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G = C5×C8.6D6order 480 = 25·3·5

Direct product of C5 and C8.6D6

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5×C8.6D6, C30.55D8, C40.58D6, C1515SD32, D24.2C10, C60.138D4, C120.65C22, C3⋊C163C10, C33(C5×SD32), C8.6(S3×C10), Q161(C5×S3), (C5×Q16)⋊5S3, C12.5(C5×D4), C6.10(C5×D8), C24.4(C2×C10), (C3×Q16)⋊1C10, (C15×Q16)⋊8C2, (C5×D24).4C2, C10.26(D4⋊S3), C20.68(C3⋊D4), (C5×C3⋊C16)⋊10C2, C2.6(C5×D4⋊S3), C4.3(C5×C3⋊D4), SmallGroup(480,147)

Series: Derived Chief Lower central Upper central

C1C24 — C5×C8.6D6
C1C3C6C12C24C120C5×D24 — C5×C8.6D6
C3C6C12C24 — C5×C8.6D6
C1C10C20C40C5×Q16

Generators and relations for C5×C8.6D6
 G = < a,b,c,d | a5=b8=1, c6=b4, d2=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c5 >

24C2
4C4
12C22
8S3
24C10
2Q8
6D4
4C12
4D6
4C20
12C2×C10
8C5×S3
3D8
3C16
2C3×Q8
2D12
2C5×Q8
6C5×D4
4S3×C10
4C60
3SD32
3C5×D8
3C80
2Q8×C15
2C5×D12
3C5×SD32

Smallest permutation representation of C5×C8.6D6
On 240 points
Generators in S240
(1 144 108 48 84)(2 129 109 33 85)(3 130 110 34 86)(4 131 111 35 87)(5 132 112 36 88)(6 133 97 37 89)(7 134 98 38 90)(8 135 99 39 91)(9 136 100 40 92)(10 137 101 41 93)(11 138 102 42 94)(12 139 103 43 95)(13 140 104 44 96)(14 141 105 45 81)(15 142 106 46 82)(16 143 107 47 83)(17 75 211 116 185)(18 76 212 117 186)(19 77 213 118 187)(20 78 214 119 188)(21 79 215 120 189)(22 80 216 121 190)(23 65 217 122 191)(24 66 218 123 192)(25 67 219 124 177)(26 68 220 125 178)(27 69 221 126 179)(28 70 222 127 180)(29 71 223 128 181)(30 72 224 113 182)(31 73 209 114 183)(32 74 210 115 184)(49 165 203 231 147)(50 166 204 232 148)(51 167 205 233 149)(52 168 206 234 150)(53 169 207 235 151)(54 170 208 236 152)(55 171 193 237 153)(56 172 194 238 154)(57 173 195 239 155)(58 174 196 240 156)(59 175 197 225 157)(60 176 198 226 158)(61 161 199 227 159)(62 162 200 228 160)(63 163 201 229 145)(64 164 202 230 146)
(1 7 13 3 9 15 5 11)(2 8 14 4 10 16 6 12)(17 23 29 19 25 31 21 27)(18 24 30 20 26 32 22 28)(33 39 45 35 41 47 37 43)(34 40 46 36 42 48 38 44)(49 55 61 51 57 63 53 59)(50 56 62 52 58 64 54 60)(65 71 77 67 73 79 69 75)(66 72 78 68 74 80 70 76)(81 87 93 83 89 95 85 91)(82 88 94 84 90 96 86 92)(97 103 109 99 105 111 101 107)(98 104 110 100 106 112 102 108)(113 119 125 115 121 127 117 123)(114 120 126 116 122 128 118 124)(129 135 141 131 137 143 133 139)(130 136 142 132 138 144 134 140)(145 151 157 147 153 159 149 155)(146 152 158 148 154 160 150 156)(161 167 173 163 169 175 165 171)(162 168 174 164 170 176 166 172)(177 183 189 179 185 191 181 187)(178 184 190 180 186 192 182 188)(193 199 205 195 201 207 197 203)(194 200 206 196 202 208 198 204)(209 215 221 211 217 223 213 219)(210 216 222 212 218 224 214 220)(225 231 237 227 233 239 229 235)(226 232 238 228 234 240 230 236)
(1 66 168 169 65 2 9 74 176 161 73 10)(3 80 170 167 67 16 11 72 162 175 75 8)(4 7 76 174 163 71 12 15 68 166 171 79)(5 78 172 165 69 14 13 70 164 173 77 6)(17 91 86 22 54 51 25 83 94 30 62 59)(18 58 63 29 95 82 26 50 55 21 87 90)(19 89 88 20 56 49 27 81 96 28 64 57)(23 85 92 32 60 61 31 93 84 24 52 53)(33 40 184 158 159 183 41 48 192 150 151 191)(34 190 152 149 177 47 42 182 160 157 185 39)(35 38 186 156 145 181 43 46 178 148 153 189)(36 188 154 147 179 45 44 180 146 155 187 37)(97 112 119 238 231 126 105 104 127 230 239 118)(98 117 240 229 128 103 106 125 232 237 120 111)(99 110 121 236 233 124 107 102 113 228 225 116)(100 115 226 227 114 101 108 123 234 235 122 109)(129 136 210 198 199 209 137 144 218 206 207 217)(130 216 208 205 219 143 138 224 200 197 211 135)(131 134 212 196 201 223 139 142 220 204 193 215)(132 214 194 203 221 141 140 222 202 195 213 133)
(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)

G:=sub<Sym(240)| (1,144,108,48,84)(2,129,109,33,85)(3,130,110,34,86)(4,131,111,35,87)(5,132,112,36,88)(6,133,97,37,89)(7,134,98,38,90)(8,135,99,39,91)(9,136,100,40,92)(10,137,101,41,93)(11,138,102,42,94)(12,139,103,43,95)(13,140,104,44,96)(14,141,105,45,81)(15,142,106,46,82)(16,143,107,47,83)(17,75,211,116,185)(18,76,212,117,186)(19,77,213,118,187)(20,78,214,119,188)(21,79,215,120,189)(22,80,216,121,190)(23,65,217,122,191)(24,66,218,123,192)(25,67,219,124,177)(26,68,220,125,178)(27,69,221,126,179)(28,70,222,127,180)(29,71,223,128,181)(30,72,224,113,182)(31,73,209,114,183)(32,74,210,115,184)(49,165,203,231,147)(50,166,204,232,148)(51,167,205,233,149)(52,168,206,234,150)(53,169,207,235,151)(54,170,208,236,152)(55,171,193,237,153)(56,172,194,238,154)(57,173,195,239,155)(58,174,196,240,156)(59,175,197,225,157)(60,176,198,226,158)(61,161,199,227,159)(62,162,200,228,160)(63,163,201,229,145)(64,164,202,230,146), (1,7,13,3,9,15,5,11)(2,8,14,4,10,16,6,12)(17,23,29,19,25,31,21,27)(18,24,30,20,26,32,22,28)(33,39,45,35,41,47,37,43)(34,40,46,36,42,48,38,44)(49,55,61,51,57,63,53,59)(50,56,62,52,58,64,54,60)(65,71,77,67,73,79,69,75)(66,72,78,68,74,80,70,76)(81,87,93,83,89,95,85,91)(82,88,94,84,90,96,86,92)(97,103,109,99,105,111,101,107)(98,104,110,100,106,112,102,108)(113,119,125,115,121,127,117,123)(114,120,126,116,122,128,118,124)(129,135,141,131,137,143,133,139)(130,136,142,132,138,144,134,140)(145,151,157,147,153,159,149,155)(146,152,158,148,154,160,150,156)(161,167,173,163,169,175,165,171)(162,168,174,164,170,176,166,172)(177,183,189,179,185,191,181,187)(178,184,190,180,186,192,182,188)(193,199,205,195,201,207,197,203)(194,200,206,196,202,208,198,204)(209,215,221,211,217,223,213,219)(210,216,222,212,218,224,214,220)(225,231,237,227,233,239,229,235)(226,232,238,228,234,240,230,236), (1,66,168,169,65,2,9,74,176,161,73,10)(3,80,170,167,67,16,11,72,162,175,75,8)(4,7,76,174,163,71,12,15,68,166,171,79)(5,78,172,165,69,14,13,70,164,173,77,6)(17,91,86,22,54,51,25,83,94,30,62,59)(18,58,63,29,95,82,26,50,55,21,87,90)(19,89,88,20,56,49,27,81,96,28,64,57)(23,85,92,32,60,61,31,93,84,24,52,53)(33,40,184,158,159,183,41,48,192,150,151,191)(34,190,152,149,177,47,42,182,160,157,185,39)(35,38,186,156,145,181,43,46,178,148,153,189)(36,188,154,147,179,45,44,180,146,155,187,37)(97,112,119,238,231,126,105,104,127,230,239,118)(98,117,240,229,128,103,106,125,232,237,120,111)(99,110,121,236,233,124,107,102,113,228,225,116)(100,115,226,227,114,101,108,123,234,235,122,109)(129,136,210,198,199,209,137,144,218,206,207,217)(130,216,208,205,219,143,138,224,200,197,211,135)(131,134,212,196,201,223,139,142,220,204,193,215)(132,214,194,203,221,141,140,222,202,195,213,133), (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)>;

G:=Group( (1,144,108,48,84)(2,129,109,33,85)(3,130,110,34,86)(4,131,111,35,87)(5,132,112,36,88)(6,133,97,37,89)(7,134,98,38,90)(8,135,99,39,91)(9,136,100,40,92)(10,137,101,41,93)(11,138,102,42,94)(12,139,103,43,95)(13,140,104,44,96)(14,141,105,45,81)(15,142,106,46,82)(16,143,107,47,83)(17,75,211,116,185)(18,76,212,117,186)(19,77,213,118,187)(20,78,214,119,188)(21,79,215,120,189)(22,80,216,121,190)(23,65,217,122,191)(24,66,218,123,192)(25,67,219,124,177)(26,68,220,125,178)(27,69,221,126,179)(28,70,222,127,180)(29,71,223,128,181)(30,72,224,113,182)(31,73,209,114,183)(32,74,210,115,184)(49,165,203,231,147)(50,166,204,232,148)(51,167,205,233,149)(52,168,206,234,150)(53,169,207,235,151)(54,170,208,236,152)(55,171,193,237,153)(56,172,194,238,154)(57,173,195,239,155)(58,174,196,240,156)(59,175,197,225,157)(60,176,198,226,158)(61,161,199,227,159)(62,162,200,228,160)(63,163,201,229,145)(64,164,202,230,146), (1,7,13,3,9,15,5,11)(2,8,14,4,10,16,6,12)(17,23,29,19,25,31,21,27)(18,24,30,20,26,32,22,28)(33,39,45,35,41,47,37,43)(34,40,46,36,42,48,38,44)(49,55,61,51,57,63,53,59)(50,56,62,52,58,64,54,60)(65,71,77,67,73,79,69,75)(66,72,78,68,74,80,70,76)(81,87,93,83,89,95,85,91)(82,88,94,84,90,96,86,92)(97,103,109,99,105,111,101,107)(98,104,110,100,106,112,102,108)(113,119,125,115,121,127,117,123)(114,120,126,116,122,128,118,124)(129,135,141,131,137,143,133,139)(130,136,142,132,138,144,134,140)(145,151,157,147,153,159,149,155)(146,152,158,148,154,160,150,156)(161,167,173,163,169,175,165,171)(162,168,174,164,170,176,166,172)(177,183,189,179,185,191,181,187)(178,184,190,180,186,192,182,188)(193,199,205,195,201,207,197,203)(194,200,206,196,202,208,198,204)(209,215,221,211,217,223,213,219)(210,216,222,212,218,224,214,220)(225,231,237,227,233,239,229,235)(226,232,238,228,234,240,230,236), (1,66,168,169,65,2,9,74,176,161,73,10)(3,80,170,167,67,16,11,72,162,175,75,8)(4,7,76,174,163,71,12,15,68,166,171,79)(5,78,172,165,69,14,13,70,164,173,77,6)(17,91,86,22,54,51,25,83,94,30,62,59)(18,58,63,29,95,82,26,50,55,21,87,90)(19,89,88,20,56,49,27,81,96,28,64,57)(23,85,92,32,60,61,31,93,84,24,52,53)(33,40,184,158,159,183,41,48,192,150,151,191)(34,190,152,149,177,47,42,182,160,157,185,39)(35,38,186,156,145,181,43,46,178,148,153,189)(36,188,154,147,179,45,44,180,146,155,187,37)(97,112,119,238,231,126,105,104,127,230,239,118)(98,117,240,229,128,103,106,125,232,237,120,111)(99,110,121,236,233,124,107,102,113,228,225,116)(100,115,226,227,114,101,108,123,234,235,122,109)(129,136,210,198,199,209,137,144,218,206,207,217)(130,216,208,205,219,143,138,224,200,197,211,135)(131,134,212,196,201,223,139,142,220,204,193,215)(132,214,194,203,221,141,140,222,202,195,213,133), (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) );

G=PermutationGroup([(1,144,108,48,84),(2,129,109,33,85),(3,130,110,34,86),(4,131,111,35,87),(5,132,112,36,88),(6,133,97,37,89),(7,134,98,38,90),(8,135,99,39,91),(9,136,100,40,92),(10,137,101,41,93),(11,138,102,42,94),(12,139,103,43,95),(13,140,104,44,96),(14,141,105,45,81),(15,142,106,46,82),(16,143,107,47,83),(17,75,211,116,185),(18,76,212,117,186),(19,77,213,118,187),(20,78,214,119,188),(21,79,215,120,189),(22,80,216,121,190),(23,65,217,122,191),(24,66,218,123,192),(25,67,219,124,177),(26,68,220,125,178),(27,69,221,126,179),(28,70,222,127,180),(29,71,223,128,181),(30,72,224,113,182),(31,73,209,114,183),(32,74,210,115,184),(49,165,203,231,147),(50,166,204,232,148),(51,167,205,233,149),(52,168,206,234,150),(53,169,207,235,151),(54,170,208,236,152),(55,171,193,237,153),(56,172,194,238,154),(57,173,195,239,155),(58,174,196,240,156),(59,175,197,225,157),(60,176,198,226,158),(61,161,199,227,159),(62,162,200,228,160),(63,163,201,229,145),(64,164,202,230,146)], [(1,7,13,3,9,15,5,11),(2,8,14,4,10,16,6,12),(17,23,29,19,25,31,21,27),(18,24,30,20,26,32,22,28),(33,39,45,35,41,47,37,43),(34,40,46,36,42,48,38,44),(49,55,61,51,57,63,53,59),(50,56,62,52,58,64,54,60),(65,71,77,67,73,79,69,75),(66,72,78,68,74,80,70,76),(81,87,93,83,89,95,85,91),(82,88,94,84,90,96,86,92),(97,103,109,99,105,111,101,107),(98,104,110,100,106,112,102,108),(113,119,125,115,121,127,117,123),(114,120,126,116,122,128,118,124),(129,135,141,131,137,143,133,139),(130,136,142,132,138,144,134,140),(145,151,157,147,153,159,149,155),(146,152,158,148,154,160,150,156),(161,167,173,163,169,175,165,171),(162,168,174,164,170,176,166,172),(177,183,189,179,185,191,181,187),(178,184,190,180,186,192,182,188),(193,199,205,195,201,207,197,203),(194,200,206,196,202,208,198,204),(209,215,221,211,217,223,213,219),(210,216,222,212,218,224,214,220),(225,231,237,227,233,239,229,235),(226,232,238,228,234,240,230,236)], [(1,66,168,169,65,2,9,74,176,161,73,10),(3,80,170,167,67,16,11,72,162,175,75,8),(4,7,76,174,163,71,12,15,68,166,171,79),(5,78,172,165,69,14,13,70,164,173,77,6),(17,91,86,22,54,51,25,83,94,30,62,59),(18,58,63,29,95,82,26,50,55,21,87,90),(19,89,88,20,56,49,27,81,96,28,64,57),(23,85,92,32,60,61,31,93,84,24,52,53),(33,40,184,158,159,183,41,48,192,150,151,191),(34,190,152,149,177,47,42,182,160,157,185,39),(35,38,186,156,145,181,43,46,178,148,153,189),(36,188,154,147,179,45,44,180,146,155,187,37),(97,112,119,238,231,126,105,104,127,230,239,118),(98,117,240,229,128,103,106,125,232,237,120,111),(99,110,121,236,233,124,107,102,113,228,225,116),(100,115,226,227,114,101,108,123,234,235,122,109),(129,136,210,198,199,209,137,144,218,206,207,217),(130,216,208,205,219,143,138,224,200,197,211,135),(131,134,212,196,201,223,139,142,220,204,193,215),(132,214,194,203,221,141,140,222,202,195,213,133)], [(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)])

90 conjugacy classes

class 1 2A2B 3 4A4B5A5B5C5D 6 8A8B10A10B10C10D10E10F10G10H12A12B12C15A15B15C15D16A16B16C16D20A20B20C20D20E20F20G20H24A24B30A30B30C30D40A···40H60A60B60C60D60E···60L80A···80P120A···120H
order122344555568810101010101010101212121515151516161616202020202020202024243030303040···406060606060···6080···80120···120
size1124228111122211112424242448822226666222288884422222···244448···86···64···4

90 irreducible representations

dim111111112222222222224444
type++++++++++
imageC1C2C2C2C5C10C10C10S3D4D6D8C3⋊D4C5×S3SD32C5×D4S3×C10C5×D8C5×C3⋊D4C5×SD32D4⋊S3C8.6D6C5×D4⋊S3C5×C8.6D6
kernelC5×C8.6D6C5×C3⋊C16C5×D24C15×Q16C8.6D6C3⋊C16D24C3×Q16C5×Q16C60C40C30C20Q16C15C12C8C6C4C3C10C5C2C1
# reps1111444411122444488161248

Matrix representation of C5×C8.6D6 in GL4(𝔽241) generated by

1000
0100
00910
00091
,
1000
0100
0021922
002300
,
0100
24024000
006282
00103179
,
0100
1000
006282
00200144
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,91,0,0,0,0,91],[1,0,0,0,0,1,0,0,0,0,219,230,0,0,22,0],[0,240,0,0,1,240,0,0,0,0,62,103,0,0,82,179],[0,1,0,0,1,0,0,0,0,0,62,200,0,0,82,144] >;

C5×C8.6D6 in GAP, Magma, Sage, TeX

C_5\times C_8._6D_6
% in TeX

G:=Group("C5xC8.6D6");
// GroupNames label

G:=SmallGroup(480,147);
// by ID

G=gap.SmallGroup(480,147);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,309,568,1683,850,192,4204,2111,102,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=1,c^6=b^4,d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^5>;
// generators/relations

Export

Subgroup lattice of C5×C8.6D6 in TeX

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