direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D5×C47, C5⋊C94, C235⋊3C2, SmallGroup(470,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C47 |
Generators and relations for D5×C47
G = < a,b,c | a47=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C47
►On 235 points
Generators in S235
(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)
(1 186 140 229 52)(2 187 141 230 53)(3 188 95 231 54)(4 142 96 232 55)(5 143 97 233 56)(6 144 98 234 57)(7 145 99 235 58)(8 146 100 189 59)(9 147 101 190 60)(10 148 102 191 61)(11 149 103 192 62)(12 150 104 193 63)(13 151 105 194 64)(14 152 106 195 65)(15 153 107 196 66)(16 154 108 197 67)(17 155 109 198 68)(18 156 110 199 69)(19 157 111 200 70)(20 158 112 201 71)(21 159 113 202 72)(22 160 114 203 73)(23 161 115 204 74)(24 162 116 205 75)(25 163 117 206 76)(26 164 118 207 77)(27 165 119 208 78)(28 166 120 209 79)(29 167 121 210 80)(30 168 122 211 81)(31 169 123 212 82)(32 170 124 213 83)(33 171 125 214 84)(34 172 126 215 85)(35 173 127 216 86)(36 174 128 217 87)(37 175 129 218 88)(38 176 130 219 89)(39 177 131 220 90)(40 178 132 221 91)(41 179 133 222 92)(42 180 134 223 93)(43 181 135 224 94)(44 182 136 225 48)(45 183 137 226 49)(46 184 138 227 50)(47 185 139 228 51)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 81)(31 82)(32 83)(33 84)(34 85)(35 86)(36 87)(37 88)(38 89)(39 90)(40 91)(41 92)(42 93)(43 94)(44 48)(45 49)(46 50)(47 51)(142 232)(143 233)(144 234)(145 235)(146 189)(147 190)(148 191)(149 192)(150 193)(151 194)(152 195)(153 196)(154 197)(155 198)(156 199)(157 200)(158 201)(159 202)(160 203)(161 204)(162 205)(163 206)(164 207)(165 208)(166 209)(167 210)(168 211)(169 212)(170 213)(171 214)(172 215)(173 216)(174 217)(175 218)(176 219)(177 220)(178 221)(179 222)(180 223)(181 224)(182 225)(183 226)(184 227)(185 228)(186 229)(187 230)(188 231)
G:=sub<Sym(235)| (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), (1,186,140,229,52)(2,187,141,230,53)(3,188,95,231,54)(4,142,96,232,55)(5,143,97,233,56)(6,144,98,234,57)(7,145,99,235,58)(8,146,100,189,59)(9,147,101,190,60)(10,148,102,191,61)(11,149,103,192,62)(12,150,104,193,63)(13,151,105,194,64)(14,152,106,195,65)(15,153,107,196,66)(16,154,108,197,67)(17,155,109,198,68)(18,156,110,199,69)(19,157,111,200,70)(20,158,112,201,71)(21,159,113,202,72)(22,160,114,203,73)(23,161,115,204,74)(24,162,116,205,75)(25,163,117,206,76)(26,164,118,207,77)(27,165,119,208,78)(28,166,120,209,79)(29,167,121,210,80)(30,168,122,211,81)(31,169,123,212,82)(32,170,124,213,83)(33,171,125,214,84)(34,172,126,215,85)(35,173,127,216,86)(36,174,128,217,87)(37,175,129,218,88)(38,176,130,219,89)(39,177,131,220,90)(40,178,132,221,91)(41,179,133,222,92)(42,180,134,223,93)(43,181,135,224,94)(44,182,136,225,48)(45,183,137,226,49)(46,184,138,227,50)(47,185,139,228,51), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,81)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,48)(45,49)(46,50)(47,51)(142,232)(143,233)(144,234)(145,235)(146,189)(147,190)(148,191)(149,192)(150,193)(151,194)(152,195)(153,196)(154,197)(155,198)(156,199)(157,200)(158,201)(159,202)(160,203)(161,204)(162,205)(163,206)(164,207)(165,208)(166,209)(167,210)(168,211)(169,212)(170,213)(171,214)(172,215)(173,216)(174,217)(175,218)(176,219)(177,220)(178,221)(179,222)(180,223)(181,224)(182,225)(183,226)(184,227)(185,228)(186,229)(187,230)(188,231)>;
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)(189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235), (1,186,140,229,52)(2,187,141,230,53)(3,188,95,231,54)(4,142,96,232,55)(5,143,97,233,56)(6,144,98,234,57)(7,145,99,235,58)(8,146,100,189,59)(9,147,101,190,60)(10,148,102,191,61)(11,149,103,192,62)(12,150,104,193,63)(13,151,105,194,64)(14,152,106,195,65)(15,153,107,196,66)(16,154,108,197,67)(17,155,109,198,68)(18,156,110,199,69)(19,157,111,200,70)(20,158,112,201,71)(21,159,113,202,72)(22,160,114,203,73)(23,161,115,204,74)(24,162,116,205,75)(25,163,117,206,76)(26,164,118,207,77)(27,165,119,208,78)(28,166,120,209,79)(29,167,121,210,80)(30,168,122,211,81)(31,169,123,212,82)(32,170,124,213,83)(33,171,125,214,84)(34,172,126,215,85)(35,173,127,216,86)(36,174,128,217,87)(37,175,129,218,88)(38,176,130,219,89)(39,177,131,220,90)(40,178,132,221,91)(41,179,133,222,92)(42,180,134,223,93)(43,181,135,224,94)(44,182,136,225,48)(45,183,137,226,49)(46,184,138,227,50)(47,185,139,228,51), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,81)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,48)(45,49)(46,50)(47,51)(142,232)(143,233)(144,234)(145,235)(146,189)(147,190)(148,191)(149,192)(150,193)(151,194)(152,195)(153,196)(154,197)(155,198)(156,199)(157,200)(158,201)(159,202)(160,203)(161,204)(162,205)(163,206)(164,207)(165,208)(166,209)(167,210)(168,211)(169,212)(170,213)(171,214)(172,215)(173,216)(174,217)(175,218)(176,219)(177,220)(178,221)(179,222)(180,223)(181,224)(182,225)(183,226)(184,227)(185,228)(186,229)(187,230)(188,231) );
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),(189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235)], [(1,186,140,229,52),(2,187,141,230,53),(3,188,95,231,54),(4,142,96,232,55),(5,143,97,233,56),(6,144,98,234,57),(7,145,99,235,58),(8,146,100,189,59),(9,147,101,190,60),(10,148,102,191,61),(11,149,103,192,62),(12,150,104,193,63),(13,151,105,194,64),(14,152,106,195,65),(15,153,107,196,66),(16,154,108,197,67),(17,155,109,198,68),(18,156,110,199,69),(19,157,111,200,70),(20,158,112,201,71),(21,159,113,202,72),(22,160,114,203,73),(23,161,115,204,74),(24,162,116,205,75),(25,163,117,206,76),(26,164,118,207,77),(27,165,119,208,78),(28,166,120,209,79),(29,167,121,210,80),(30,168,122,211,81),(31,169,123,212,82),(32,170,124,213,83),(33,171,125,214,84),(34,172,126,215,85),(35,173,127,216,86),(36,174,128,217,87),(37,175,129,218,88),(38,176,130,219,89),(39,177,131,220,90),(40,178,132,221,91),(41,179,133,222,92),(42,180,134,223,93),(43,181,135,224,94),(44,182,136,225,48),(45,183,137,226,49),(46,184,138,227,50),(47,185,139,228,51)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,61),(11,62),(12,63),(13,64),(14,65),(15,66),(16,67),(17,68),(18,69),(19,70),(20,71),(21,72),(22,73),(23,74),(24,75),(25,76),(26,77),(27,78),(28,79),(29,80),(30,81),(31,82),(32,83),(33,84),(34,85),(35,86),(36,87),(37,88),(38,89),(39,90),(40,91),(41,92),(42,93),(43,94),(44,48),(45,49),(46,50),(47,51),(142,232),(143,233),(144,234),(145,235),(146,189),(147,190),(148,191),(149,192),(150,193),(151,194),(152,195),(153,196),(154,197),(155,198),(156,199),(157,200),(158,201),(159,202),(160,203),(161,204),(162,205),(163,206),(164,207),(165,208),(166,209),(167,210),(168,211),(169,212),(170,213),(171,214),(172,215),(173,216),(174,217),(175,218),(176,219),(177,220),(178,221),(179,222),(180,223),(181,224),(182,225),(183,226),(184,227),(185,228),(186,229),(187,230),(188,231)]])
188 conjugacy classes
| class | 1 | 2 | 5A | 5B | 47A | ··· | 47AT | 94A | ··· | 94AT | 235A | ··· | 235CN |
| order | 1 | 2 | 5 | 5 | 47 | ··· | 47 | 94 | ··· | 94 | 235 | ··· | 235 |
| size | 1 | 5 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 |
188 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C47 | C94 | D5 | D5×C47 |
| kernel | D5×C47 | C235 | D5 | C5 | C47 | C1 |
| # reps | 1 | 1 | 46 | 46 | 2 | 92 |
Matrix representation of D5×C47 ►in GL2(𝔽941) generated by
| 34 | 0 |
| 0 | 34 |
| 940 | 1 |
| 712 | 228 |
| 940 | 0 |
| 712 | 1 |
G:=sub<GL(2,GF(941))| [34,0,0,34],[940,712,1,228],[940,712,0,1] >;
D5×C47 in GAP, Magma, Sage, TeX
D_5\times C_{47} % in TeX
G:=Group("D5xC47"); // GroupNames label
G:=SmallGroup(470,1);
// by ID
G=gap.SmallGroup(470,1);
# by ID
G:=PCGroup([3,-2,-47,-5,3386]);
// Polycyclic
G:=Group<a,b,c|a^47=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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