metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊3D29, C232⋊4C2, D58.1C4, C4.13D58, C29⋊3M4(2), Dic29.1C4, C116.13C22, C29⋊2C8⋊4C2, C58.9(C2×C4), C2.3(C4×D29), (C4×D29).2C2, SmallGroup(464,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊D29
G = < a,b,c | a8=b29=c2=1, ab=ba, cac=a5, cbc=b-1 >
Smallest permutation representation of C8⋊D29
►On 232 points
Generators in S232
(1 208 93 156 58 183 85 142)(2 209 94 157 30 184 86 143)(3 210 95 158 31 185 87 144)(4 211 96 159 32 186 59 145)(5 212 97 160 33 187 60 117)(6 213 98 161 34 188 61 118)(7 214 99 162 35 189 62 119)(8 215 100 163 36 190 63 120)(9 216 101 164 37 191 64 121)(10 217 102 165 38 192 65 122)(11 218 103 166 39 193 66 123)(12 219 104 167 40 194 67 124)(13 220 105 168 41 195 68 125)(14 221 106 169 42 196 69 126)(15 222 107 170 43 197 70 127)(16 223 108 171 44 198 71 128)(17 224 109 172 45 199 72 129)(18 225 110 173 46 200 73 130)(19 226 111 174 47 201 74 131)(20 227 112 146 48 202 75 132)(21 228 113 147 49 203 76 133)(22 229 114 148 50 175 77 134)(23 230 115 149 51 176 78 135)(24 231 116 150 52 177 79 136)(25 232 88 151 53 178 80 137)(26 204 89 152 54 179 81 138)(27 205 90 153 55 180 82 139)(28 206 91 154 56 181 83 140)(29 207 92 155 57 182 84 141)
(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 56)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)(57 58)(59 81)(60 80)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(82 87)(83 86)(84 85)(88 97)(89 96)(90 95)(91 94)(92 93)(98 116)(99 115)(100 114)(101 113)(102 112)(103 111)(104 110)(105 109)(106 108)(117 151)(118 150)(119 149)(120 148)(121 147)(122 146)(123 174)(124 173)(125 172)(126 171)(127 170)(128 169)(129 168)(130 167)(131 166)(132 165)(133 164)(134 163)(135 162)(136 161)(137 160)(138 159)(139 158)(140 157)(141 156)(142 155)(143 154)(144 153)(145 152)(175 215)(176 214)(177 213)(178 212)(179 211)(180 210)(181 209)(182 208)(183 207)(184 206)(185 205)(186 204)(187 232)(188 231)(189 230)(190 229)(191 228)(192 227)(193 226)(194 225)(195 224)(196 223)(197 222)(198 221)(199 220)(200 219)(201 218)(202 217)(203 216)
G:=sub<Sym(232)| (1,208,93,156,58,183,85,142)(2,209,94,157,30,184,86,143)(3,210,95,158,31,185,87,144)(4,211,96,159,32,186,59,145)(5,212,97,160,33,187,60,117)(6,213,98,161,34,188,61,118)(7,214,99,162,35,189,62,119)(8,215,100,163,36,190,63,120)(9,216,101,164,37,191,64,121)(10,217,102,165,38,192,65,122)(11,218,103,166,39,193,66,123)(12,219,104,167,40,194,67,124)(13,220,105,168,41,195,68,125)(14,221,106,169,42,196,69,126)(15,222,107,170,43,197,70,127)(16,223,108,171,44,198,71,128)(17,224,109,172,45,199,72,129)(18,225,110,173,46,200,73,130)(19,226,111,174,47,201,74,131)(20,227,112,146,48,202,75,132)(21,228,113,147,49,203,76,133)(22,229,114,148,50,175,77,134)(23,230,115,149,51,176,78,135)(24,231,116,150,52,177,79,136)(25,232,88,151,53,178,80,137)(26,204,89,152,54,179,81,138)(27,205,90,153,55,180,82,139)(28,206,91,154,56,181,83,140)(29,207,92,155,57,182,84,141), (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)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(57,58)(59,81)(60,80)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(82,87)(83,86)(84,85)(88,97)(89,96)(90,95)(91,94)(92,93)(98,116)(99,115)(100,114)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108)(117,151)(118,150)(119,149)(120,148)(121,147)(122,146)(123,174)(124,173)(125,172)(126,171)(127,170)(128,169)(129,168)(130,167)(131,166)(132,165)(133,164)(134,163)(135,162)(136,161)(137,160)(138,159)(139,158)(140,157)(141,156)(142,155)(143,154)(144,153)(145,152)(175,215)(176,214)(177,213)(178,212)(179,211)(180,210)(181,209)(182,208)(183,207)(184,206)(185,205)(186,204)(187,232)(188,231)(189,230)(190,229)(191,228)(192,227)(193,226)(194,225)(195,224)(196,223)(197,222)(198,221)(199,220)(200,219)(201,218)(202,217)(203,216)>;
G:=Group( (1,208,93,156,58,183,85,142)(2,209,94,157,30,184,86,143)(3,210,95,158,31,185,87,144)(4,211,96,159,32,186,59,145)(5,212,97,160,33,187,60,117)(6,213,98,161,34,188,61,118)(7,214,99,162,35,189,62,119)(8,215,100,163,36,190,63,120)(9,216,101,164,37,191,64,121)(10,217,102,165,38,192,65,122)(11,218,103,166,39,193,66,123)(12,219,104,167,40,194,67,124)(13,220,105,168,41,195,68,125)(14,221,106,169,42,196,69,126)(15,222,107,170,43,197,70,127)(16,223,108,171,44,198,71,128)(17,224,109,172,45,199,72,129)(18,225,110,173,46,200,73,130)(19,226,111,174,47,201,74,131)(20,227,112,146,48,202,75,132)(21,228,113,147,49,203,76,133)(22,229,114,148,50,175,77,134)(23,230,115,149,51,176,78,135)(24,231,116,150,52,177,79,136)(25,232,88,151,53,178,80,137)(26,204,89,152,54,179,81,138)(27,205,90,153,55,180,82,139)(28,206,91,154,56,181,83,140)(29,207,92,155,57,182,84,141), (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)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(57,58)(59,81)(60,80)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(82,87)(83,86)(84,85)(88,97)(89,96)(90,95)(91,94)(92,93)(98,116)(99,115)(100,114)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108)(117,151)(118,150)(119,149)(120,148)(121,147)(122,146)(123,174)(124,173)(125,172)(126,171)(127,170)(128,169)(129,168)(130,167)(131,166)(132,165)(133,164)(134,163)(135,162)(136,161)(137,160)(138,159)(139,158)(140,157)(141,156)(142,155)(143,154)(144,153)(145,152)(175,215)(176,214)(177,213)(178,212)(179,211)(180,210)(181,209)(182,208)(183,207)(184,206)(185,205)(186,204)(187,232)(188,231)(189,230)(190,229)(191,228)(192,227)(193,226)(194,225)(195,224)(196,223)(197,222)(198,221)(199,220)(200,219)(201,218)(202,217)(203,216) );
G=PermutationGroup([[(1,208,93,156,58,183,85,142),(2,209,94,157,30,184,86,143),(3,210,95,158,31,185,87,144),(4,211,96,159,32,186,59,145),(5,212,97,160,33,187,60,117),(6,213,98,161,34,188,61,118),(7,214,99,162,35,189,62,119),(8,215,100,163,36,190,63,120),(9,216,101,164,37,191,64,121),(10,217,102,165,38,192,65,122),(11,218,103,166,39,193,66,123),(12,219,104,167,40,194,67,124),(13,220,105,168,41,195,68,125),(14,221,106,169,42,196,69,126),(15,222,107,170,43,197,70,127),(16,223,108,171,44,198,71,128),(17,224,109,172,45,199,72,129),(18,225,110,173,46,200,73,130),(19,226,111,174,47,201,74,131),(20,227,112,146,48,202,75,132),(21,228,113,147,49,203,76,133),(22,229,114,148,50,175,77,134),(23,230,115,149,51,176,78,135),(24,231,116,150,52,177,79,136),(25,232,88,151,53,178,80,137),(26,204,89,152,54,179,81,138),(27,205,90,153,55,180,82,139),(28,206,91,154,56,181,83,140),(29,207,92,155,57,182,84,141)], [(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),(30,56),(31,55),(32,54),(33,53),(34,52),(35,51),(36,50),(37,49),(38,48),(39,47),(40,46),(41,45),(42,44),(57,58),(59,81),(60,80),(61,79),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71),(82,87),(83,86),(84,85),(88,97),(89,96),(90,95),(91,94),(92,93),(98,116),(99,115),(100,114),(101,113),(102,112),(103,111),(104,110),(105,109),(106,108),(117,151),(118,150),(119,149),(120,148),(121,147),(122,146),(123,174),(124,173),(125,172),(126,171),(127,170),(128,169),(129,168),(130,167),(131,166),(132,165),(133,164),(134,163),(135,162),(136,161),(137,160),(138,159),(139,158),(140,157),(141,156),(142,155),(143,154),(144,153),(145,152),(175,215),(176,214),(177,213),(178,212),(179,211),(180,210),(181,209),(182,208),(183,207),(184,206),(185,205),(186,204),(187,232),(188,231),(189,230),(190,229),(191,228),(192,227),(193,226),(194,225),(195,224),(196,223),(197,222),(198,221),(199,220),(200,219),(201,218),(202,217),(203,216)]])
122 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 29A | ··· | 29N | 58A | ··· | 58N | 116A | ··· | 116AB | 232A | ··· | 232BD |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 | 232 | ··· | 232 |
| size | 1 | 1 | 58 | 1 | 1 | 58 | 2 | 2 | 58 | 58 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
122 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | M4(2) | D29 | D58 | C4×D29 | C8⋊D29 |
| kernel | C8⋊D29 | C29⋊2C8 | C232 | C4×D29 | Dic29 | D58 | C29 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 14 | 14 | 28 | 56 |
Matrix representation of C8⋊D29 ►in GL2(𝔽233) generated by
| 173 | 4 |
| 229 | 60 |
| 203 | 1 |
| 232 | 0 |
| 145 | 44 |
| 200 | 88 |
G:=sub<GL(2,GF(233))| [173,229,4,60],[203,232,1,0],[145,200,44,88] >;
C8⋊D29 in GAP, Magma, Sage, TeX
C_8\rtimes D_{29} % in TeX
G:=Group("C8:D29"); // GroupNames label
G:=SmallGroup(464,5);
// by ID
G=gap.SmallGroup(464,5);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-29,101,26,42,11204]);
// Polycyclic
G:=Group<a,b,c|a^8=b^29=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations
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