metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D29⋊C8, C116.3C4, D58.2C4, Dic29.4C22, C29⋊C8⋊3C2, C29⋊1(C2×C8), C4.3(C29⋊C4), C58.1(C2×C4), (C4×D29).5C2, C2.1(C2×C29⋊C4), SmallGroup(464,28)
Series: Derived ►Chief ►Lower central ►Upper central
| C29 — D29⋊C8 |
Generators and relations for D29⋊C8
G = < a,b,c | a29=b2=c8=1, bab=a-1, cac-1=a17, cbc-1=a16b >
Smallest permutation representation of D29⋊C8
►On 232 points
Generators in S232
(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 40)(31 39)(32 38)(33 37)(34 36)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(59 87)(60 86)(61 85)(62 84)(63 83)(64 82)(65 81)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(88 116)(89 115)(90 114)(91 113)(92 112)(93 111)(94 110)(95 109)(96 108)(97 107)(98 106)(99 105)(100 104)(101 103)(117 122)(118 121)(119 120)(123 145)(124 144)(125 143)(126 142)(127 141)(128 140)(129 139)(130 138)(131 137)(132 136)(133 135)(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)(176 203)(177 202)(178 201)(179 200)(180 199)(181 198)(182 197)(183 196)(184 195)(185 194)(186 193)(187 192)(188 191)(189 190)(204 228)(205 227)(206 226)(207 225)(208 224)(209 223)(210 222)(211 221)(212 220)(213 219)(214 218)(215 217)(229 232)(230 231)
(1 231 88 172 50 190 59 120)(2 214 116 160 51 202 87 137)(3 226 115 148 52 185 86 125)(4 209 114 165 53 197 85 142)(5 221 113 153 54 180 84 130)(6 204 112 170 55 192 83 118)(7 216 111 158 56 175 82 135)(8 228 110 146 57 187 81 123)(9 211 109 163 58 199 80 140)(10 223 108 151 30 182 79 128)(11 206 107 168 31 194 78 145)(12 218 106 156 32 177 77 133)(13 230 105 173 33 189 76 121)(14 213 104 161 34 201 75 138)(15 225 103 149 35 184 74 126)(16 208 102 166 36 196 73 143)(17 220 101 154 37 179 72 131)(18 232 100 171 38 191 71 119)(19 215 99 159 39 203 70 136)(20 227 98 147 40 186 69 124)(21 210 97 164 41 198 68 141)(22 222 96 152 42 181 67 129)(23 205 95 169 43 193 66 117)(24 217 94 157 44 176 65 134)(25 229 93 174 45 188 64 122)(26 212 92 162 46 200 63 139)(27 224 91 150 47 183 62 127)(28 207 90 167 48 195 61 144)(29 219 89 155 49 178 60 132)
G:=sub<Sym(232)| (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)(30,40)(31,39)(32,38)(33,37)(34,36)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,87)(60,86)(61,85)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(88,116)(89,115)(90,114)(91,113)(92,112)(93,111)(94,110)(95,109)(96,108)(97,107)(98,106)(99,105)(100,104)(101,103)(117,122)(118,121)(119,120)(123,145)(124,144)(125,143)(126,142)(127,141)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(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)(176,203)(177,202)(178,201)(179,200)(180,199)(181,198)(182,197)(183,196)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190)(204,228)(205,227)(206,226)(207,225)(208,224)(209,223)(210,222)(211,221)(212,220)(213,219)(214,218)(215,217)(229,232)(230,231), (1,231,88,172,50,190,59,120)(2,214,116,160,51,202,87,137)(3,226,115,148,52,185,86,125)(4,209,114,165,53,197,85,142)(5,221,113,153,54,180,84,130)(6,204,112,170,55,192,83,118)(7,216,111,158,56,175,82,135)(8,228,110,146,57,187,81,123)(9,211,109,163,58,199,80,140)(10,223,108,151,30,182,79,128)(11,206,107,168,31,194,78,145)(12,218,106,156,32,177,77,133)(13,230,105,173,33,189,76,121)(14,213,104,161,34,201,75,138)(15,225,103,149,35,184,74,126)(16,208,102,166,36,196,73,143)(17,220,101,154,37,179,72,131)(18,232,100,171,38,191,71,119)(19,215,99,159,39,203,70,136)(20,227,98,147,40,186,69,124)(21,210,97,164,41,198,68,141)(22,222,96,152,42,181,67,129)(23,205,95,169,43,193,66,117)(24,217,94,157,44,176,65,134)(25,229,93,174,45,188,64,122)(26,212,92,162,46,200,63,139)(27,224,91,150,47,183,62,127)(28,207,90,167,48,195,61,144)(29,219,89,155,49,178,60,132)>;
G:=Group( (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)(30,40)(31,39)(32,38)(33,37)(34,36)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,87)(60,86)(61,85)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(88,116)(89,115)(90,114)(91,113)(92,112)(93,111)(94,110)(95,109)(96,108)(97,107)(98,106)(99,105)(100,104)(101,103)(117,122)(118,121)(119,120)(123,145)(124,144)(125,143)(126,142)(127,141)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(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)(176,203)(177,202)(178,201)(179,200)(180,199)(181,198)(182,197)(183,196)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190)(204,228)(205,227)(206,226)(207,225)(208,224)(209,223)(210,222)(211,221)(212,220)(213,219)(214,218)(215,217)(229,232)(230,231), (1,231,88,172,50,190,59,120)(2,214,116,160,51,202,87,137)(3,226,115,148,52,185,86,125)(4,209,114,165,53,197,85,142)(5,221,113,153,54,180,84,130)(6,204,112,170,55,192,83,118)(7,216,111,158,56,175,82,135)(8,228,110,146,57,187,81,123)(9,211,109,163,58,199,80,140)(10,223,108,151,30,182,79,128)(11,206,107,168,31,194,78,145)(12,218,106,156,32,177,77,133)(13,230,105,173,33,189,76,121)(14,213,104,161,34,201,75,138)(15,225,103,149,35,184,74,126)(16,208,102,166,36,196,73,143)(17,220,101,154,37,179,72,131)(18,232,100,171,38,191,71,119)(19,215,99,159,39,203,70,136)(20,227,98,147,40,186,69,124)(21,210,97,164,41,198,68,141)(22,222,96,152,42,181,67,129)(23,205,95,169,43,193,66,117)(24,217,94,157,44,176,65,134)(25,229,93,174,45,188,64,122)(26,212,92,162,46,200,63,139)(27,224,91,150,47,183,62,127)(28,207,90,167,48,195,61,144)(29,219,89,155,49,178,60,132) );
G=PermutationGroup([[(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),(30,40),(31,39),(32,38),(33,37),(34,36),(41,58),(42,57),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50),(59,87),(60,86),(61,85),(62,84),(63,83),(64,82),(65,81),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74),(88,116),(89,115),(90,114),(91,113),(92,112),(93,111),(94,110),(95,109),(96,108),(97,107),(98,106),(99,105),(100,104),(101,103),(117,122),(118,121),(119,120),(123,145),(124,144),(125,143),(126,142),(127,141),(128,140),(129,139),(130,138),(131,137),(132,136),(133,135),(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),(176,203),(177,202),(178,201),(179,200),(180,199),(181,198),(182,197),(183,196),(184,195),(185,194),(186,193),(187,192),(188,191),(189,190),(204,228),(205,227),(206,226),(207,225),(208,224),(209,223),(210,222),(211,221),(212,220),(213,219),(214,218),(215,217),(229,232),(230,231)], [(1,231,88,172,50,190,59,120),(2,214,116,160,51,202,87,137),(3,226,115,148,52,185,86,125),(4,209,114,165,53,197,85,142),(5,221,113,153,54,180,84,130),(6,204,112,170,55,192,83,118),(7,216,111,158,56,175,82,135),(8,228,110,146,57,187,81,123),(9,211,109,163,58,199,80,140),(10,223,108,151,30,182,79,128),(11,206,107,168,31,194,78,145),(12,218,106,156,32,177,77,133),(13,230,105,173,33,189,76,121),(14,213,104,161,34,201,75,138),(15,225,103,149,35,184,74,126),(16,208,102,166,36,196,73,143),(17,220,101,154,37,179,72,131),(18,232,100,171,38,191,71,119),(19,215,99,159,39,203,70,136),(20,227,98,147,40,186,69,124),(21,210,97,164,41,198,68,141),(22,222,96,152,42,181,67,129),(23,205,95,169,43,193,66,117),(24,217,94,157,44,176,65,134),(25,229,93,174,45,188,64,122),(26,212,92,162,46,200,63,139),(27,224,91,150,47,183,62,127),(28,207,90,167,48,195,61,144),(29,219,89,155,49,178,60,132)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | ··· | 8H | 29A | ··· | 29G | 58A | ··· | 58G | 116A | ··· | 116N |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 29 | 29 | 1 | 1 | 29 | 29 | 29 | ··· | 29 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 |
| type | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C29⋊C4 | C2×C29⋊C4 | D29⋊C8 |
| kernel | D29⋊C8 | C29⋊C8 | C4×D29 | C116 | D58 | D29 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 7 | 7 | 14 |
Matrix representation of D29⋊C8 ►in GL4(𝔽233) generated by
| 232 | 1 | 0 | 0 |
| 232 | 0 | 1 | 0 |
| 232 | 0 | 0 | 1 |
| 167 | 213 | 20 | 65 |
| 232 | 0 | 0 | 0 |
| 19 | 213 | 168 | 1 |
| 114 | 33 | 203 | 65 |
| 120 | 115 | 12 | 51 |
| 179 | 90 | 189 | 214 |
| 89 | 208 | 25 | 144 |
| 68 | 159 | 121 | 191 |
| 189 | 38 | 47 | 191 |
G:=sub<GL(4,GF(233))| [232,232,232,167,1,0,0,213,0,1,0,20,0,0,1,65],[232,19,114,120,0,213,33,115,0,168,203,12,0,1,65,51],[179,89,68,189,90,208,159,38,189,25,121,47,214,144,191,191] >;
D29⋊C8 in GAP, Magma, Sage, TeX
D_{29}\rtimes C_8 % in TeX
G:=Group("D29:C8"); // GroupNames label
G:=SmallGroup(464,28);
// by ID
G=gap.SmallGroup(464,28);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-29,20,46,42,4804,2814]);
// Polycyclic
G:=Group<a,b,c|a^29=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^17,c*b*c^-1=a^16*b>;
// generators/relations
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