metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D58⋊1C4, C58.6D4, C2.2D116, C22.6D58, (C2×C4)⋊1D29, (C2×C116)⋊1C2, C2.5(C4×D29), C29⋊2(C22⋊C4), C58.12(C2×C4), (C2×Dic29)⋊1C2, C2.2(C29⋊D4), (C2×C58).6C22, (C22×D29).1C2, SmallGroup(464,14)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D58⋊C4
G = < a,b,c | a58=b2=c4=1, bab=a-1, ac=ca, cbc-1=a29b >
Smallest permutation representation of D58⋊C4
►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 58)(2 57)(3 56)(4 55)(5 54)(6 53)(7 52)(8 51)(9 50)(10 49)(11 48)(12 47)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 40)(20 39)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 30)(59 88)(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)(89 116)(90 115)(91 114)(92 113)(93 112)(94 111)(95 110)(96 109)(97 108)(98 107)(99 106)(100 105)(101 104)(102 103)(117 161)(118 160)(119 159)(120 158)(121 157)(122 156)(123 155)(124 154)(125 153)(126 152)(127 151)(128 150)(129 149)(130 148)(131 147)(132 146)(133 145)(134 144)(135 143)(136 142)(137 141)(138 140)(162 174)(163 173)(164 172)(165 171)(166 170)(167 169)(175 187)(176 186)(177 185)(178 184)(179 183)(180 182)(188 232)(189 231)(190 230)(191 229)(192 228)(193 227)(194 226)(195 225)(196 224)(197 223)(198 222)(199 221)(200 220)(201 219)(202 218)(203 217)(204 216)(205 215)(206 214)(207 213)(208 212)(209 211)
(1 196 74 125)(2 197 75 126)(3 198 76 127)(4 199 77 128)(5 200 78 129)(6 201 79 130)(7 202 80 131)(8 203 81 132)(9 204 82 133)(10 205 83 134)(11 206 84 135)(12 207 85 136)(13 208 86 137)(14 209 87 138)(15 210 88 139)(16 211 89 140)(17 212 90 141)(18 213 91 142)(19 214 92 143)(20 215 93 144)(21 216 94 145)(22 217 95 146)(23 218 96 147)(24 219 97 148)(25 220 98 149)(26 221 99 150)(27 222 100 151)(28 223 101 152)(29 224 102 153)(30 225 103 154)(31 226 104 155)(32 227 105 156)(33 228 106 157)(34 229 107 158)(35 230 108 159)(36 231 109 160)(37 232 110 161)(38 175 111 162)(39 176 112 163)(40 177 113 164)(41 178 114 165)(42 179 115 166)(43 180 116 167)(44 181 59 168)(45 182 60 169)(46 183 61 170)(47 184 62 171)(48 185 63 172)(49 186 64 173)(50 187 65 174)(51 188 66 117)(52 189 67 118)(53 190 68 119)(54 191 69 120)(55 192 70 121)(56 193 71 122)(57 194 72 123)(58 195 73 124)
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), (1,58)(2,57)(3,56)(4,55)(5,54)(6,53)(7,52)(8,51)(9,50)(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(59,88)(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)(89,116)(90,115)(91,114)(92,113)(93,112)(94,111)(95,110)(96,109)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)(117,161)(118,160)(119,159)(120,158)(121,157)(122,156)(123,155)(124,154)(125,153)(126,152)(127,151)(128,150)(129,149)(130,148)(131,147)(132,146)(133,145)(134,144)(135,143)(136,142)(137,141)(138,140)(162,174)(163,173)(164,172)(165,171)(166,170)(167,169)(175,187)(176,186)(177,185)(178,184)(179,183)(180,182)(188,232)(189,231)(190,230)(191,229)(192,228)(193,227)(194,226)(195,225)(196,224)(197,223)(198,222)(199,221)(200,220)(201,219)(202,218)(203,217)(204,216)(205,215)(206,214)(207,213)(208,212)(209,211), (1,196,74,125)(2,197,75,126)(3,198,76,127)(4,199,77,128)(5,200,78,129)(6,201,79,130)(7,202,80,131)(8,203,81,132)(9,204,82,133)(10,205,83,134)(11,206,84,135)(12,207,85,136)(13,208,86,137)(14,209,87,138)(15,210,88,139)(16,211,89,140)(17,212,90,141)(18,213,91,142)(19,214,92,143)(20,215,93,144)(21,216,94,145)(22,217,95,146)(23,218,96,147)(24,219,97,148)(25,220,98,149)(26,221,99,150)(27,222,100,151)(28,223,101,152)(29,224,102,153)(30,225,103,154)(31,226,104,155)(32,227,105,156)(33,228,106,157)(34,229,107,158)(35,230,108,159)(36,231,109,160)(37,232,110,161)(38,175,111,162)(39,176,112,163)(40,177,113,164)(41,178,114,165)(42,179,115,166)(43,180,116,167)(44,181,59,168)(45,182,60,169)(46,183,61,170)(47,184,62,171)(48,185,63,172)(49,186,64,173)(50,187,65,174)(51,188,66,117)(52,189,67,118)(53,190,68,119)(54,191,69,120)(55,192,70,121)(56,193,71,122)(57,194,72,123)(58,195,73,124)>;
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), (1,58)(2,57)(3,56)(4,55)(5,54)(6,53)(7,52)(8,51)(9,50)(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(59,88)(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)(89,116)(90,115)(91,114)(92,113)(93,112)(94,111)(95,110)(96,109)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)(117,161)(118,160)(119,159)(120,158)(121,157)(122,156)(123,155)(124,154)(125,153)(126,152)(127,151)(128,150)(129,149)(130,148)(131,147)(132,146)(133,145)(134,144)(135,143)(136,142)(137,141)(138,140)(162,174)(163,173)(164,172)(165,171)(166,170)(167,169)(175,187)(176,186)(177,185)(178,184)(179,183)(180,182)(188,232)(189,231)(190,230)(191,229)(192,228)(193,227)(194,226)(195,225)(196,224)(197,223)(198,222)(199,221)(200,220)(201,219)(202,218)(203,217)(204,216)(205,215)(206,214)(207,213)(208,212)(209,211), (1,196,74,125)(2,197,75,126)(3,198,76,127)(4,199,77,128)(5,200,78,129)(6,201,79,130)(7,202,80,131)(8,203,81,132)(9,204,82,133)(10,205,83,134)(11,206,84,135)(12,207,85,136)(13,208,86,137)(14,209,87,138)(15,210,88,139)(16,211,89,140)(17,212,90,141)(18,213,91,142)(19,214,92,143)(20,215,93,144)(21,216,94,145)(22,217,95,146)(23,218,96,147)(24,219,97,148)(25,220,98,149)(26,221,99,150)(27,222,100,151)(28,223,101,152)(29,224,102,153)(30,225,103,154)(31,226,104,155)(32,227,105,156)(33,228,106,157)(34,229,107,158)(35,230,108,159)(36,231,109,160)(37,232,110,161)(38,175,111,162)(39,176,112,163)(40,177,113,164)(41,178,114,165)(42,179,115,166)(43,180,116,167)(44,181,59,168)(45,182,60,169)(46,183,61,170)(47,184,62,171)(48,185,63,172)(49,186,64,173)(50,187,65,174)(51,188,66,117)(52,189,67,118)(53,190,68,119)(54,191,69,120)(55,192,70,121)(56,193,71,122)(57,194,72,123)(58,195,73,124) );
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)], [(1,58),(2,57),(3,56),(4,55),(5,54),(6,53),(7,52),(8,51),(9,50),(10,49),(11,48),(12,47),(13,46),(14,45),(15,44),(16,43),(17,42),(18,41),(19,40),(20,39),(21,38),(22,37),(23,36),(24,35),(25,34),(26,33),(27,32),(28,31),(29,30),(59,88),(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),(89,116),(90,115),(91,114),(92,113),(93,112),(94,111),(95,110),(96,109),(97,108),(98,107),(99,106),(100,105),(101,104),(102,103),(117,161),(118,160),(119,159),(120,158),(121,157),(122,156),(123,155),(124,154),(125,153),(126,152),(127,151),(128,150),(129,149),(130,148),(131,147),(132,146),(133,145),(134,144),(135,143),(136,142),(137,141),(138,140),(162,174),(163,173),(164,172),(165,171),(166,170),(167,169),(175,187),(176,186),(177,185),(178,184),(179,183),(180,182),(188,232),(189,231),(190,230),(191,229),(192,228),(193,227),(194,226),(195,225),(196,224),(197,223),(198,222),(199,221),(200,220),(201,219),(202,218),(203,217),(204,216),(205,215),(206,214),(207,213),(208,212),(209,211)], [(1,196,74,125),(2,197,75,126),(3,198,76,127),(4,199,77,128),(5,200,78,129),(6,201,79,130),(7,202,80,131),(8,203,81,132),(9,204,82,133),(10,205,83,134),(11,206,84,135),(12,207,85,136),(13,208,86,137),(14,209,87,138),(15,210,88,139),(16,211,89,140),(17,212,90,141),(18,213,91,142),(19,214,92,143),(20,215,93,144),(21,216,94,145),(22,217,95,146),(23,218,96,147),(24,219,97,148),(25,220,98,149),(26,221,99,150),(27,222,100,151),(28,223,101,152),(29,224,102,153),(30,225,103,154),(31,226,104,155),(32,227,105,156),(33,228,106,157),(34,229,107,158),(35,230,108,159),(36,231,109,160),(37,232,110,161),(38,175,111,162),(39,176,112,163),(40,177,113,164),(41,178,114,165),(42,179,115,166),(43,180,116,167),(44,181,59,168),(45,182,60,169),(46,183,61,170),(47,184,62,171),(48,185,63,172),(49,186,64,173),(50,187,65,174),(51,188,66,117),(52,189,67,118),(53,190,68,119),(54,191,69,120),(55,192,70,121),(56,193,71,122),(57,194,72,123),(58,195,73,124)]])
122 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 29A | ··· | 29N | 58A | ··· | 58AP | 116A | ··· | 116BD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 1 | 1 | 58 | 58 | 2 | 2 | 58 | 58 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
122 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D29 | D58 | C4×D29 | D116 | C29⋊D4 |
| kernel | D58⋊C4 | C2×Dic29 | C2×C116 | C22×D29 | D58 | C58 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 14 | 14 | 28 | 28 | 28 |
Matrix representation of D58⋊C4 ►in GL3(𝔽233) generated by
| 1 | 0 | 0 |
| 0 | 19 | 160 |
| 0 | 167 | 131 |
| 232 | 0 | 0 |
| 0 | 104 | 171 |
| 0 | 88 | 129 |
| 89 | 0 | 0 |
| 0 | 129 | 164 |
| 0 | 177 | 104 |
G:=sub<GL(3,GF(233))| [1,0,0,0,19,167,0,160,131],[232,0,0,0,104,88,0,171,129],[89,0,0,0,129,177,0,164,104] >;
D58⋊C4 in GAP, Magma, Sage, TeX
D_{58}\rtimes C_4 % in TeX
G:=Group("D58:C4"); // GroupNames label
G:=SmallGroup(464,14);
// by ID
G=gap.SmallGroup(464,14);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-29,101,26,11204]);
// Polycyclic
G:=Group<a,b,c|a^58=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^29*b>;
// generators/relations
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