direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×D47, C47⋊C10, C235⋊2C2, SmallGroup(470,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C47 — C5×D47 |
Generators and relations for C5×D47
G = < a,b,c | a5=b47=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D47
►On 235 points
Generators in S235
(1 192 147 106 86)(2 193 148 107 87)(3 194 149 108 88)(4 195 150 109 89)(5 196 151 110 90)(6 197 152 111 91)(7 198 153 112 92)(8 199 154 113 93)(9 200 155 114 94)(10 201 156 115 48)(11 202 157 116 49)(12 203 158 117 50)(13 204 159 118 51)(14 205 160 119 52)(15 206 161 120 53)(16 207 162 121 54)(17 208 163 122 55)(18 209 164 123 56)(19 210 165 124 57)(20 211 166 125 58)(21 212 167 126 59)(22 213 168 127 60)(23 214 169 128 61)(24 215 170 129 62)(25 216 171 130 63)(26 217 172 131 64)(27 218 173 132 65)(28 219 174 133 66)(29 220 175 134 67)(30 221 176 135 68)(31 222 177 136 69)(32 223 178 137 70)(33 224 179 138 71)(34 225 180 139 72)(35 226 181 140 73)(36 227 182 141 74)(37 228 183 95 75)(38 229 184 96 76)(39 230 185 97 77)(40 231 186 98 78)(41 232 187 99 79)(42 233 188 100 80)(43 234 142 101 81)(44 235 143 102 82)(45 189 144 103 83)(46 190 145 104 84)(47 191 146 105 85)
(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)(189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235)
(1 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(48 76)(49 75)(50 74)(51 73)(52 72)(53 71)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)(61 63)(77 94)(78 93)(79 92)(80 91)(81 90)(82 89)(83 88)(84 87)(85 86)(95 116)(96 115)(97 114)(98 113)(99 112)(100 111)(101 110)(102 109)(103 108)(104 107)(105 106)(117 141)(118 140)(119 139)(120 138)(121 137)(122 136)(123 135)(124 134)(125 133)(126 132)(127 131)(128 130)(142 151)(143 150)(144 149)(145 148)(146 147)(152 188)(153 187)(154 186)(155 185)(156 184)(157 183)(158 182)(159 181)(160 180)(161 179)(162 178)(163 177)(164 176)(165 175)(166 174)(167 173)(168 172)(169 171)(189 194)(190 193)(191 192)(195 235)(196 234)(197 233)(198 232)(199 231)(200 230)(201 229)(202 228)(203 227)(204 226)(205 225)(206 224)(207 223)(208 222)(209 221)(210 220)(211 219)(212 218)(213 217)(214 216)
G:=sub<Sym(235)| (1,192,147,106,86)(2,193,148,107,87)(3,194,149,108,88)(4,195,150,109,89)(5,196,151,110,90)(6,197,152,111,91)(7,198,153,112,92)(8,199,154,113,93)(9,200,155,114,94)(10,201,156,115,48)(11,202,157,116,49)(12,203,158,117,50)(13,204,159,118,51)(14,205,160,119,52)(15,206,161,120,53)(16,207,162,121,54)(17,208,163,122,55)(18,209,164,123,56)(19,210,165,124,57)(20,211,166,125,58)(21,212,167,126,59)(22,213,168,127,60)(23,214,169,128,61)(24,215,170,129,62)(25,216,171,130,63)(26,217,172,131,64)(27,218,173,132,65)(28,219,174,133,66)(29,220,175,134,67)(30,221,176,135,68)(31,222,177,136,69)(32,223,178,137,70)(33,224,179,138,71)(34,225,180,139,72)(35,226,181,140,73)(36,227,182,141,74)(37,228,183,95,75)(38,229,184,96,76)(39,230,185,97,77)(40,231,186,98,78)(41,232,187,99,79)(42,233,188,100,80)(43,234,142,101,81)(44,235,143,102,82)(45,189,144,103,83)(46,190,145,104,84)(47,191,146,105,85), (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)(189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(48,76)(49,75)(50,74)(51,73)(52,72)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)(61,63)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)(117,141)(118,140)(119,139)(120,138)(121,137)(122,136)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(142,151)(143,150)(144,149)(145,148)(146,147)(152,188)(153,187)(154,186)(155,185)(156,184)(157,183)(158,182)(159,181)(160,180)(161,179)(162,178)(163,177)(164,176)(165,175)(166,174)(167,173)(168,172)(169,171)(189,194)(190,193)(191,192)(195,235)(196,234)(197,233)(198,232)(199,231)(200,230)(201,229)(202,228)(203,227)(204,226)(205,225)(206,224)(207,223)(208,222)(209,221)(210,220)(211,219)(212,218)(213,217)(214,216)>;
G:=Group( (1,192,147,106,86)(2,193,148,107,87)(3,194,149,108,88)(4,195,150,109,89)(5,196,151,110,90)(6,197,152,111,91)(7,198,153,112,92)(8,199,154,113,93)(9,200,155,114,94)(10,201,156,115,48)(11,202,157,116,49)(12,203,158,117,50)(13,204,159,118,51)(14,205,160,119,52)(15,206,161,120,53)(16,207,162,121,54)(17,208,163,122,55)(18,209,164,123,56)(19,210,165,124,57)(20,211,166,125,58)(21,212,167,126,59)(22,213,168,127,60)(23,214,169,128,61)(24,215,170,129,62)(25,216,171,130,63)(26,217,172,131,64)(27,218,173,132,65)(28,219,174,133,66)(29,220,175,134,67)(30,221,176,135,68)(31,222,177,136,69)(32,223,178,137,70)(33,224,179,138,71)(34,225,180,139,72)(35,226,181,140,73)(36,227,182,141,74)(37,228,183,95,75)(38,229,184,96,76)(39,230,185,97,77)(40,231,186,98,78)(41,232,187,99,79)(42,233,188,100,80)(43,234,142,101,81)(44,235,143,102,82)(45,189,144,103,83)(46,190,145,104,84)(47,191,146,105,85), (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)(189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(48,76)(49,75)(50,74)(51,73)(52,72)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)(61,63)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)(117,141)(118,140)(119,139)(120,138)(121,137)(122,136)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(142,151)(143,150)(144,149)(145,148)(146,147)(152,188)(153,187)(154,186)(155,185)(156,184)(157,183)(158,182)(159,181)(160,180)(161,179)(162,178)(163,177)(164,176)(165,175)(166,174)(167,173)(168,172)(169,171)(189,194)(190,193)(191,192)(195,235)(196,234)(197,233)(198,232)(199,231)(200,230)(201,229)(202,228)(203,227)(204,226)(205,225)(206,224)(207,223)(208,222)(209,221)(210,220)(211,219)(212,218)(213,217)(214,216) );
G=PermutationGroup([[(1,192,147,106,86),(2,193,148,107,87),(3,194,149,108,88),(4,195,150,109,89),(5,196,151,110,90),(6,197,152,111,91),(7,198,153,112,92),(8,199,154,113,93),(9,200,155,114,94),(10,201,156,115,48),(11,202,157,116,49),(12,203,158,117,50),(13,204,159,118,51),(14,205,160,119,52),(15,206,161,120,53),(16,207,162,121,54),(17,208,163,122,55),(18,209,164,123,56),(19,210,165,124,57),(20,211,166,125,58),(21,212,167,126,59),(22,213,168,127,60),(23,214,169,128,61),(24,215,170,129,62),(25,216,171,130,63),(26,217,172,131,64),(27,218,173,132,65),(28,219,174,133,66),(29,220,175,134,67),(30,221,176,135,68),(31,222,177,136,69),(32,223,178,137,70),(33,224,179,138,71),(34,225,180,139,72),(35,226,181,140,73),(36,227,182,141,74),(37,228,183,95,75),(38,229,184,96,76),(39,230,185,97,77),(40,231,186,98,78),(41,232,187,99,79),(42,233,188,100,80),(43,234,142,101,81),(44,235,143,102,82),(45,189,144,103,83),(46,190,145,104,84),(47,191,146,105,85)], [(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),(189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235)], [(1,47),(2,46),(3,45),(4,44),(5,43),(6,42),(7,41),(8,40),(9,39),(10,38),(11,37),(12,36),(13,35),(14,34),(15,33),(16,32),(17,31),(18,30),(19,29),(20,28),(21,27),(22,26),(23,25),(48,76),(49,75),(50,74),(51,73),(52,72),(53,71),(54,70),(55,69),(56,68),(57,67),(58,66),(59,65),(60,64),(61,63),(77,94),(78,93),(79,92),(80,91),(81,90),(82,89),(83,88),(84,87),(85,86),(95,116),(96,115),(97,114),(98,113),(99,112),(100,111),(101,110),(102,109),(103,108),(104,107),(105,106),(117,141),(118,140),(119,139),(120,138),(121,137),(122,136),(123,135),(124,134),(125,133),(126,132),(127,131),(128,130),(142,151),(143,150),(144,149),(145,148),(146,147),(152,188),(153,187),(154,186),(155,185),(156,184),(157,183),(158,182),(159,181),(160,180),(161,179),(162,178),(163,177),(164,176),(165,175),(166,174),(167,173),(168,172),(169,171),(189,194),(190,193),(191,192),(195,235),(196,234),(197,233),(198,232),(199,231),(200,230),(201,229),(202,228),(203,227),(204,226),(205,225),(206,224),(207,223),(208,222),(209,221),(210,220),(211,219),(212,218),(213,217),(214,216)]])
125 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 47A | ··· | 47W | 235A | ··· | 235CN |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 47 | ··· | 47 | 235 | ··· | 235 |
| size | 1 | 47 | 1 | 1 | 1 | 1 | 47 | 47 | 47 | 47 | 2 | ··· | 2 | 2 | ··· | 2 |
125 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C5 | C10 | D47 | C5×D47 |
| kernel | C5×D47 | C235 | D47 | C47 | C5 | C1 |
| # reps | 1 | 1 | 4 | 4 | 23 | 92 |
Matrix representation of C5×D47 ►in GL2(𝔽941) generated by
| 412 | 0 |
| 0 | 412 |
| 141 | 1 |
| 940 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(941))| [412,0,0,412],[141,940,1,0],[0,1,1,0] >;
C5×D47 in GAP, Magma, Sage, TeX
C_5\times D_{47} % in TeX
G:=Group("C5xD47"); // GroupNames label
G:=SmallGroup(470,2);
// by ID
G=gap.SmallGroup(470,2);
# by ID
G:=PCGroup([3,-2,-5,-47,4142]);
// Polycyclic
G:=Group<a,b,c|a^5=b^47=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export