metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C80⋊2S3, C48⋊2D5, C240⋊2C2, C16⋊2D15, C6.2D40, C4.2D60, C15⋊7SD32, C2.4D120, C10.2D24, C30.23D8, C40.66D6, C8.14D30, Dic60⋊1C2, D120.1C2, C24.66D10, C60.158D4, C20.27D12, C12.27D20, C120.79C22, C3⋊1(C16⋊D5), C5⋊1(C48⋊C2), SmallGroup(480,160)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C48⋊D5
G = < a,b,c | a48=b5=c2=1, ab=ba, cac=a23, cbc=b-1 >
120C2
60C22
60C4
40S3
24D5
30D4
30Q8
20D6
20Dic3
12Dic5
12D10
8D15
15Q16
15D8
10D12
10Dic6
6D20
4D30
15SD32
5D24
3D40
2D60
Smallest permutation representation of C48⋊D5
►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)
(1 185 198 104 65)(2 186 199 105 66)(3 187 200 106 67)(4 188 201 107 68)(5 189 202 108 69)(6 190 203 109 70)(7 191 204 110 71)(8 192 205 111 72)(9 145 206 112 73)(10 146 207 113 74)(11 147 208 114 75)(12 148 209 115 76)(13 149 210 116 77)(14 150 211 117 78)(15 151 212 118 79)(16 152 213 119 80)(17 153 214 120 81)(18 154 215 121 82)(19 155 216 122 83)(20 156 217 123 84)(21 157 218 124 85)(22 158 219 125 86)(23 159 220 126 87)(24 160 221 127 88)(25 161 222 128 89)(26 162 223 129 90)(27 163 224 130 91)(28 164 225 131 92)(29 165 226 132 93)(30 166 227 133 94)(31 167 228 134 95)(32 168 229 135 96)(33 169 230 136 49)(34 170 231 137 50)(35 171 232 138 51)(36 172 233 139 52)(37 173 234 140 53)(38 174 235 141 54)(39 175 236 142 55)(40 176 237 143 56)(41 177 238 144 57)(42 178 239 97 58)(43 179 240 98 59)(44 180 193 99 60)(45 181 194 100 61)(46 182 195 101 62)(47 183 196 102 63)(48 184 197 103 64)
(1 65)(2 88)(3 63)(4 86)(5 61)(6 84)(7 59)(8 82)(9 57)(10 80)(11 55)(12 78)(13 53)(14 76)(15 51)(16 74)(17 49)(18 72)(19 95)(20 70)(21 93)(22 68)(23 91)(24 66)(25 89)(26 64)(27 87)(28 62)(29 85)(30 60)(31 83)(32 58)(33 81)(34 56)(35 79)(36 54)(37 77)(38 52)(39 75)(40 50)(41 73)(42 96)(43 71)(44 94)(45 69)(46 92)(47 67)(48 90)(97 168)(98 191)(99 166)(100 189)(101 164)(102 187)(103 162)(104 185)(105 160)(106 183)(107 158)(108 181)(109 156)(110 179)(111 154)(112 177)(113 152)(114 175)(115 150)(116 173)(117 148)(118 171)(119 146)(120 169)(121 192)(122 167)(123 190)(124 165)(125 188)(126 163)(127 186)(128 161)(129 184)(130 159)(131 182)(132 157)(133 180)(134 155)(135 178)(136 153)(137 176)(138 151)(139 174)(140 149)(141 172)(142 147)(143 170)(144 145)(193 227)(194 202)(195 225)(196 200)(197 223)(199 221)(201 219)(203 217)(204 240)(205 215)(206 238)(207 213)(208 236)(209 211)(210 234)(212 232)(214 230)(216 228)(218 226)(220 224)(229 239)(231 237)(233 235)
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), (1,185,198,104,65)(2,186,199,105,66)(3,187,200,106,67)(4,188,201,107,68)(5,189,202,108,69)(6,190,203,109,70)(7,191,204,110,71)(8,192,205,111,72)(9,145,206,112,73)(10,146,207,113,74)(11,147,208,114,75)(12,148,209,115,76)(13,149,210,116,77)(14,150,211,117,78)(15,151,212,118,79)(16,152,213,119,80)(17,153,214,120,81)(18,154,215,121,82)(19,155,216,122,83)(20,156,217,123,84)(21,157,218,124,85)(22,158,219,125,86)(23,159,220,126,87)(24,160,221,127,88)(25,161,222,128,89)(26,162,223,129,90)(27,163,224,130,91)(28,164,225,131,92)(29,165,226,132,93)(30,166,227,133,94)(31,167,228,134,95)(32,168,229,135,96)(33,169,230,136,49)(34,170,231,137,50)(35,171,232,138,51)(36,172,233,139,52)(37,173,234,140,53)(38,174,235,141,54)(39,175,236,142,55)(40,176,237,143,56)(41,177,238,144,57)(42,178,239,97,58)(43,179,240,98,59)(44,180,193,99,60)(45,181,194,100,61)(46,182,195,101,62)(47,183,196,102,63)(48,184,197,103,64), (1,65)(2,88)(3,63)(4,86)(5,61)(6,84)(7,59)(8,82)(9,57)(10,80)(11,55)(12,78)(13,53)(14,76)(15,51)(16,74)(17,49)(18,72)(19,95)(20,70)(21,93)(22,68)(23,91)(24,66)(25,89)(26,64)(27,87)(28,62)(29,85)(30,60)(31,83)(32,58)(33,81)(34,56)(35,79)(36,54)(37,77)(38,52)(39,75)(40,50)(41,73)(42,96)(43,71)(44,94)(45,69)(46,92)(47,67)(48,90)(97,168)(98,191)(99,166)(100,189)(101,164)(102,187)(103,162)(104,185)(105,160)(106,183)(107,158)(108,181)(109,156)(110,179)(111,154)(112,177)(113,152)(114,175)(115,150)(116,173)(117,148)(118,171)(119,146)(120,169)(121,192)(122,167)(123,190)(124,165)(125,188)(126,163)(127,186)(128,161)(129,184)(130,159)(131,182)(132,157)(133,180)(134,155)(135,178)(136,153)(137,176)(138,151)(139,174)(140,149)(141,172)(142,147)(143,170)(144,145)(193,227)(194,202)(195,225)(196,200)(197,223)(199,221)(201,219)(203,217)(204,240)(205,215)(206,238)(207,213)(208,236)(209,211)(210,234)(212,232)(214,230)(216,228)(218,226)(220,224)(229,239)(231,237)(233,235)>;
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), (1,185,198,104,65)(2,186,199,105,66)(3,187,200,106,67)(4,188,201,107,68)(5,189,202,108,69)(6,190,203,109,70)(7,191,204,110,71)(8,192,205,111,72)(9,145,206,112,73)(10,146,207,113,74)(11,147,208,114,75)(12,148,209,115,76)(13,149,210,116,77)(14,150,211,117,78)(15,151,212,118,79)(16,152,213,119,80)(17,153,214,120,81)(18,154,215,121,82)(19,155,216,122,83)(20,156,217,123,84)(21,157,218,124,85)(22,158,219,125,86)(23,159,220,126,87)(24,160,221,127,88)(25,161,222,128,89)(26,162,223,129,90)(27,163,224,130,91)(28,164,225,131,92)(29,165,226,132,93)(30,166,227,133,94)(31,167,228,134,95)(32,168,229,135,96)(33,169,230,136,49)(34,170,231,137,50)(35,171,232,138,51)(36,172,233,139,52)(37,173,234,140,53)(38,174,235,141,54)(39,175,236,142,55)(40,176,237,143,56)(41,177,238,144,57)(42,178,239,97,58)(43,179,240,98,59)(44,180,193,99,60)(45,181,194,100,61)(46,182,195,101,62)(47,183,196,102,63)(48,184,197,103,64), (1,65)(2,88)(3,63)(4,86)(5,61)(6,84)(7,59)(8,82)(9,57)(10,80)(11,55)(12,78)(13,53)(14,76)(15,51)(16,74)(17,49)(18,72)(19,95)(20,70)(21,93)(22,68)(23,91)(24,66)(25,89)(26,64)(27,87)(28,62)(29,85)(30,60)(31,83)(32,58)(33,81)(34,56)(35,79)(36,54)(37,77)(38,52)(39,75)(40,50)(41,73)(42,96)(43,71)(44,94)(45,69)(46,92)(47,67)(48,90)(97,168)(98,191)(99,166)(100,189)(101,164)(102,187)(103,162)(104,185)(105,160)(106,183)(107,158)(108,181)(109,156)(110,179)(111,154)(112,177)(113,152)(114,175)(115,150)(116,173)(117,148)(118,171)(119,146)(120,169)(121,192)(122,167)(123,190)(124,165)(125,188)(126,163)(127,186)(128,161)(129,184)(130,159)(131,182)(132,157)(133,180)(134,155)(135,178)(136,153)(137,176)(138,151)(139,174)(140,149)(141,172)(142,147)(143,170)(144,145)(193,227)(194,202)(195,225)(196,200)(197,223)(199,221)(201,219)(203,217)(204,240)(205,215)(206,238)(207,213)(208,236)(209,211)(210,234)(212,232)(214,230)(216,228)(218,226)(220,224)(229,239)(231,237)(233,235) );
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)], [(1,185,198,104,65),(2,186,199,105,66),(3,187,200,106,67),(4,188,201,107,68),(5,189,202,108,69),(6,190,203,109,70),(7,191,204,110,71),(8,192,205,111,72),(9,145,206,112,73),(10,146,207,113,74),(11,147,208,114,75),(12,148,209,115,76),(13,149,210,116,77),(14,150,211,117,78),(15,151,212,118,79),(16,152,213,119,80),(17,153,214,120,81),(18,154,215,121,82),(19,155,216,122,83),(20,156,217,123,84),(21,157,218,124,85),(22,158,219,125,86),(23,159,220,126,87),(24,160,221,127,88),(25,161,222,128,89),(26,162,223,129,90),(27,163,224,130,91),(28,164,225,131,92),(29,165,226,132,93),(30,166,227,133,94),(31,167,228,134,95),(32,168,229,135,96),(33,169,230,136,49),(34,170,231,137,50),(35,171,232,138,51),(36,172,233,139,52),(37,173,234,140,53),(38,174,235,141,54),(39,175,236,142,55),(40,176,237,143,56),(41,177,238,144,57),(42,178,239,97,58),(43,179,240,98,59),(44,180,193,99,60),(45,181,194,100,61),(46,182,195,101,62),(47,183,196,102,63),(48,184,197,103,64)], [(1,65),(2,88),(3,63),(4,86),(5,61),(6,84),(7,59),(8,82),(9,57),(10,80),(11,55),(12,78),(13,53),(14,76),(15,51),(16,74),(17,49),(18,72),(19,95),(20,70),(21,93),(22,68),(23,91),(24,66),(25,89),(26,64),(27,87),(28,62),(29,85),(30,60),(31,83),(32,58),(33,81),(34,56),(35,79),(36,54),(37,77),(38,52),(39,75),(40,50),(41,73),(42,96),(43,71),(44,94),(45,69),(46,92),(47,67),(48,90),(97,168),(98,191),(99,166),(100,189),(101,164),(102,187),(103,162),(104,185),(105,160),(106,183),(107,158),(108,181),(109,156),(110,179),(111,154),(112,177),(113,152),(114,175),(115,150),(116,173),(117,148),(118,171),(119,146),(120,169),(121,192),(122,167),(123,190),(124,165),(125,188),(126,163),(127,186),(128,161),(129,184),(130,159),(131,182),(132,157),(133,180),(134,155),(135,178),(136,153),(137,176),(138,151),(139,174),(140,149),(141,172),(142,147),(143,170),(144,145),(193,227),(194,202),(195,225),(196,200),(197,223),(199,221),(201,219),(203,217),(204,240),(205,215),(206,238),(207,213),(208,236),(209,211),(210,234),(212,232),(214,230),(216,228),(218,226),(220,224),(229,239),(231,237),(233,235)]])
123 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 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 | 3 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 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 | 120 | 2 | 2 | 120 | 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 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
123 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D8 | D10 | D12 | D15 | SD32 | D20 | D24 | D30 | D40 | C48⋊C2 | D60 | C16⋊D5 | D120 | C48⋊D5 |
| kernel | C48⋊D5 | C240 | D120 | Dic60 | C80 | C60 | C48 | C40 | C30 | C24 | C20 | C16 | C15 | C12 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of C48⋊D5 ►in GL4(𝔽241) generated by
| 34 | 116 | 0 | 0 |
| 125 | 150 | 0 | 0 |
| 0 | 0 | 113 | 163 |
| 0 | 0 | 78 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 189 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(241))| [34,125,0,0,116,150,0,0,0,0,113,78,0,0,163,72],[1,0,0,0,0,1,0,0,0,0,0,240,0,0,1,189],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C48⋊D5 in GAP, Magma, Sage, TeX
C_{48}\rtimes D_5 % in TeX
G:=Group("C48:D5"); // GroupNames label
G:=SmallGroup(480,160);
// by ID
G=gap.SmallGroup(480,160);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,85,92,590,58,675,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^48=b^5=c^2=1,a*b=b*a,c*a*c=a^23,c*b*c=b^-1>;
// generators/relations
Export