metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊2D48, C15⋊3D16, D24⋊1D5, C30.3D8, C40.5D6, C10.6D24, C60.49D4, C20.1D12, D120⋊10C2, C24.40D10, C120.17C22, C5⋊2C16⋊1S3, (C5×D24)⋊1C2, C3⋊1(C5⋊D16), C8.16(S3×D5), C6.1(D4⋊D5), C2.4(C5⋊D24), C4.1(C5⋊D12), C12.51(C5⋊D4), (C3×C5⋊2C16)⋊1C2, SmallGroup(480,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊D48
G = < a,b,c | a5=b48=c2=1, bab-1=cac=a-1, cbc=b-1 >
24C2
120C2
12C22
60C22
8S3
40S3
24D5
24C10
6D4
30D4
4D6
20D6
12D10
12C2×C10
8D15
3D8
5C16
15D8
2D12
10D12
6D20
4D30
15D16
5C48
5D24
3D40
2D60
5D48
Smallest permutation representation of C5⋊D48
►On 240 points
Generators in S240
(1 148 133 96 236)(2 237 49 134 149)(3 150 135 50 238)(4 239 51 136 151)(5 152 137 52 240)(6 193 53 138 153)(7 154 139 54 194)(8 195 55 140 155)(9 156 141 56 196)(10 197 57 142 157)(11 158 143 58 198)(12 199 59 144 159)(13 160 97 60 200)(14 201 61 98 161)(15 162 99 62 202)(16 203 63 100 163)(17 164 101 64 204)(18 205 65 102 165)(19 166 103 66 206)(20 207 67 104 167)(21 168 105 68 208)(22 209 69 106 169)(23 170 107 70 210)(24 211 71 108 171)(25 172 109 72 212)(26 213 73 110 173)(27 174 111 74 214)(28 215 75 112 175)(29 176 113 76 216)(30 217 77 114 177)(31 178 115 78 218)(32 219 79 116 179)(33 180 117 80 220)(34 221 81 118 181)(35 182 119 82 222)(36 223 83 120 183)(37 184 121 84 224)(38 225 85 122 185)(39 186 123 86 226)(40 227 87 124 187)(41 188 125 88 228)(42 229 89 126 189)(43 190 127 90 230)(44 231 91 128 191)(45 192 129 92 232)(46 233 93 130 145)(47 146 131 94 234)(48 235 95 132 147)
(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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(49 116)(50 115)(51 114)(52 113)(53 112)(54 111)(55 110)(56 109)(57 108)(58 107)(59 106)(60 105)(61 104)(62 103)(63 102)(64 101)(65 100)(66 99)(67 98)(68 97)(69 144)(70 143)(71 142)(72 141)(73 140)(74 139)(75 138)(76 137)(77 136)(78 135)(79 134)(80 133)(81 132)(82 131)(83 130)(84 129)(85 128)(86 127)(87 126)(88 125)(89 124)(90 123)(91 122)(92 121)(93 120)(94 119)(95 118)(96 117)(145 223)(146 222)(147 221)(148 220)(149 219)(150 218)(151 217)(152 216)(153 215)(154 214)(155 213)(156 212)(157 211)(158 210)(159 209)(160 208)(161 207)(162 206)(163 205)(164 204)(165 203)(166 202)(167 201)(168 200)(169 199)(170 198)(171 197)(172 196)(173 195)(174 194)(175 193)(176 240)(177 239)(178 238)(179 237)(180 236)(181 235)(182 234)(183 233)(184 232)(185 231)(186 230)(187 229)(188 228)(189 227)(190 226)(191 225)(192 224)
G:=sub<Sym(240)| (1,148,133,96,236)(2,237,49,134,149)(3,150,135,50,238)(4,239,51,136,151)(5,152,137,52,240)(6,193,53,138,153)(7,154,139,54,194)(8,195,55,140,155)(9,156,141,56,196)(10,197,57,142,157)(11,158,143,58,198)(12,199,59,144,159)(13,160,97,60,200)(14,201,61,98,161)(15,162,99,62,202)(16,203,63,100,163)(17,164,101,64,204)(18,205,65,102,165)(19,166,103,66,206)(20,207,67,104,167)(21,168,105,68,208)(22,209,69,106,169)(23,170,107,70,210)(24,211,71,108,171)(25,172,109,72,212)(26,213,73,110,173)(27,174,111,74,214)(28,215,75,112,175)(29,176,113,76,216)(30,217,77,114,177)(31,178,115,78,218)(32,219,79,116,179)(33,180,117,80,220)(34,221,81,118,181)(35,182,119,82,222)(36,223,83,120,183)(37,184,121,84,224)(38,225,85,122,185)(39,186,123,86,226)(40,227,87,124,187)(41,188,125,88,228)(42,229,89,126,189)(43,190,127,90,230)(44,231,91,128,191)(45,192,129,92,232)(46,233,93,130,145)(47,146,131,94,234)(48,235,95,132,147), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,105)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,144)(70,143)(71,142)(72,141)(73,140)(74,139)(75,138)(76,137)(77,136)(78,135)(79,134)(80,133)(81,132)(82,131)(83,130)(84,129)(85,128)(86,127)(87,126)(88,125)(89,124)(90,123)(91,122)(92,121)(93,120)(94,119)(95,118)(96,117)(145,223)(146,222)(147,221)(148,220)(149,219)(150,218)(151,217)(152,216)(153,215)(154,214)(155,213)(156,212)(157,211)(158,210)(159,209)(160,208)(161,207)(162,206)(163,205)(164,204)(165,203)(166,202)(167,201)(168,200)(169,199)(170,198)(171,197)(172,196)(173,195)(174,194)(175,193)(176,240)(177,239)(178,238)(179,237)(180,236)(181,235)(182,234)(183,233)(184,232)(185,231)(186,230)(187,229)(188,228)(189,227)(190,226)(191,225)(192,224)>;
G:=Group( (1,148,133,96,236)(2,237,49,134,149)(3,150,135,50,238)(4,239,51,136,151)(5,152,137,52,240)(6,193,53,138,153)(7,154,139,54,194)(8,195,55,140,155)(9,156,141,56,196)(10,197,57,142,157)(11,158,143,58,198)(12,199,59,144,159)(13,160,97,60,200)(14,201,61,98,161)(15,162,99,62,202)(16,203,63,100,163)(17,164,101,64,204)(18,205,65,102,165)(19,166,103,66,206)(20,207,67,104,167)(21,168,105,68,208)(22,209,69,106,169)(23,170,107,70,210)(24,211,71,108,171)(25,172,109,72,212)(26,213,73,110,173)(27,174,111,74,214)(28,215,75,112,175)(29,176,113,76,216)(30,217,77,114,177)(31,178,115,78,218)(32,219,79,116,179)(33,180,117,80,220)(34,221,81,118,181)(35,182,119,82,222)(36,223,83,120,183)(37,184,121,84,224)(38,225,85,122,185)(39,186,123,86,226)(40,227,87,124,187)(41,188,125,88,228)(42,229,89,126,189)(43,190,127,90,230)(44,231,91,128,191)(45,192,129,92,232)(46,233,93,130,145)(47,146,131,94,234)(48,235,95,132,147), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,105)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,144)(70,143)(71,142)(72,141)(73,140)(74,139)(75,138)(76,137)(77,136)(78,135)(79,134)(80,133)(81,132)(82,131)(83,130)(84,129)(85,128)(86,127)(87,126)(88,125)(89,124)(90,123)(91,122)(92,121)(93,120)(94,119)(95,118)(96,117)(145,223)(146,222)(147,221)(148,220)(149,219)(150,218)(151,217)(152,216)(153,215)(154,214)(155,213)(156,212)(157,211)(158,210)(159,209)(160,208)(161,207)(162,206)(163,205)(164,204)(165,203)(166,202)(167,201)(168,200)(169,199)(170,198)(171,197)(172,196)(173,195)(174,194)(175,193)(176,240)(177,239)(178,238)(179,237)(180,236)(181,235)(182,234)(183,233)(184,232)(185,231)(186,230)(187,229)(188,228)(189,227)(190,226)(191,225)(192,224) );
G=PermutationGroup([[(1,148,133,96,236),(2,237,49,134,149),(3,150,135,50,238),(4,239,51,136,151),(5,152,137,52,240),(6,193,53,138,153),(7,154,139,54,194),(8,195,55,140,155),(9,156,141,56,196),(10,197,57,142,157),(11,158,143,58,198),(12,199,59,144,159),(13,160,97,60,200),(14,201,61,98,161),(15,162,99,62,202),(16,203,63,100,163),(17,164,101,64,204),(18,205,65,102,165),(19,166,103,66,206),(20,207,67,104,167),(21,168,105,68,208),(22,209,69,106,169),(23,170,107,70,210),(24,211,71,108,171),(25,172,109,72,212),(26,213,73,110,173),(27,174,111,74,214),(28,215,75,112,175),(29,176,113,76,216),(30,217,77,114,177),(31,178,115,78,218),(32,219,79,116,179),(33,180,117,80,220),(34,221,81,118,181),(35,182,119,82,222),(36,223,83,120,183),(37,184,121,84,224),(38,225,85,122,185),(39,186,123,86,226),(40,227,87,124,187),(41,188,125,88,228),(42,229,89,126,189),(43,190,127,90,230),(44,231,91,128,191),(45,192,129,92,232),(46,233,93,130,145),(47,146,131,94,234),(48,235,95,132,147)], [(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)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(49,116),(50,115),(51,114),(52,113),(53,112),(54,111),(55,110),(56,109),(57,108),(58,107),(59,106),(60,105),(61,104),(62,103),(63,102),(64,101),(65,100),(66,99),(67,98),(68,97),(69,144),(70,143),(71,142),(72,141),(73,140),(74,139),(75,138),(76,137),(77,136),(78,135),(79,134),(80,133),(81,132),(82,131),(83,130),(84,129),(85,128),(86,127),(87,126),(88,125),(89,124),(90,123),(91,122),(92,121),(93,120),(94,119),(95,118),(96,117),(145,223),(146,222),(147,221),(148,220),(149,219),(150,218),(151,217),(152,216),(153,215),(154,214),(155,213),(156,212),(157,211),(158,210),(159,209),(160,208),(161,207),(162,206),(163,205),(164,204),(165,203),(166,202),(167,201),(168,200),(169,199),(170,198),(171,197),(172,196),(173,195),(174,194),(175,193),(176,240),(177,239),(178,238),(179,237),(180,236),(181,235),(182,234),(183,233),(184,232),(185,231),(186,230),(187,229),(188,228),(189,227),(190,226),(191,225),(192,224)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 15A | 15B | 16A | 16B | 16C | 16D | 20A | 20B | 24A | 24B | 24C | 24D | 30A | 30B | 40A | 40B | 40C | 40D | 48A | ··· | 48H | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 48 | ··· | 48 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 24 | 120 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 24 | 24 | 24 | 24 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D8 | D10 | D12 | D16 | C5⋊D4 | D24 | D48 | S3×D5 | D4⋊D5 | C5⋊D12 | C5⋊D16 | C5⋊D24 | C5⋊D48 |
| kernel | C5⋊D48 | C3×C5⋊2C16 | C5×D24 | D120 | C5⋊2C16 | C60 | D24 | C40 | C30 | C24 | C20 | C15 | C12 | C10 | C5 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C5⋊D48 ►in GL4(𝔽241) generated by
| 240 | 1 | 0 | 0 |
| 50 | 190 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 172 | 214 | 0 | 0 |
| 221 | 69 | 0 | 0 |
| 0 | 0 | 222 | 28 |
| 0 | 0 | 117 | 170 |
| 51 | 1 | 0 | 0 |
| 51 | 190 | 0 | 0 |
| 0 | 0 | 239 | 54 |
| 0 | 0 | 174 | 2 |
G:=sub<GL(4,GF(241))| [240,50,0,0,1,190,0,0,0,0,1,0,0,0,0,1],[172,221,0,0,214,69,0,0,0,0,222,117,0,0,28,170],[51,51,0,0,1,190,0,0,0,0,239,174,0,0,54,2] >;
C5⋊D48 in GAP, Magma, Sage, TeX
C_5\rtimes D_{48} % in TeX
G:=Group("C5:D48"); // GroupNames label
G:=SmallGroup(480,15);
// by ID
G=gap.SmallGroup(480,15);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,85,135,142,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^5=b^48=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export