direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8×D29, C4.6D58, Dic58⋊4C2, C58.7C23, C116.6C22, D58.9C22, Dic29.3C22, C29⋊2(C2×Q8), (Q8×C29)⋊2C2, (C4×D29).1C2, C2.8(C22×D29), SmallGroup(464,41)
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=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of Q8×D29
►On 232 points
Generators in S232
(1 96 51 74)(2 97 52 75)(3 98 53 76)(4 99 54 77)(5 100 55 78)(6 101 56 79)(7 102 57 80)(8 103 58 81)(9 104 30 82)(10 105 31 83)(11 106 32 84)(12 107 33 85)(13 108 34 86)(14 109 35 87)(15 110 36 59)(16 111 37 60)(17 112 38 61)(18 113 39 62)(19 114 40 63)(20 115 41 64)(21 116 42 65)(22 88 43 66)(23 89 44 67)(24 90 45 68)(25 91 46 69)(26 92 47 70)(27 93 48 71)(28 94 49 72)(29 95 50 73)(117 201 172 230)(118 202 173 231)(119 203 174 232)(120 175 146 204)(121 176 147 205)(122 177 148 206)(123 178 149 207)(124 179 150 208)(125 180 151 209)(126 181 152 210)(127 182 153 211)(128 183 154 212)(129 184 155 213)(130 185 156 214)(131 186 157 215)(132 187 158 216)(133 188 159 217)(134 189 160 218)(135 190 161 219)(136 191 162 220)(137 192 163 221)(138 193 164 222)(139 194 165 223)(140 195 166 224)(141 196 167 225)(142 197 168 226)(143 198 169 227)(144 199 170 228)(145 200 171 229)
(1 172 51 117)(2 173 52 118)(3 174 53 119)(4 146 54 120)(5 147 55 121)(6 148 56 122)(7 149 57 123)(8 150 58 124)(9 151 30 125)(10 152 31 126)(11 153 32 127)(12 154 33 128)(13 155 34 129)(14 156 35 130)(15 157 36 131)(16 158 37 132)(17 159 38 133)(18 160 39 134)(19 161 40 135)(20 162 41 136)(21 163 42 137)(22 164 43 138)(23 165 44 139)(24 166 45 140)(25 167 46 141)(26 168 47 142)(27 169 48 143)(28 170 49 144)(29 171 50 145)(59 215 110 186)(60 216 111 187)(61 217 112 188)(62 218 113 189)(63 219 114 190)(64 220 115 191)(65 221 116 192)(66 222 88 193)(67 223 89 194)(68 224 90 195)(69 225 91 196)(70 226 92 197)(71 227 93 198)(72 228 94 199)(73 229 95 200)(74 230 96 201)(75 231 97 202)(76 232 98 203)(77 204 99 175)(78 205 100 176)(79 206 101 177)(80 207 102 178)(81 208 103 179)(82 209 104 180)(83 210 105 181)(84 211 106 182)(85 212 107 183)(86 213 108 184)(87 214 109 185)
(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)(60 87)(61 86)(62 85)(63 84)(64 83)(65 82)(66 81)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(88 103)(89 102)(90 101)(91 100)(92 99)(93 98)(94 97)(95 96)(104 116)(105 115)(106 114)(107 113)(108 112)(109 111)(117 145)(118 144)(119 143)(120 142)(121 141)(122 140)(123 139)(124 138)(125 137)(126 136)(127 135)(128 134)(129 133)(130 132)(146 168)(147 167)(148 166)(149 165)(150 164)(151 163)(152 162)(153 161)(154 160)(155 159)(156 158)(169 174)(170 173)(171 172)(175 197)(176 196)(177 195)(178 194)(179 193)(180 192)(181 191)(182 190)(183 189)(184 188)(185 187)(198 203)(199 202)(200 201)(204 226)(205 225)(206 224)(207 223)(208 222)(209 221)(210 220)(211 219)(212 218)(213 217)(214 216)(227 232)(228 231)(229 230)
G:=sub<Sym(232)| (1,96,51,74)(2,97,52,75)(3,98,53,76)(4,99,54,77)(5,100,55,78)(6,101,56,79)(7,102,57,80)(8,103,58,81)(9,104,30,82)(10,105,31,83)(11,106,32,84)(12,107,33,85)(13,108,34,86)(14,109,35,87)(15,110,36,59)(16,111,37,60)(17,112,38,61)(18,113,39,62)(19,114,40,63)(20,115,41,64)(21,116,42,65)(22,88,43,66)(23,89,44,67)(24,90,45,68)(25,91,46,69)(26,92,47,70)(27,93,48,71)(28,94,49,72)(29,95,50,73)(117,201,172,230)(118,202,173,231)(119,203,174,232)(120,175,146,204)(121,176,147,205)(122,177,148,206)(123,178,149,207)(124,179,150,208)(125,180,151,209)(126,181,152,210)(127,182,153,211)(128,183,154,212)(129,184,155,213)(130,185,156,214)(131,186,157,215)(132,187,158,216)(133,188,159,217)(134,189,160,218)(135,190,161,219)(136,191,162,220)(137,192,163,221)(138,193,164,222)(139,194,165,223)(140,195,166,224)(141,196,167,225)(142,197,168,226)(143,198,169,227)(144,199,170,228)(145,200,171,229), (1,172,51,117)(2,173,52,118)(3,174,53,119)(4,146,54,120)(5,147,55,121)(6,148,56,122)(7,149,57,123)(8,150,58,124)(9,151,30,125)(10,152,31,126)(11,153,32,127)(12,154,33,128)(13,155,34,129)(14,156,35,130)(15,157,36,131)(16,158,37,132)(17,159,38,133)(18,160,39,134)(19,161,40,135)(20,162,41,136)(21,163,42,137)(22,164,43,138)(23,165,44,139)(24,166,45,140)(25,167,46,141)(26,168,47,142)(27,169,48,143)(28,170,49,144)(29,171,50,145)(59,215,110,186)(60,216,111,187)(61,217,112,188)(62,218,113,189)(63,219,114,190)(64,220,115,191)(65,221,116,192)(66,222,88,193)(67,223,89,194)(68,224,90,195)(69,225,91,196)(70,226,92,197)(71,227,93,198)(72,228,94,199)(73,229,95,200)(74,230,96,201)(75,231,97,202)(76,232,98,203)(77,204,99,175)(78,205,100,176)(79,206,101,177)(80,207,102,178)(81,208,103,179)(82,209,104,180)(83,210,105,181)(84,211,106,182)(85,212,107,183)(86,213,108,184)(87,214,109,185), (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)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111)(117,145)(118,144)(119,143)(120,142)(121,141)(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132)(146,168)(147,167)(148,166)(149,165)(150,164)(151,163)(152,162)(153,161)(154,160)(155,159)(156,158)(169,174)(170,173)(171,172)(175,197)(176,196)(177,195)(178,194)(179,193)(180,192)(181,191)(182,190)(183,189)(184,188)(185,187)(198,203)(199,202)(200,201)(204,226)(205,225)(206,224)(207,223)(208,222)(209,221)(210,220)(211,219)(212,218)(213,217)(214,216)(227,232)(228,231)(229,230)>;
G:=Group( (1,96,51,74)(2,97,52,75)(3,98,53,76)(4,99,54,77)(5,100,55,78)(6,101,56,79)(7,102,57,80)(8,103,58,81)(9,104,30,82)(10,105,31,83)(11,106,32,84)(12,107,33,85)(13,108,34,86)(14,109,35,87)(15,110,36,59)(16,111,37,60)(17,112,38,61)(18,113,39,62)(19,114,40,63)(20,115,41,64)(21,116,42,65)(22,88,43,66)(23,89,44,67)(24,90,45,68)(25,91,46,69)(26,92,47,70)(27,93,48,71)(28,94,49,72)(29,95,50,73)(117,201,172,230)(118,202,173,231)(119,203,174,232)(120,175,146,204)(121,176,147,205)(122,177,148,206)(123,178,149,207)(124,179,150,208)(125,180,151,209)(126,181,152,210)(127,182,153,211)(128,183,154,212)(129,184,155,213)(130,185,156,214)(131,186,157,215)(132,187,158,216)(133,188,159,217)(134,189,160,218)(135,190,161,219)(136,191,162,220)(137,192,163,221)(138,193,164,222)(139,194,165,223)(140,195,166,224)(141,196,167,225)(142,197,168,226)(143,198,169,227)(144,199,170,228)(145,200,171,229), (1,172,51,117)(2,173,52,118)(3,174,53,119)(4,146,54,120)(5,147,55,121)(6,148,56,122)(7,149,57,123)(8,150,58,124)(9,151,30,125)(10,152,31,126)(11,153,32,127)(12,154,33,128)(13,155,34,129)(14,156,35,130)(15,157,36,131)(16,158,37,132)(17,159,38,133)(18,160,39,134)(19,161,40,135)(20,162,41,136)(21,163,42,137)(22,164,43,138)(23,165,44,139)(24,166,45,140)(25,167,46,141)(26,168,47,142)(27,169,48,143)(28,170,49,144)(29,171,50,145)(59,215,110,186)(60,216,111,187)(61,217,112,188)(62,218,113,189)(63,219,114,190)(64,220,115,191)(65,221,116,192)(66,222,88,193)(67,223,89,194)(68,224,90,195)(69,225,91,196)(70,226,92,197)(71,227,93,198)(72,228,94,199)(73,229,95,200)(74,230,96,201)(75,231,97,202)(76,232,98,203)(77,204,99,175)(78,205,100,176)(79,206,101,177)(80,207,102,178)(81,208,103,179)(82,209,104,180)(83,210,105,181)(84,211,106,182)(85,212,107,183)(86,213,108,184)(87,214,109,185), (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)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111)(117,145)(118,144)(119,143)(120,142)(121,141)(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132)(146,168)(147,167)(148,166)(149,165)(150,164)(151,163)(152,162)(153,161)(154,160)(155,159)(156,158)(169,174)(170,173)(171,172)(175,197)(176,196)(177,195)(178,194)(179,193)(180,192)(181,191)(182,190)(183,189)(184,188)(185,187)(198,203)(199,202)(200,201)(204,226)(205,225)(206,224)(207,223)(208,222)(209,221)(210,220)(211,219)(212,218)(213,217)(214,216)(227,232)(228,231)(229,230) );
G=PermutationGroup([[(1,96,51,74),(2,97,52,75),(3,98,53,76),(4,99,54,77),(5,100,55,78),(6,101,56,79),(7,102,57,80),(8,103,58,81),(9,104,30,82),(10,105,31,83),(11,106,32,84),(12,107,33,85),(13,108,34,86),(14,109,35,87),(15,110,36,59),(16,111,37,60),(17,112,38,61),(18,113,39,62),(19,114,40,63),(20,115,41,64),(21,116,42,65),(22,88,43,66),(23,89,44,67),(24,90,45,68),(25,91,46,69),(26,92,47,70),(27,93,48,71),(28,94,49,72),(29,95,50,73),(117,201,172,230),(118,202,173,231),(119,203,174,232),(120,175,146,204),(121,176,147,205),(122,177,148,206),(123,178,149,207),(124,179,150,208),(125,180,151,209),(126,181,152,210),(127,182,153,211),(128,183,154,212),(129,184,155,213),(130,185,156,214),(131,186,157,215),(132,187,158,216),(133,188,159,217),(134,189,160,218),(135,190,161,219),(136,191,162,220),(137,192,163,221),(138,193,164,222),(139,194,165,223),(140,195,166,224),(141,196,167,225),(142,197,168,226),(143,198,169,227),(144,199,170,228),(145,200,171,229)], [(1,172,51,117),(2,173,52,118),(3,174,53,119),(4,146,54,120),(5,147,55,121),(6,148,56,122),(7,149,57,123),(8,150,58,124),(9,151,30,125),(10,152,31,126),(11,153,32,127),(12,154,33,128),(13,155,34,129),(14,156,35,130),(15,157,36,131),(16,158,37,132),(17,159,38,133),(18,160,39,134),(19,161,40,135),(20,162,41,136),(21,163,42,137),(22,164,43,138),(23,165,44,139),(24,166,45,140),(25,167,46,141),(26,168,47,142),(27,169,48,143),(28,170,49,144),(29,171,50,145),(59,215,110,186),(60,216,111,187),(61,217,112,188),(62,218,113,189),(63,219,114,190),(64,220,115,191),(65,221,116,192),(66,222,88,193),(67,223,89,194),(68,224,90,195),(69,225,91,196),(70,226,92,197),(71,227,93,198),(72,228,94,199),(73,229,95,200),(74,230,96,201),(75,231,97,202),(76,232,98,203),(77,204,99,175),(78,205,100,176),(79,206,101,177),(80,207,102,178),(81,208,103,179),(82,209,104,180),(83,210,105,181),(84,211,106,182),(85,212,107,183),(86,213,108,184),(87,214,109,185)], [(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),(60,87),(61,86),(62,85),(63,84),(64,83),(65,82),(66,81),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(88,103),(89,102),(90,101),(91,100),(92,99),(93,98),(94,97),(95,96),(104,116),(105,115),(106,114),(107,113),(108,112),(109,111),(117,145),(118,144),(119,143),(120,142),(121,141),(122,140),(123,139),(124,138),(125,137),(126,136),(127,135),(128,134),(129,133),(130,132),(146,168),(147,167),(148,166),(149,165),(150,164),(151,163),(152,162),(153,161),(154,160),(155,159),(156,158),(169,174),(170,173),(171,172),(175,197),(176,196),(177,195),(178,194),(179,193),(180,192),(181,191),(182,190),(183,189),(184,188),(185,187),(198,203),(199,202),(200,201),(204,226),(205,225),(206,224),(207,223),(208,222),(209,221),(210,220),(211,219),(212,218),(213,217),(214,216),(227,232),(228,231),(229,230)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 29A | ··· | 29N | 58A | ··· | 58N | 116A | ··· | 116AP |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 29 | 29 | 2 | 2 | 2 | 58 | 58 | 58 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | Q8 | D29 | D58 | Q8×D29 |
| kernel | Q8×D29 | Dic58 | C4×D29 | Q8×C29 | D29 | Q8 | C4 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 14 | 42 | 14 |
Matrix representation of Q8×D29 ►in GL4(𝔽233) generated by
| 232 | 0 | 0 | 0 |
| 0 | 232 | 0 | 0 |
| 0 | 0 | 189 | 162 |
| 0 | 0 | 24 | 44 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 131 | 189 |
| 0 | 0 | 210 | 102 |
| 75 | 1 | 0 | 0 |
| 43 | 100 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 52 | 174 | 0 | 0 |
| 109 | 181 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(233))| [232,0,0,0,0,232,0,0,0,0,189,24,0,0,162,44],[1,0,0,0,0,1,0,0,0,0,131,210,0,0,189,102],[75,43,0,0,1,100,0,0,0,0,1,0,0,0,0,1],[52,109,0,0,174,181,0,0,0,0,1,0,0,0,0,1] >;
Q8×D29 in GAP, Magma, Sage, TeX
Q_8\times D_{29} % in TeX
G:=Group("Q8xD29"); // GroupNames label
G:=SmallGroup(464,41);
// by ID
G=gap.SmallGroup(464,41);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-29,46,97,42,11204]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^29=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export