metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊2D29, C4.7D58, D116⋊4C2, C58.8C23, C116.7C22, D58.3C22, Dic29.9C22, (C4×D29)⋊3C2, C29⋊3(C4○D4), (Q8×C29)⋊3C2, C2.9(C22×D29), SmallGroup(464,42)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊2D29
G = < a,b,c,d | a4=c29=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of Q8⋊2D29
►On 232 points
Generators in S232
(1 95 31 64)(2 96 32 65)(3 97 33 66)(4 98 34 67)(5 99 35 68)(6 100 36 69)(7 101 37 70)(8 102 38 71)(9 103 39 72)(10 104 40 73)(11 105 41 74)(12 106 42 75)(13 107 43 76)(14 108 44 77)(15 109 45 78)(16 110 46 79)(17 111 47 80)(18 112 48 81)(19 113 49 82)(20 114 50 83)(21 115 51 84)(22 116 52 85)(23 88 53 86)(24 89 54 87)(25 90 55 59)(26 91 56 60)(27 92 57 61)(28 93 58 62)(29 94 30 63)(117 202 146 225)(118 203 147 226)(119 175 148 227)(120 176 149 228)(121 177 150 229)(122 178 151 230)(123 179 152 231)(124 180 153 232)(125 181 154 204)(126 182 155 205)(127 183 156 206)(128 184 157 207)(129 185 158 208)(130 186 159 209)(131 187 160 210)(132 188 161 211)(133 189 162 212)(134 190 163 213)(135 191 164 214)(136 192 165 215)(137 193 166 216)(138 194 167 217)(139 195 168 218)(140 196 169 219)(141 197 170 220)(142 198 171 221)(143 199 172 222)(144 200 173 223)(145 201 174 224)
(1 153 31 124)(2 154 32 125)(3 155 33 126)(4 156 34 127)(5 157 35 128)(6 158 36 129)(7 159 37 130)(8 160 38 131)(9 161 39 132)(10 162 40 133)(11 163 41 134)(12 164 42 135)(13 165 43 136)(14 166 44 137)(15 167 45 138)(16 168 46 139)(17 169 47 140)(18 170 48 141)(19 171 49 142)(20 172 50 143)(21 173 51 144)(22 174 52 145)(23 146 53 117)(24 147 54 118)(25 148 55 119)(26 149 56 120)(27 150 57 121)(28 151 58 122)(29 152 30 123)(59 227 90 175)(60 228 91 176)(61 229 92 177)(62 230 93 178)(63 231 94 179)(64 232 95 180)(65 204 96 181)(66 205 97 182)(67 206 98 183)(68 207 99 184)(69 208 100 185)(70 209 101 186)(71 210 102 187)(72 211 103 188)(73 212 104 189)(74 213 105 190)(75 214 106 191)(76 215 107 192)(77 216 108 193)(78 217 109 194)(79 218 110 195)(80 219 111 196)(81 220 112 197)(82 221 113 198)(83 222 114 199)(84 223 115 200)(85 224 116 201)(86 225 88 202)(87 226 89 203)
(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 63)(2 62)(3 61)(4 60)(5 59)(6 87)(7 86)(8 85)(9 84)(10 83)(11 82)(12 81)(13 80)(14 79)(15 78)(16 77)(17 76)(18 75)(19 74)(20 73)(21 72)(22 71)(23 70)(24 69)(25 68)(26 67)(27 66)(28 65)(29 64)(30 95)(31 94)(32 93)(33 92)(34 91)(35 90)(36 89)(37 88)(38 116)(39 115)(40 114)(41 113)(42 112)(43 111)(44 110)(45 109)(46 108)(47 107)(48 106)(49 105)(50 104)(51 103)(52 102)(53 101)(54 100)(55 99)(56 98)(57 97)(58 96)(117 186)(118 185)(119 184)(120 183)(121 182)(122 181)(123 180)(124 179)(125 178)(126 177)(127 176)(128 175)(129 203)(130 202)(131 201)(132 200)(133 199)(134 198)(135 197)(136 196)(137 195)(138 194)(139 193)(140 192)(141 191)(142 190)(143 189)(144 188)(145 187)(146 209)(147 208)(148 207)(149 206)(150 205)(151 204)(152 232)(153 231)(154 230)(155 229)(156 228)(157 227)(158 226)(159 225)(160 224)(161 223)(162 222)(163 221)(164 220)(165 219)(166 218)(167 217)(168 216)(169 215)(170 214)(171 213)(172 212)(173 211)(174 210)
G:=sub<Sym(232)| (1,95,31,64)(2,96,32,65)(3,97,33,66)(4,98,34,67)(5,99,35,68)(6,100,36,69)(7,101,37,70)(8,102,38,71)(9,103,39,72)(10,104,40,73)(11,105,41,74)(12,106,42,75)(13,107,43,76)(14,108,44,77)(15,109,45,78)(16,110,46,79)(17,111,47,80)(18,112,48,81)(19,113,49,82)(20,114,50,83)(21,115,51,84)(22,116,52,85)(23,88,53,86)(24,89,54,87)(25,90,55,59)(26,91,56,60)(27,92,57,61)(28,93,58,62)(29,94,30,63)(117,202,146,225)(118,203,147,226)(119,175,148,227)(120,176,149,228)(121,177,150,229)(122,178,151,230)(123,179,152,231)(124,180,153,232)(125,181,154,204)(126,182,155,205)(127,183,156,206)(128,184,157,207)(129,185,158,208)(130,186,159,209)(131,187,160,210)(132,188,161,211)(133,189,162,212)(134,190,163,213)(135,191,164,214)(136,192,165,215)(137,193,166,216)(138,194,167,217)(139,195,168,218)(140,196,169,219)(141,197,170,220)(142,198,171,221)(143,199,172,222)(144,200,173,223)(145,201,174,224), (1,153,31,124)(2,154,32,125)(3,155,33,126)(4,156,34,127)(5,157,35,128)(6,158,36,129)(7,159,37,130)(8,160,38,131)(9,161,39,132)(10,162,40,133)(11,163,41,134)(12,164,42,135)(13,165,43,136)(14,166,44,137)(15,167,45,138)(16,168,46,139)(17,169,47,140)(18,170,48,141)(19,171,49,142)(20,172,50,143)(21,173,51,144)(22,174,52,145)(23,146,53,117)(24,147,54,118)(25,148,55,119)(26,149,56,120)(27,150,57,121)(28,151,58,122)(29,152,30,123)(59,227,90,175)(60,228,91,176)(61,229,92,177)(62,230,93,178)(63,231,94,179)(64,232,95,180)(65,204,96,181)(66,205,97,182)(67,206,98,183)(68,207,99,184)(69,208,100,185)(70,209,101,186)(71,210,102,187)(72,211,103,188)(73,212,104,189)(74,213,105,190)(75,214,106,191)(76,215,107,192)(77,216,108,193)(78,217,109,194)(79,218,110,195)(80,219,111,196)(81,220,112,197)(82,221,113,198)(83,222,114,199)(84,223,115,200)(85,224,116,201)(86,225,88,202)(87,226,89,203), (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), (1,63)(2,62)(3,61)(4,60)(5,59)(6,87)(7,86)(8,85)(9,84)(10,83)(11,82)(12,81)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,88)(38,116)(39,115)(40,114)(41,113)(42,112)(43,111)(44,110)(45,109)(46,108)(47,107)(48,106)(49,105)(50,104)(51,103)(52,102)(53,101)(54,100)(55,99)(56,98)(57,97)(58,96)(117,186)(118,185)(119,184)(120,183)(121,182)(122,181)(123,180)(124,179)(125,178)(126,177)(127,176)(128,175)(129,203)(130,202)(131,201)(132,200)(133,199)(134,198)(135,197)(136,196)(137,195)(138,194)(139,193)(140,192)(141,191)(142,190)(143,189)(144,188)(145,187)(146,209)(147,208)(148,207)(149,206)(150,205)(151,204)(152,232)(153,231)(154,230)(155,229)(156,228)(157,227)(158,226)(159,225)(160,224)(161,223)(162,222)(163,221)(164,220)(165,219)(166,218)(167,217)(168,216)(169,215)(170,214)(171,213)(172,212)(173,211)(174,210)>;
G:=Group( (1,95,31,64)(2,96,32,65)(3,97,33,66)(4,98,34,67)(5,99,35,68)(6,100,36,69)(7,101,37,70)(8,102,38,71)(9,103,39,72)(10,104,40,73)(11,105,41,74)(12,106,42,75)(13,107,43,76)(14,108,44,77)(15,109,45,78)(16,110,46,79)(17,111,47,80)(18,112,48,81)(19,113,49,82)(20,114,50,83)(21,115,51,84)(22,116,52,85)(23,88,53,86)(24,89,54,87)(25,90,55,59)(26,91,56,60)(27,92,57,61)(28,93,58,62)(29,94,30,63)(117,202,146,225)(118,203,147,226)(119,175,148,227)(120,176,149,228)(121,177,150,229)(122,178,151,230)(123,179,152,231)(124,180,153,232)(125,181,154,204)(126,182,155,205)(127,183,156,206)(128,184,157,207)(129,185,158,208)(130,186,159,209)(131,187,160,210)(132,188,161,211)(133,189,162,212)(134,190,163,213)(135,191,164,214)(136,192,165,215)(137,193,166,216)(138,194,167,217)(139,195,168,218)(140,196,169,219)(141,197,170,220)(142,198,171,221)(143,199,172,222)(144,200,173,223)(145,201,174,224), (1,153,31,124)(2,154,32,125)(3,155,33,126)(4,156,34,127)(5,157,35,128)(6,158,36,129)(7,159,37,130)(8,160,38,131)(9,161,39,132)(10,162,40,133)(11,163,41,134)(12,164,42,135)(13,165,43,136)(14,166,44,137)(15,167,45,138)(16,168,46,139)(17,169,47,140)(18,170,48,141)(19,171,49,142)(20,172,50,143)(21,173,51,144)(22,174,52,145)(23,146,53,117)(24,147,54,118)(25,148,55,119)(26,149,56,120)(27,150,57,121)(28,151,58,122)(29,152,30,123)(59,227,90,175)(60,228,91,176)(61,229,92,177)(62,230,93,178)(63,231,94,179)(64,232,95,180)(65,204,96,181)(66,205,97,182)(67,206,98,183)(68,207,99,184)(69,208,100,185)(70,209,101,186)(71,210,102,187)(72,211,103,188)(73,212,104,189)(74,213,105,190)(75,214,106,191)(76,215,107,192)(77,216,108,193)(78,217,109,194)(79,218,110,195)(80,219,111,196)(81,220,112,197)(82,221,113,198)(83,222,114,199)(84,223,115,200)(85,224,116,201)(86,225,88,202)(87,226,89,203), (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), (1,63)(2,62)(3,61)(4,60)(5,59)(6,87)(7,86)(8,85)(9,84)(10,83)(11,82)(12,81)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,88)(38,116)(39,115)(40,114)(41,113)(42,112)(43,111)(44,110)(45,109)(46,108)(47,107)(48,106)(49,105)(50,104)(51,103)(52,102)(53,101)(54,100)(55,99)(56,98)(57,97)(58,96)(117,186)(118,185)(119,184)(120,183)(121,182)(122,181)(123,180)(124,179)(125,178)(126,177)(127,176)(128,175)(129,203)(130,202)(131,201)(132,200)(133,199)(134,198)(135,197)(136,196)(137,195)(138,194)(139,193)(140,192)(141,191)(142,190)(143,189)(144,188)(145,187)(146,209)(147,208)(148,207)(149,206)(150,205)(151,204)(152,232)(153,231)(154,230)(155,229)(156,228)(157,227)(158,226)(159,225)(160,224)(161,223)(162,222)(163,221)(164,220)(165,219)(166,218)(167,217)(168,216)(169,215)(170,214)(171,213)(172,212)(173,211)(174,210) );
G=PermutationGroup([[(1,95,31,64),(2,96,32,65),(3,97,33,66),(4,98,34,67),(5,99,35,68),(6,100,36,69),(7,101,37,70),(8,102,38,71),(9,103,39,72),(10,104,40,73),(11,105,41,74),(12,106,42,75),(13,107,43,76),(14,108,44,77),(15,109,45,78),(16,110,46,79),(17,111,47,80),(18,112,48,81),(19,113,49,82),(20,114,50,83),(21,115,51,84),(22,116,52,85),(23,88,53,86),(24,89,54,87),(25,90,55,59),(26,91,56,60),(27,92,57,61),(28,93,58,62),(29,94,30,63),(117,202,146,225),(118,203,147,226),(119,175,148,227),(120,176,149,228),(121,177,150,229),(122,178,151,230),(123,179,152,231),(124,180,153,232),(125,181,154,204),(126,182,155,205),(127,183,156,206),(128,184,157,207),(129,185,158,208),(130,186,159,209),(131,187,160,210),(132,188,161,211),(133,189,162,212),(134,190,163,213),(135,191,164,214),(136,192,165,215),(137,193,166,216),(138,194,167,217),(139,195,168,218),(140,196,169,219),(141,197,170,220),(142,198,171,221),(143,199,172,222),(144,200,173,223),(145,201,174,224)], [(1,153,31,124),(2,154,32,125),(3,155,33,126),(4,156,34,127),(5,157,35,128),(6,158,36,129),(7,159,37,130),(8,160,38,131),(9,161,39,132),(10,162,40,133),(11,163,41,134),(12,164,42,135),(13,165,43,136),(14,166,44,137),(15,167,45,138),(16,168,46,139),(17,169,47,140),(18,170,48,141),(19,171,49,142),(20,172,50,143),(21,173,51,144),(22,174,52,145),(23,146,53,117),(24,147,54,118),(25,148,55,119),(26,149,56,120),(27,150,57,121),(28,151,58,122),(29,152,30,123),(59,227,90,175),(60,228,91,176),(61,229,92,177),(62,230,93,178),(63,231,94,179),(64,232,95,180),(65,204,96,181),(66,205,97,182),(67,206,98,183),(68,207,99,184),(69,208,100,185),(70,209,101,186),(71,210,102,187),(72,211,103,188),(73,212,104,189),(74,213,105,190),(75,214,106,191),(76,215,107,192),(77,216,108,193),(78,217,109,194),(79,218,110,195),(80,219,111,196),(81,220,112,197),(82,221,113,198),(83,222,114,199),(84,223,115,200),(85,224,116,201),(86,225,88,202),(87,226,89,203)], [(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)], [(1,63),(2,62),(3,61),(4,60),(5,59),(6,87),(7,86),(8,85),(9,84),(10,83),(11,82),(12,81),(13,80),(14,79),(15,78),(16,77),(17,76),(18,75),(19,74),(20,73),(21,72),(22,71),(23,70),(24,69),(25,68),(26,67),(27,66),(28,65),(29,64),(30,95),(31,94),(32,93),(33,92),(34,91),(35,90),(36,89),(37,88),(38,116),(39,115),(40,114),(41,113),(42,112),(43,111),(44,110),(45,109),(46,108),(47,107),(48,106),(49,105),(50,104),(51,103),(52,102),(53,101),(54,100),(55,99),(56,98),(57,97),(58,96),(117,186),(118,185),(119,184),(120,183),(121,182),(122,181),(123,180),(124,179),(125,178),(126,177),(127,176),(128,175),(129,203),(130,202),(131,201),(132,200),(133,199),(134,198),(135,197),(136,196),(137,195),(138,194),(139,193),(140,192),(141,191),(142,190),(143,189),(144,188),(145,187),(146,209),(147,208),(148,207),(149,206),(150,205),(151,204),(152,232),(153,231),(154,230),(155,229),(156,228),(157,227),(158,226),(159,225),(160,224),(161,223),(162,222),(163,221),(164,220),(165,219),(166,218),(167,217),(168,216),(169,215),(170,214),(171,213),(172,212),(173,211),(174,210)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 29A | ··· | 29N | 58A | ··· | 58N | 116A | ··· | 116AP |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 58 | 58 | 58 | 2 | 2 | 2 | 29 | 29 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C4○D4 | D29 | D58 | Q8⋊2D29 |
| kernel | Q8⋊2D29 | C4×D29 | D116 | Q8×C29 | C29 | Q8 | C4 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 14 | 42 | 14 |
Matrix representation of Q8⋊2D29 ►in GL4(𝔽233) generated by
| 232 | 0 | 0 | 0 |
| 0 | 232 | 0 | 0 |
| 0 | 0 | 232 | 56 |
| 0 | 0 | 208 | 1 |
| 232 | 0 | 0 | 0 |
| 0 | 232 | 0 | 0 |
| 0 | 0 | 89 | 142 |
| 0 | 0 | 0 | 144 |
| 1 | 1 | 0 | 0 |
| 173 | 174 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 222 | 38 | 0 | 0 |
| 144 | 11 | 0 | 0 |
| 0 | 0 | 232 | 56 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(233))| [232,0,0,0,0,232,0,0,0,0,232,208,0,0,56,1],[232,0,0,0,0,232,0,0,0,0,89,0,0,0,142,144],[1,173,0,0,1,174,0,0,0,0,1,0,0,0,0,1],[222,144,0,0,38,11,0,0,0,0,232,0,0,0,56,1] >;
Q8⋊2D29 in GAP, Magma, Sage, TeX
Q_8\rtimes_2D_{29} % in TeX
G:=Group("Q8:2D29"); // GroupNames label
G:=SmallGroup(464,42);
// by ID
G=gap.SmallGroup(464,42);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-29,46,182,97,42,11204]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export