direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C48, C80⋊4C6, C240⋊10C2, D10.4C24, C24.80D10, Dic5.4C24, C120.98C22, C5⋊3(C2×C48), C5⋊2C16⋊6C6, C15⋊11(C2×C16), (C6×D5).8C8, C6.16(C8×D5), C2.1(D5×C24), C8.19(C6×D5), C5⋊2C8.7C12, C30.48(C2×C8), C40.19(C2×C6), (C8×D5).11C6, (C4×D5).9C12, C4.16(D5×C12), C12.86(C4×D5), C10.10(C2×C24), C20.42(C2×C12), C60.208(C2×C4), (D5×C12).19C4, (D5×C24).22C2, (C3×Dic5).8C8, (C3×C5⋊2C16)⋊13C2, (C3×C5⋊2C8).14C4, SmallGroup(480,75)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C48 |
Generators and relations for D5×C48
G = < a,b,c | a48=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C48
►On 240 points
Generators in S240
(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)
(1 116 66 209 150)(2 117 67 210 151)(3 118 68 211 152)(4 119 69 212 153)(5 120 70 213 154)(6 121 71 214 155)(7 122 72 215 156)(8 123 73 216 157)(9 124 74 217 158)(10 125 75 218 159)(11 126 76 219 160)(12 127 77 220 161)(13 128 78 221 162)(14 129 79 222 163)(15 130 80 223 164)(16 131 81 224 165)(17 132 82 225 166)(18 133 83 226 167)(19 134 84 227 168)(20 135 85 228 169)(21 136 86 229 170)(22 137 87 230 171)(23 138 88 231 172)(24 139 89 232 173)(25 140 90 233 174)(26 141 91 234 175)(27 142 92 235 176)(28 143 93 236 177)(29 144 94 237 178)(30 97 95 238 179)(31 98 96 239 180)(32 99 49 240 181)(33 100 50 193 182)(34 101 51 194 183)(35 102 52 195 184)(36 103 53 196 185)(37 104 54 197 186)(38 105 55 198 187)(39 106 56 199 188)(40 107 57 200 189)(41 108 58 201 190)(42 109 59 202 191)(43 110 60 203 192)(44 111 61 204 145)(45 112 62 205 146)(46 113 63 206 147)(47 114 64 207 148)(48 115 65 208 149)
(1 174)(2 175)(3 176)(4 177)(5 178)(6 179)(7 180)(8 181)(9 182)(10 183)(11 184)(12 185)(13 186)(14 187)(15 188)(16 189)(17 190)(18 191)(19 192)(20 145)(21 146)(22 147)(23 148)(24 149)(25 150)(26 151)(27 152)(28 153)(29 154)(30 155)(31 156)(32 157)(33 158)(34 159)(35 160)(36 161)(37 162)(38 163)(39 164)(40 165)(41 166)(42 167)(43 168)(44 169)(45 170)(46 171)(47 172)(48 173)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(57 81)(58 82)(59 83)(60 84)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)(97 214)(98 215)(99 216)(100 217)(101 218)(102 219)(103 220)(104 221)(105 222)(106 223)(107 224)(108 225)(109 226)(110 227)(111 228)(112 229)(113 230)(114 231)(115 232)(116 233)(117 234)(118 235)(119 236)(120 237)(121 238)(122 239)(123 240)(124 193)(125 194)(126 195)(127 196)(128 197)(129 198)(130 199)(131 200)(132 201)(133 202)(134 203)(135 204)(136 205)(137 206)(138 207)(139 208)(140 209)(141 210)(142 211)(143 212)(144 213)
G:=sub<Sym(240)| (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), (1,116,66,209,150)(2,117,67,210,151)(3,118,68,211,152)(4,119,69,212,153)(5,120,70,213,154)(6,121,71,214,155)(7,122,72,215,156)(8,123,73,216,157)(9,124,74,217,158)(10,125,75,218,159)(11,126,76,219,160)(12,127,77,220,161)(13,128,78,221,162)(14,129,79,222,163)(15,130,80,223,164)(16,131,81,224,165)(17,132,82,225,166)(18,133,83,226,167)(19,134,84,227,168)(20,135,85,228,169)(21,136,86,229,170)(22,137,87,230,171)(23,138,88,231,172)(24,139,89,232,173)(25,140,90,233,174)(26,141,91,234,175)(27,142,92,235,176)(28,143,93,236,177)(29,144,94,237,178)(30,97,95,238,179)(31,98,96,239,180)(32,99,49,240,181)(33,100,50,193,182)(34,101,51,194,183)(35,102,52,195,184)(36,103,53,196,185)(37,104,54,197,186)(38,105,55,198,187)(39,106,56,199,188)(40,107,57,200,189)(41,108,58,201,190)(42,109,59,202,191)(43,110,60,203,192)(44,111,61,204,145)(45,112,62,205,146)(46,113,63,206,147)(47,114,64,207,148)(48,115,65,208,149), (1,174)(2,175)(3,176)(4,177)(5,178)(6,179)(7,180)(8,181)(9,182)(10,183)(11,184)(12,185)(13,186)(14,187)(15,188)(16,189)(17,190)(18,191)(19,192)(20,145)(21,146)(22,147)(23,148)(24,149)(25,150)(26,151)(27,152)(28,153)(29,154)(30,155)(31,156)(32,157)(33,158)(34,159)(35,160)(36,161)(37,162)(38,163)(39,164)(40,165)(41,166)(42,167)(43,168)(44,169)(45,170)(46,171)(47,172)(48,173)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(97,214)(98,215)(99,216)(100,217)(101,218)(102,219)(103,220)(104,221)(105,222)(106,223)(107,224)(108,225)(109,226)(110,227)(111,228)(112,229)(113,230)(114,231)(115,232)(116,233)(117,234)(118,235)(119,236)(120,237)(121,238)(122,239)(123,240)(124,193)(125,194)(126,195)(127,196)(128,197)(129,198)(130,199)(131,200)(132,201)(133,202)(134,203)(135,204)(136,205)(137,206)(138,207)(139,208)(140,209)(141,210)(142,211)(143,212)(144,213)>;
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)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,116,66,209,150)(2,117,67,210,151)(3,118,68,211,152)(4,119,69,212,153)(5,120,70,213,154)(6,121,71,214,155)(7,122,72,215,156)(8,123,73,216,157)(9,124,74,217,158)(10,125,75,218,159)(11,126,76,219,160)(12,127,77,220,161)(13,128,78,221,162)(14,129,79,222,163)(15,130,80,223,164)(16,131,81,224,165)(17,132,82,225,166)(18,133,83,226,167)(19,134,84,227,168)(20,135,85,228,169)(21,136,86,229,170)(22,137,87,230,171)(23,138,88,231,172)(24,139,89,232,173)(25,140,90,233,174)(26,141,91,234,175)(27,142,92,235,176)(28,143,93,236,177)(29,144,94,237,178)(30,97,95,238,179)(31,98,96,239,180)(32,99,49,240,181)(33,100,50,193,182)(34,101,51,194,183)(35,102,52,195,184)(36,103,53,196,185)(37,104,54,197,186)(38,105,55,198,187)(39,106,56,199,188)(40,107,57,200,189)(41,108,58,201,190)(42,109,59,202,191)(43,110,60,203,192)(44,111,61,204,145)(45,112,62,205,146)(46,113,63,206,147)(47,114,64,207,148)(48,115,65,208,149), (1,174)(2,175)(3,176)(4,177)(5,178)(6,179)(7,180)(8,181)(9,182)(10,183)(11,184)(12,185)(13,186)(14,187)(15,188)(16,189)(17,190)(18,191)(19,192)(20,145)(21,146)(22,147)(23,148)(24,149)(25,150)(26,151)(27,152)(28,153)(29,154)(30,155)(31,156)(32,157)(33,158)(34,159)(35,160)(36,161)(37,162)(38,163)(39,164)(40,165)(41,166)(42,167)(43,168)(44,169)(45,170)(46,171)(47,172)(48,173)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(97,214)(98,215)(99,216)(100,217)(101,218)(102,219)(103,220)(104,221)(105,222)(106,223)(107,224)(108,225)(109,226)(110,227)(111,228)(112,229)(113,230)(114,231)(115,232)(116,233)(117,234)(118,235)(119,236)(120,237)(121,238)(122,239)(123,240)(124,193)(125,194)(126,195)(127,196)(128,197)(129,198)(130,199)(131,200)(132,201)(133,202)(134,203)(135,204)(136,205)(137,206)(138,207)(139,208)(140,209)(141,210)(142,211)(143,212)(144,213) );
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),(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,116,66,209,150),(2,117,67,210,151),(3,118,68,211,152),(4,119,69,212,153),(5,120,70,213,154),(6,121,71,214,155),(7,122,72,215,156),(8,123,73,216,157),(9,124,74,217,158),(10,125,75,218,159),(11,126,76,219,160),(12,127,77,220,161),(13,128,78,221,162),(14,129,79,222,163),(15,130,80,223,164),(16,131,81,224,165),(17,132,82,225,166),(18,133,83,226,167),(19,134,84,227,168),(20,135,85,228,169),(21,136,86,229,170),(22,137,87,230,171),(23,138,88,231,172),(24,139,89,232,173),(25,140,90,233,174),(26,141,91,234,175),(27,142,92,235,176),(28,143,93,236,177),(29,144,94,237,178),(30,97,95,238,179),(31,98,96,239,180),(32,99,49,240,181),(33,100,50,193,182),(34,101,51,194,183),(35,102,52,195,184),(36,103,53,196,185),(37,104,54,197,186),(38,105,55,198,187),(39,106,56,199,188),(40,107,57,200,189),(41,108,58,201,190),(42,109,59,202,191),(43,110,60,203,192),(44,111,61,204,145),(45,112,62,205,146),(46,113,63,206,147),(47,114,64,207,148),(48,115,65,208,149)], [(1,174),(2,175),(3,176),(4,177),(5,178),(6,179),(7,180),(8,181),(9,182),(10,183),(11,184),(12,185),(13,186),(14,187),(15,188),(16,189),(17,190),(18,191),(19,192),(20,145),(21,146),(22,147),(23,148),(24,149),(25,150),(26,151),(27,152),(28,153),(29,154),(30,155),(31,156),(32,157),(33,158),(34,159),(35,160),(36,161),(37,162),(38,163),(39,164),(40,165),(41,166),(42,167),(43,168),(44,169),(45,170),(46,171),(47,172),(48,173),(49,73),(50,74),(51,75),(52,76),(53,77),(54,78),(55,79),(56,80),(57,81),(58,82),(59,83),(60,84),(61,85),(62,86),(63,87),(64,88),(65,89),(66,90),(67,91),(68,92),(69,93),(70,94),(71,95),(72,96),(97,214),(98,215),(99,216),(100,217),(101,218),(102,219),(103,220),(104,221),(105,222),(106,223),(107,224),(108,225),(109,226),(110,227),(111,228),(112,229),(113,230),(114,231),(115,232),(116,233),(117,234),(118,235),(119,236),(120,237),(121,238),(122,239),(123,240),(124,193),(125,194),(126,195),(127,196),(128,197),(129,198),(130,199),(131,200),(132,201),(133,202),(134,203),(135,204),(136,205),(137,206),(138,207),(139,208),(140,209),(141,210),(142,211),(143,212),(144,213)]])
192 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 15C | 15D | 16A | ··· | 16H | 16I | ··· | 16P | 20A | 20B | 20C | 20D | 24A | ··· | 24H | 24I | ··· | 24P | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 48A | ··· | 48P | 48Q | ··· | 48AF | 60A | ··· | 60H | 80A | ··· | 80P | 120A | ··· | 120P | 240A | ··· | 240AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 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 | 48 | ··· | 48 | 60 | ··· | 60 | 80 | ··· | 80 | 120 | ··· | 120 | 240 | ··· | 240 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 1 | 1 | 5 | 5 | 2 | 2 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
192 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C8 | C8 | C12 | C12 | C16 | C24 | C24 | C48 | D5 | D10 | C3×D5 | C4×D5 | C6×D5 | C8×D5 | D5×C12 | D5×C16 | D5×C24 | D5×C48 |
| kernel | D5×C48 | C3×C5⋊2C16 | C240 | D5×C24 | D5×C16 | C3×C5⋊2C8 | D5×C12 | C5⋊2C16 | C80 | C8×D5 | C3×Dic5 | C6×D5 | C5⋊2C8 | C4×D5 | C3×D5 | Dic5 | D10 | D5 | C48 | C24 | C16 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 16 | 8 | 8 | 32 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of D5×C48 ►in GL3(𝔽241) generated by
| 197 | 0 | 0 |
| 0 | 32 | 0 |
| 0 | 0 | 32 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 240 | 189 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(241))| [197,0,0,0,32,0,0,0,32],[1,0,0,0,0,240,0,1,189],[1,0,0,0,0,1,0,1,0] >;
D5×C48 in GAP, Magma, Sage, TeX
D_5\times C_{48} % in TeX
G:=Group("D5xC48"); // GroupNames label
G:=SmallGroup(480,75);
// by ID
G=gap.SmallGroup(480,75);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,92,80,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^48=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export