163,208)(127,187,164,209)(128,188,165,210)(129,189,166,211)(130,190,167,212)(131,191,168,213)(132,192,169,214)(133,193,170,215)(134,194,171,216)(135,195,172,217)(136,196,173,218)(137,197,174,219)(138,198,146,220)(139,199,147,221)(140,200,148,222)(141,201,149,223)(142,202,150,224)(143,203,151,225)(144,175,152,226)(145,176,153,227), (1,220)(2,221)(3,222)(4,223)(5,224)(6,225)(7,226)(8,227)(9,228)(10,229)(11,230)(12,231)(13,232)(14,204)(15,205)(16,206)(17,207)(18,208)(19,209)(20,210)(21,211)(22,212)(23,213)(24,214)(25,215)(26,216)(27,217)(28,218)(29,219)(30,177)(31,178)(32,179)(33,180)(34,181)(35,182)(36,183)(37,184)(38,185)(39,186)(40,187)(41,188)(42,189)(43,190)(44,191)(45,192)(46,193)(47,194)(48,195)(49,196)(50,197)(51,198)(52,199)(53,200)(54,201)(55,202)(56,203)(57,175)(58,176)(59,121)(60,122)(61,123)(62,124)(63,125)(64,126)(65,127)(66,128)(67,129)(68,130)(69,131)(70,132)(71,133)(72,134)(73,135)(74,136)(75,137)(76,138)(77,139)(78,140)(79,141)(80,142)(81,143)(82,144)(83,145)(84,117)(85,118)(86,119)(87,120)(88,154)(89,155)(90,156)(91,157)(92,158)(93,159)(94,160)(95,161)(96,162)(97,163)(98,164)(99,165)(100,166)(101,167)(102,168)(103,169)(104,170)(105,171)(106,172)(107,173)(108,174)(109,146)(110,147)(111,148)(112,149)(113,150)(114,151)(115,152)(116,153), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,116)(69,115)(70,114)(71,113)(72,112)(73,111)(74,110)(75,109)(76,108)(77,107)(78,106)(79,105)(80,104)(81,103)(82,102)(83,101)(84,100)(85,99)(86,98)(87,97)(117,189)(118,188)(119,187)(120,186)(121,185)(122,184)(123,183)(124,182)(125,181)(126,180)(127,179)(128,178)(129,177)(130,176)(131,175)(132,203)(133,202)(134,201)(135,200)(136,199)(137,198)(138,197)(139,196)(140,195)(141,194)(142,193)(143,192)(144,191)(145,190)(146,219)(147,218)(148,217)(149,216)(150,215)(151,214)(152,213)(153,212)(154,211)(155,210)(156,209)(157,208)(158,207)(159,206)(160,205)(161,204)(162,232)(163,231)(164,230)(165,229)(166,228)(167,227)(168,226)(169,225)(170,224)(171,223)(172,222)(173,221)(174,220) );
G=PermutationGroup([[(1,109,51,76),(2,110,52,77),(3,111,53,78),(4,112,54,79),(5,113,55,80),(6,114,56,81),(7,115,57,82),(8,116,58,83),(9,88,30,84),(10,89,31,85),(11,90,32,86),(12,91,33,87),(13,92,34,59),(14,93,35,60),(15,94,36,61),(16,95,37,62),(17,96,38,63),(18,97,39,64),(19,98,40,65),(20,99,41,66),(21,100,42,67),(22,101,43,68),(23,102,44,69),(24,103,45,70),(25,104,46,71),(26,105,47,72),(27,106,48,73),(28,107,49,74),(29,108,50,75),(117,177,154,228),(118,178,155,229),(119,179,156,230),(120,180,157,231),(121,181,158,232),(122,182,159,204),(123,183,160,205),(124,184,161,206),(125,185,162,207),(126,186,163,208),(127,187,164,209),(128,188,165,210),(129,189,166,211),(130,190,167,212),(131,191,168,213),(132,192,169,214),(133,193,170,215),(134,194,171,216),(135,195,172,217),(136,196,173,218),(137,197,174,219),(138,198,146,220),(139,199,147,221),(140,200,148,222),(141,201,149,223),(142,202,150,224),(143,203,151,225),(144,175,152,226),(145,176,153,227)], [(1,220),(2,221),(3,222),(4,223),(5,224),(6,225),(7,226),(8,227),(9,228),(10,229),(11,230),(12,231),(13,232),(14,204),(15,205),(16,206),(17,207),(18,208),(19,209),(20,210),(21,211),(22,212),(23,213),(24,214),(25,215),(26,216),(27,217),(28,218),(29,219),(30,177),(31,178),(32,179),(33,180),(34,181),(35,182),(36,183),(37,184),(38,185),(39,186),(40,187),(41,188),(42,189),(43,190),(44,191),(45,192),(46,193),(47,194),(48,195),(49,196),(50,197),(51,198),(52,199),(53,200),(54,201),(55,202),(56,203),(57,175),(58,176),(59,121),(60,122),(61,123),(62,124),(63,125),(64,126),(65,127),(66,128),(67,129),(68,130),(69,131),(70,132),(71,133),(72,134),(73,135),(74,136),(75,137),(76,138),(77,139),(78,140),(79,141),(80,142),(81,143),(82,144),(83,145),(84,117),(85,118),(86,119),(87,120),(88,154),(89,155),(90,156),(91,157),(92,158),(93,159),(94,160),(95,161),(96,162),(97,163),(98,164),(99,165),(100,166),(101,167),(102,168),(103,169),(104,170),(105,171),(106,172),(107,173),(108,174),(109,146),(110,147),(111,148),(112,149),(113,150),(114,151),(115,152),(116,153)], [(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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,20),(11,19),(12,18),(13,17),(14,16),(30,42),(31,41),(32,40),(33,39),(34,38),(35,37),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(59,96),(60,95),(61,94),(62,93),(63,92),(64,91),(65,90),(66,89),(67,88),(68,116),(69,115),(70,114),(71,113),(72,112),(73,111),(74,110),(75,109),(76,108),(77,107),(78,106),(79,105),(80,104),(81,103),(82,102),(83,101),(84,100),(85,99),(86,98),(87,97),(117,189),(118,188),(119,187),(120,186),(121,185),(122,184),(123,183),(124,182),(125,181),(126,180),(127,179),(128,178),(129,177),(130,176),(131,175),(132,203),(133,202),(134,201),(135,200),(136,199),(137,198),(138,197),(139,196),(140,195),(141,194),(142,193),(143,192),(144,191),(145,190),(146,219),(147,218),(148,217),(149,216),(150,215),(151,214),(152,213),(153,212),(154,211),(155,210),(156,209),(157,208),(158,207),(159,206),(160,205),(161,204),(162,232),(163,231),(164,230),(165,229),(166,228),(167,227),(168,226),(169,225),(170,224),(171,223),(172,222),(173,221),(174,220)]])
77 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 8A | 8B | 29A | ··· | 29N | 58A | ··· | 58N | 58O | ··· | 58AP | 116A | ··· | 116N |
| order | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 29 | ··· | 29 | 58 | ··· | 58 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 4 | 116 | 2 | 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 | D8 | D29 | D58 | C29⋊D4 | D4⋊D29 |
| kernel | D4⋊D29 | C29⋊2C8 | D116 | 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 | 232 | 14 |
| 0 | 0 | 133 | 1 |
| 232 | 0 | 0 | 0 |
| 0 | 232 | 0 | 0 |
| 0 | 0 | 148 | 129 |
| 0 | 0 | 177 | 85 |
| 159 | 1 | 0 | 0 |
| 213 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 93 | 195 | 0 | 0 |
| 154 | 140 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 100 | 232 |
G:=sub<GL(4,GF(233))| [1,0,0,0,0,1,0,0,0,0,232,133,0,0,14,1],[232,0,0,0,0,232,0,0,0,0,148,177,0,0,129,85],[159,213,0,0,1,16,0,0,0,0,1,0,0,0,0,1],[93,154,0,0,195,140,0,0,0,0,1,100,0,0,0,232] >;
D4⋊D29 in GAP, Magma, Sage, TeX
D_4\rtimes D_{29} % in TeX
G:=Group("D4:D29"); // GroupNames label
G:=SmallGroup(464,15);
// by ID
G=gap.SmallGroup(464,15);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-29,61,182,97,42,11204]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^29=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations
Export