direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D48, C80⋊3S3, C15⋊6D16, C240⋊4C2, C48⋊1C10, D24⋊1C10, C40.75D6, C30.32D8, C60.177D4, C10.13D24, C20.39D12, C120.93C22, C3⋊1(C5×D16), C16⋊1(C5×S3), C6.1(C5×D8), (C5×D24)⋊9C2, C2.3(C5×D24), C4.1(C5×D12), C8.13(S3×C10), C12.24(C5×D4), C24.14(C2×C10), SmallGroup(480,118)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D48
G = < a,b,c | a5=b48=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D48
►On 240 points
Generators in S240
(1 119 188 218 58)(2 120 189 219 59)(3 121 190 220 60)(4 122 191 221 61)(5 123 192 222 62)(6 124 145 223 63)(7 125 146 224 64)(8 126 147 225 65)(9 127 148 226 66)(10 128 149 227 67)(11 129 150 228 68)(12 130 151 229 69)(13 131 152 230 70)(14 132 153 231 71)(15 133 154 232 72)(16 134 155 233 73)(17 135 156 234 74)(18 136 157 235 75)(19 137 158 236 76)(20 138 159 237 77)(21 139 160 238 78)(22 140 161 239 79)(23 141 162 240 80)(24 142 163 193 81)(25 143 164 194 82)(26 144 165 195 83)(27 97 166 196 84)(28 98 167 197 85)(29 99 168 198 86)(30 100 169 199 87)(31 101 170 200 88)(32 102 171 201 89)(33 103 172 202 90)(34 104 173 203 91)(35 105 174 204 92)(36 106 175 205 93)(37 107 176 206 94)(38 108 177 207 95)(39 109 178 208 96)(40 110 179 209 49)(41 111 180 210 50)(42 112 181 211 51)(43 113 182 212 52)(44 114 183 213 53)(45 115 184 214 54)(46 116 185 215 55)(47 117 186 216 56)(48 118 187 217 57)
(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)(193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(49 81)(50 80)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 72)(59 71)(60 70)(61 69)(62 68)(63 67)(64 66)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(97 107)(98 106)(99 105)(100 104)(101 103)(108 144)(109 143)(110 142)(111 141)(112 140)(113 139)(114 138)(115 137)(116 136)(117 135)(118 134)(119 133)(120 132)(121 131)(122 130)(123 129)(124 128)(125 127)(145 149)(146 148)(150 192)(151 191)(152 190)(153 189)(154 188)(155 187)(156 186)(157 185)(158 184)(159 183)(160 182)(161 181)(162 180)(163 179)(164 178)(165 177)(166 176)(167 175)(168 174)(169 173)(170 172)(193 209)(194 208)(195 207)(196 206)(197 205)(198 204)(199 203)(200 202)(210 240)(211 239)(212 238)(213 237)(214 236)(215 235)(216 234)(217 233)(218 232)(219 231)(220 230)(221 229)(222 228)(223 227)(224 226)
G:=sub<Sym(240)| (1,119,188,218,58)(2,120,189,219,59)(3,121,190,220,60)(4,122,191,221,61)(5,123,192,222,62)(6,124,145,223,63)(7,125,146,224,64)(8,126,147,225,65)(9,127,148,226,66)(10,128,149,227,67)(11,129,150,228,68)(12,130,151,229,69)(13,131,152,230,70)(14,132,153,231,71)(15,133,154,232,72)(16,134,155,233,73)(17,135,156,234,74)(18,136,157,235,75)(19,137,158,236,76)(20,138,159,237,77)(21,139,160,238,78)(22,140,161,239,79)(23,141,162,240,80)(24,142,163,193,81)(25,143,164,194,82)(26,144,165,195,83)(27,97,166,196,84)(28,98,167,197,85)(29,99,168,198,86)(30,100,169,199,87)(31,101,170,200,88)(32,102,171,201,89)(33,103,172,202,90)(34,104,173,203,91)(35,105,174,204,92)(36,106,175,205,93)(37,107,176,206,94)(38,108,177,207,95)(39,109,178,208,96)(40,110,179,209,49)(41,111,180,210,50)(42,112,181,211,51)(43,113,182,212,52)(44,114,183,213,53)(45,115,184,214,54)(46,116,185,215,55)(47,117,186,216,56)(48,118,187,217,57), (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)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,72)(59,71)(60,70)(61,69)(62,68)(63,67)(64,66)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(97,107)(98,106)(99,105)(100,104)(101,103)(108,144)(109,143)(110,142)(111,141)(112,140)(113,139)(114,138)(115,137)(116,136)(117,135)(118,134)(119,133)(120,132)(121,131)(122,130)(123,129)(124,128)(125,127)(145,149)(146,148)(150,192)(151,191)(152,190)(153,189)(154,188)(155,187)(156,186)(157,185)(158,184)(159,183)(160,182)(161,181)(162,180)(163,179)(164,178)(165,177)(166,176)(167,175)(168,174)(169,173)(170,172)(193,209)(194,208)(195,207)(196,206)(197,205)(198,204)(199,203)(200,202)(210,240)(211,239)(212,238)(213,237)(214,236)(215,235)(216,234)(217,233)(218,232)(219,231)(220,230)(221,229)(222,228)(223,227)(224,226)>;
G:=Group( (1,119,188,218,58)(2,120,189,219,59)(3,121,190,220,60)(4,122,191,221,61)(5,123,192,222,62)(6,124,145,223,63)(7,125,146,224,64)(8,126,147,225,65)(9,127,148,226,66)(10,128,149,227,67)(11,129,150,228,68)(12,130,151,229,69)(13,131,152,230,70)(14,132,153,231,71)(15,133,154,232,72)(16,134,155,233,73)(17,135,156,234,74)(18,136,157,235,75)(19,137,158,236,76)(20,138,159,237,77)(21,139,160,238,78)(22,140,161,239,79)(23,141,162,240,80)(24,142,163,193,81)(25,143,164,194,82)(26,144,165,195,83)(27,97,166,196,84)(28,98,167,197,85)(29,99,168,198,86)(30,100,169,199,87)(31,101,170,200,88)(32,102,171,201,89)(33,103,172,202,90)(34,104,173,203,91)(35,105,174,204,92)(36,106,175,205,93)(37,107,176,206,94)(38,108,177,207,95)(39,109,178,208,96)(40,110,179,209,49)(41,111,180,210,50)(42,112,181,211,51)(43,113,182,212,52)(44,114,183,213,53)(45,115,184,214,54)(46,116,185,215,55)(47,117,186,216,56)(48,118,187,217,57), (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)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,72)(59,71)(60,70)(61,69)(62,68)(63,67)(64,66)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(97,107)(98,106)(99,105)(100,104)(101,103)(108,144)(109,143)(110,142)(111,141)(112,140)(113,139)(114,138)(115,137)(116,136)(117,135)(118,134)(119,133)(120,132)(121,131)(122,130)(123,129)(124,128)(125,127)(145,149)(146,148)(150,192)(151,191)(152,190)(153,189)(154,188)(155,187)(156,186)(157,185)(158,184)(159,183)(160,182)(161,181)(162,180)(163,179)(164,178)(165,177)(166,176)(167,175)(168,174)(169,173)(170,172)(193,209)(194,208)(195,207)(196,206)(197,205)(198,204)(199,203)(200,202)(210,240)(211,239)(212,238)(213,237)(214,236)(215,235)(216,234)(217,233)(218,232)(219,231)(220,230)(221,229)(222,228)(223,227)(224,226) );
G=PermutationGroup([[(1,119,188,218,58),(2,120,189,219,59),(3,121,190,220,60),(4,122,191,221,61),(5,123,192,222,62),(6,124,145,223,63),(7,125,146,224,64),(8,126,147,225,65),(9,127,148,226,66),(10,128,149,227,67),(11,129,150,228,68),(12,130,151,229,69),(13,131,152,230,70),(14,132,153,231,71),(15,133,154,232,72),(16,134,155,233,73),(17,135,156,234,74),(18,136,157,235,75),(19,137,158,236,76),(20,138,159,237,77),(21,139,160,238,78),(22,140,161,239,79),(23,141,162,240,80),(24,142,163,193,81),(25,143,164,194,82),(26,144,165,195,83),(27,97,166,196,84),(28,98,167,197,85),(29,99,168,198,86),(30,100,169,199,87),(31,101,170,200,88),(32,102,171,201,89),(33,103,172,202,90),(34,104,173,203,91),(35,105,174,204,92),(36,106,175,205,93),(37,107,176,206,94),(38,108,177,207,95),(39,109,178,208,96),(40,110,179,209,49),(41,111,180,210,50),(42,112,181,211,51),(43,113,182,212,52),(44,114,183,213,53),(45,115,184,214,54),(46,116,185,215,55),(47,117,186,216,56),(48,118,187,217,57)], [(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),(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,40),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(31,33),(49,81),(50,80),(51,79),(52,78),(53,77),(54,76),(55,75),(56,74),(57,73),(58,72),(59,71),(60,70),(61,69),(62,68),(63,67),(64,66),(82,96),(83,95),(84,94),(85,93),(86,92),(87,91),(88,90),(97,107),(98,106),(99,105),(100,104),(101,103),(108,144),(109,143),(110,142),(111,141),(112,140),(113,139),(114,138),(115,137),(116,136),(117,135),(118,134),(119,133),(120,132),(121,131),(122,130),(123,129),(124,128),(125,127),(145,149),(146,148),(150,192),(151,191),(152,190),(153,189),(154,188),(155,187),(156,186),(157,185),(158,184),(159,183),(160,182),(161,181),(162,180),(163,179),(164,178),(165,177),(166,176),(167,175),(168,174),(169,173),(170,172),(193,209),(194,208),(195,207),(196,206),(197,205),(198,204),(199,203),(200,202),(210,240),(211,239),(212,238),(213,237),(214,236),(215,235),(216,234),(217,233),(218,232),(219,231),(220,230),(221,229),(222,228),(223,227),(224,226)]])
135 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 5A | 5B | 5C | 5D | 6 | 8A | 8B | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 12A | 12B | 15A | 15B | 15C | 15D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 48A | ··· | 48H | 60A | ··· | 60H | 80A | ··· | 80P | 120A | ··· | 120P | 240A | ··· | 240AF |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 5 | 5 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 48 | ··· | 48 | 60 | ··· | 60 | 80 | ··· | 80 | 120 | ··· | 120 | 240 | ··· | 240 |
| size | 1 | 1 | 24 | 24 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 24 | ··· | 24 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | S3 | D4 | D6 | D8 | D12 | C5×S3 | D16 | C5×D4 | D24 | S3×C10 | C5×D8 | D48 | C5×D12 | C5×D16 | C5×D24 | C5×D48 |
| kernel | C5×D48 | C240 | C5×D24 | D48 | C48 | D24 | C80 | C60 | C40 | C30 | C20 | C16 | C15 | C12 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of C5×D48 ►in GL2(𝔽241) generated by
| 91 | 0 |
| 0 | 91 |
| 197 | 107 |
| 134 | 63 |
| 136 | 232 |
| 127 | 105 |
G:=sub<GL(2,GF(241))| [91,0,0,91],[197,134,107,63],[136,127,232,105] >;
C5×D48 in GAP, Magma, Sage, TeX
C_5\times D_{48} % in TeX
G:=Group("C5xD48"); // GroupNames label
G:=SmallGroup(480,118);
// by ID
G=gap.SmallGroup(480,118);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,309,428,1683,192,4204,102,15686]);
// Polycyclic
G:=Group<a,b,c|a^5=b^48=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export