metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊D29, C58.9D4, C4.3D58, C29⋊3SD16, D116.2C2, C116.3C22, C29⋊2C8⋊3C2, (Q8×C29)⋊1C2, C2.6(C29⋊D4), SmallGroup(464,17)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊D29
G = < a,b,c,d | a4=c29=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >
Smallest permutation representation of Q8⋊D29
►On 232 points
Generators in S232
(1 102 31 81)(2 103 32 82)(3 104 33 83)(4 105 34 84)(5 106 35 85)(6 107 36 86)(7 108 37 87)(8 109 38 59)(9 110 39 60)(10 111 40 61)(11 112 41 62)(12 113 42 63)(13 114 43 64)(14 115 44 65)(15 116 45 66)(16 88 46 67)(17 89 47 68)(18 90 48 69)(19 91 49 70)(20 92 50 71)(21 93 51 72)(22 94 52 73)(23 95 53 74)(24 96 54 75)(25 97 55 76)(26 98 56 77)(27 99 57 78)(28 100 58 79)(29 101 30 80)(117 180 168 209)(118 181 169 210)(119 182 170 211)(120 183 171 212)(121 184 172 213)(122 185 173 214)(123 186 174 215)(124 187 146 216)(125 188 147 217)(126 189 148 218)(127 190 149 219)(128 191 150 220)(129 192 151 221)(130 193 152 222)(131 194 153 223)(132 195 154 224)(133 196 155 225)(134 197 156 226)(135 198 157 227)(136 199 158 228)(137 200 159 229)(138 201 160 230)(139 202 161 231)(140 203 162 232)(141 175 163 204)(142 176 164 205)(143 177 165 206)(144 178 166 207)(145 179 167 208)
(1 151 31 129)(2 152 32 130)(3 153 33 131)(4 154 34 132)(5 155 35 133)(6 156 36 134)(7 157 37 135)(8 158 38 136)(9 159 39 137)(10 160 40 138)(11 161 41 139)(12 162 42 140)(13 163 43 141)(14 164 44 142)(15 165 45 143)(16 166 46 144)(17 167 47 145)(18 168 48 117)(19 169 49 118)(20 170 50 119)(21 171 51 120)(22 172 52 121)(23 173 53 122)(24 174 54 123)(25 146 55 124)(26 147 56 125)(27 148 57 126)(28 149 58 127)(29 150 30 128)(59 228 109 199)(60 229 110 200)(61 230 111 201)(62 231 112 202)(63 232 113 203)(64 204 114 175)(65 205 115 176)(66 206 116 177)(67 207 88 178)(68 208 89 179)(69 209 90 180)(70 210 91 181)(71 211 92 182)(72 212 93 183)(73 213 94 184)(74 214 95 185)(75 215 96 186)(76 216 97 187)(77 217 98 188)(78 218 99 189)(79 219 100 190)(80 220 101 191)(81 221 102 192)(82 222 103 193)(83 223 104 194)(84 224 105 195)(85 225 106 196)(86 226 107 197)(87 227 108 198)
(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 31)(32 58)(33 57)(34 56)(35 55)(36 54)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(59 94)(60 93)(61 92)(62 91)(63 90)(64 89)(65 88)(66 116)(67 115)(68 114)(69 113)(70 112)(71 111)(72 110)(73 109)(74 108)(75 107)(76 106)(77 105)(78 104)(79 103)(80 102)(81 101)(82 100)(83 99)(84 98)(85 97)(86 96)(87 95)(117 203)(118 202)(119 201)(120 200)(121 199)(122 198)(123 197)(124 196)(125 195)(126 194)(127 193)(128 192)(129 191)(130 190)(131 189)(132 188)(133 187)(134 186)(135 185)(136 184)(137 183)(138 182)(139 181)(140 180)(141 179)(142 178)(143 177)(144 176)(145 175)(146 225)(147 224)(148 223)(149 222)(150 221)(151 220)(152 219)(153 218)(154 217)(155 216)(156 215)(157 214)(158 213)(159 212)(160 211)(161 210)(162 209)(163 208)(164 207)(165 206)(166 205)(167 204)(168 232)(169 231)(170 230)(171 229)(172 228)(173 227)(174 226)
G:=sub<Sym(232)| (1,102,31,81)(2,103,32,82)(3,104,33,83)(4,105,34,84)(5,106,35,85)(6,107,36,86)(7,108,37,87)(8,109,38,59)(9,110,39,60)(10,111,40,61)(11,112,41,62)(12,113,42,63)(13,114,43,64)(14,115,44,65)(15,116,45,66)(16,88,46,67)(17,89,47,68)(18,90,48,69)(19,91,49,70)(20,92,50,71)(21,93,51,72)(22,94,52,73)(23,95,53,74)(24,96,54,75)(25,97,55,76)(26,98,56,77)(27,99,57,78)(28,100,58,79)(29,101,30,80)(117,180,168,209)(118,181,169,210)(119,182,170,211)(120,183,171,212)(121,184,172,213)(122,185,173,214)(123,186,174,215)(124,187,146,216)(125,188,147,217)(126,189,148,218)(127,190,149,219)(128,191,150,220)(129,192,151,221)(130,193,152,222)(131,194,153,223)(132,195,154,224)(133,196,155,225)(134,197,156,226)(135,198,157,227)(136,199,158,228)(137,200,159,229)(138,201,160,230)(139,202,161,231)(140,203,162,232)(141,175,163,204)(142,176,164,205)(143,177,165,206)(144,178,166,207)(145,179,167,208), (1,151,31,129)(2,152,32,130)(3,153,33,131)(4,154,34,132)(5,155,35,133)(6,156,36,134)(7,157,37,135)(8,158,38,136)(9,159,39,137)(10,160,40,138)(11,161,41,139)(12,162,42,140)(13,163,43,141)(14,164,44,142)(15,165,45,143)(16,166,46,144)(17,167,47,145)(18,168,48,117)(19,169,49,118)(20,170,50,119)(21,171,51,120)(22,172,52,121)(23,173,53,122)(24,174,54,123)(25,146,55,124)(26,147,56,125)(27,148,57,126)(28,149,58,127)(29,150,30,128)(59,228,109,199)(60,229,110,200)(61,230,111,201)(62,231,112,202)(63,232,113,203)(64,204,114,175)(65,205,115,176)(66,206,116,177)(67,207,88,178)(68,208,89,179)(69,209,90,180)(70,210,91,181)(71,211,92,182)(72,212,93,183)(73,213,94,184)(74,214,95,185)(75,215,96,186)(76,216,97,187)(77,217,98,188)(78,218,99,189)(79,219,100,190)(80,220,101,191)(81,221,102,192)(82,222,103,193)(83,223,104,194)(84,224,105,195)(85,225,106,196)(86,226,107,197)(87,227,108,198), (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)(30,31)(32,58)(33,57)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,116)(67,115)(68,114)(69,113)(70,112)(71,111)(72,110)(73,109)(74,108)(75,107)(76,106)(77,105)(78,104)(79,103)(80,102)(81,101)(82,100)(83,99)(84,98)(85,97)(86,96)(87,95)(117,203)(118,202)(119,201)(120,200)(121,199)(122,198)(123,197)(124,196)(125,195)(126,194)(127,193)(128,192)(129,191)(130,190)(131,189)(132,188)(133,187)(134,186)(135,185)(136,184)(137,183)(138,182)(139,181)(140,180)(141,179)(142,178)(143,177)(144,176)(145,175)(146,225)(147,224)(148,223)(149,222)(150,221)(151,220)(152,219)(153,218)(154,217)(155,216)(156,215)(157,214)(158,213)(159,212)(160,211)(161,210)(162,209)(163,208)(164,207)(165,206)(166,205)(167,204)(168,232)(169,231)(170,230)(171,229)(172,228)(173,227)(174,226)>;
G:=Group( (1,102,31,81)(2,103,32,82)(3,104,33,83)(4,105,34,84)(5,106,35,85)(6,107,36,86)(7,108,37,87)(8,109,38,59)(9,110,39,60)(10,111,40,61)(11,112,41,62)(12,113,42,63)(13,114,43,64)(14,115,44,65)(15,116,45,66)(16,88,46,67)(17,89,47,68)(18,90,48,69)(19,91,49,70)(20,92,50,71)(21,93,51,72)(22,94,52,73)(23,95,53,74)(24,96,54,75)(25,97,55,76)(26,98,56,77)(27,99,57,78)(28,100,58,79)(29,101,30,80)(117,180,168,209)(118,181,169,210)(119,182,170,211)(120,183,171,212)(121,184,172,213)(122,185,173,214)(123,186,174,215)(124,187,146,216)(125,188,147,217)(126,189,148,218)(127,190,149,219)(128,191,150,220)(129,192,151,221)(130,193,152,222)(131,194,153,223)(132,195,154,224)(133,196,155,225)(134,197,156,226)(135,198,157,227)(136,199,158,228)(137,200,159,229)(138,201,160,230)(139,202,161,231)(140,203,162,232)(141,175,163,204)(142,176,164,205)(143,177,165,206)(144,178,166,207)(145,179,167,208), (1,151,31,129)(2,152,32,130)(3,153,33,131)(4,154,34,132)(5,155,35,133)(6,156,36,134)(7,157,37,135)(8,158,38,136)(9,159,39,137)(10,160,40,138)(11,161,41,139)(12,162,42,140)(13,163,43,141)(14,164,44,142)(15,165,45,143)(16,166,46,144)(17,167,47,145)(18,168,48,117)(19,169,49,118)(20,170,50,119)(21,171,51,120)(22,172,52,121)(23,173,53,122)(24,174,54,123)(25,146,55,124)(26,147,56,125)(27,148,57,126)(28,149,58,127)(29,150,30,128)(59,228,109,199)(60,229,110,200)(61,230,111,201)(62,231,112,202)(63,232,113,203)(64,204,114,175)(65,205,115,176)(66,206,116,177)(67,207,88,178)(68,208,89,179)(69,209,90,180)(70,210,91,181)(71,211,92,182)(72,212,93,183)(73,213,94,184)(74,214,95,185)(75,215,96,186)(76,216,97,187)(77,217,98,188)(78,218,99,189)(79,219,100,190)(80,220,101,191)(81,221,102,192)(82,222,103,193)(83,223,104,194)(84,224,105,195)(85,225,106,196)(86,226,107,197)(87,227,108,198), (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)(30,31)(32,58)(33,57)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,116)(67,115)(68,114)(69,113)(70,112)(71,111)(72,110)(73,109)(74,108)(75,107)(76,106)(77,105)(78,104)(79,103)(80,102)(81,101)(82,100)(83,99)(84,98)(85,97)(86,96)(87,95)(117,203)(118,202)(119,201)(120,200)(121,199)(122,198)(123,197)(124,196)(125,195)(126,194)(127,193)(128,192)(129,191)(130,190)(131,189)(132,188)(133,187)(134,186)(135,185)(136,184)(137,183)(138,182)(139,181)(140,180)(141,179)(142,178)(143,177)(144,176)(145,175)(146,225)(147,224)(148,223)(149,222)(150,221)(151,220)(152,219)(153,218)(154,217)(155,216)(156,215)(157,214)(158,213)(159,212)(160,211)(161,210)(162,209)(163,208)(164,207)(165,206)(166,205)(167,204)(168,232)(169,231)(170,230)(171,229)(172,228)(173,227)(174,226) );
G=PermutationGroup([[(1,102,31,81),(2,103,32,82),(3,104,33,83),(4,105,34,84),(5,106,35,85),(6,107,36,86),(7,108,37,87),(8,109,38,59),(9,110,39,60),(10,111,40,61),(11,112,41,62),(12,113,42,63),(13,114,43,64),(14,115,44,65),(15,116,45,66),(16,88,46,67),(17,89,47,68),(18,90,48,69),(19,91,49,70),(20,92,50,71),(21,93,51,72),(22,94,52,73),(23,95,53,74),(24,96,54,75),(25,97,55,76),(26,98,56,77),(27,99,57,78),(28,100,58,79),(29,101,30,80),(117,180,168,209),(118,181,169,210),(119,182,170,211),(120,183,171,212),(121,184,172,213),(122,185,173,214),(123,186,174,215),(124,187,146,216),(125,188,147,217),(126,189,148,218),(127,190,149,219),(128,191,150,220),(129,192,151,221),(130,193,152,222),(131,194,153,223),(132,195,154,224),(133,196,155,225),(134,197,156,226),(135,198,157,227),(136,199,158,228),(137,200,159,229),(138,201,160,230),(139,202,161,231),(140,203,162,232),(141,175,163,204),(142,176,164,205),(143,177,165,206),(144,178,166,207),(145,179,167,208)], [(1,151,31,129),(2,152,32,130),(3,153,33,131),(4,154,34,132),(5,155,35,133),(6,156,36,134),(7,157,37,135),(8,158,38,136),(9,159,39,137),(10,160,40,138),(11,161,41,139),(12,162,42,140),(13,163,43,141),(14,164,44,142),(15,165,45,143),(16,166,46,144),(17,167,47,145),(18,168,48,117),(19,169,49,118),(20,170,50,119),(21,171,51,120),(22,172,52,121),(23,173,53,122),(24,174,54,123),(25,146,55,124),(26,147,56,125),(27,148,57,126),(28,149,58,127),(29,150,30,128),(59,228,109,199),(60,229,110,200),(61,230,111,201),(62,231,112,202),(63,232,113,203),(64,204,114,175),(65,205,115,176),(66,206,116,177),(67,207,88,178),(68,208,89,179),(69,209,90,180),(70,210,91,181),(71,211,92,182),(72,212,93,183),(73,213,94,184),(74,214,95,185),(75,215,96,186),(76,216,97,187),(77,217,98,188),(78,218,99,189),(79,219,100,190),(80,220,101,191),(81,221,102,192),(82,222,103,193),(83,223,104,194),(84,224,105,195),(85,225,106,196),(86,226,107,197),(87,227,108,198)], [(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),(30,31),(32,58),(33,57),(34,56),(35,55),(36,54),(37,53),(38,52),(39,51),(40,50),(41,49),(42,48),(43,47),(44,46),(59,94),(60,93),(61,92),(62,91),(63,90),(64,89),(65,88),(66,116),(67,115),(68,114),(69,113),(70,112),(71,111),(72,110),(73,109),(74,108),(75,107),(76,106),(77,105),(78,104),(79,103),(80,102),(81,101),(82,100),(83,99),(84,98),(85,97),(86,96),(87,95),(117,203),(118,202),(119,201),(120,200),(121,199),(122,198),(123,197),(124,196),(125,195),(126,194),(127,193),(128,192),(129,191),(130,190),(131,189),(132,188),(133,187),(134,186),(135,185),(136,184),(137,183),(138,182),(139,181),(140,180),(141,179),(142,178),(143,177),(144,176),(145,175),(146,225),(147,224),(148,223),(149,222),(150,221),(151,220),(152,219),(153,218),(154,217),(155,216),(156,215),(157,214),(158,213),(159,212),(160,211),(161,210),(162,209),(163,208),(164,207),(165,206),(166,205),(167,204),(168,232),(169,231),(170,230),(171,229),(172,228),(173,227),(174,226)]])
77 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 29A | ··· | 29N | 58A | ··· | 58N | 116A | ··· | 116AP |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 116 | 2 | 4 | 58 | 58 | 2 | ··· | 2 | 2 | ··· | 2 | 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 | Q8⋊D29 |
| kernel | Q8⋊D29 | C29⋊2C8 | D116 | Q8×C29 | C58 | C29 | Q8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 14 | 14 | 28 | 14 |
Matrix representation of Q8⋊D29 ►in GL4(𝔽233) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 47 | 55 |
| 0 | 0 | 214 | 186 |
| 232 | 0 | 0 | 0 |
| 0 | 232 | 0 | 0 |
| 0 | 0 | 180 | 148 |
| 0 | 0 | 129 | 53 |
| 196 | 1 | 0 | 0 |
| 162 | 216 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 36 | 38 | 0 | 0 |
| 156 | 197 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 11 | 232 |
G:=sub<GL(4,GF(233))| [1,0,0,0,0,1,0,0,0,0,47,214,0,0,55,186],[232,0,0,0,0,232,0,0,0,0,180,129,0,0,148,53],[196,162,0,0,1,216,0,0,0,0,1,0,0,0,0,1],[36,156,0,0,38,197,0,0,0,0,1,11,0,0,0,232] >;
Q8⋊D29 in GAP, Magma, Sage, TeX
Q_8\rtimes D_{29} % in TeX
G:=Group("Q8:D29"); // GroupNames label
G:=SmallGroup(464,17);
// by ID
G=gap.SmallGroup(464,17);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-29,61,46,182,97,42,11204]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^29=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations
Export