metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.32D4, C6.10D40, C30.14D8, D20⋊4Dic3, C30.2SD16, (C3×D20)⋊7C4, C12.2(C4×D5), C60.90(C2×C4), (C2×D20).1S3, C60⋊5C4⋊22C2, (C6×D20).2C2, (C2×C30).12D4, (C2×C6).29D20, C4.1(D5×Dic3), C15⋊4(D4⋊C4), C3⋊3(D20⋊5C4), C10.5(D4⋊S3), (C2×C12).48D10, (C2×C20).279D6, C6.6(C40⋊C2), C2.1(C3⋊D40), C5⋊2(D4⋊Dic3), C4.21(C15⋊D4), C20.79(C3⋊D4), C12.11(C5⋊D4), C10.1(D4.S3), (C2×C60).98C22, C20.37(C2×Dic3), C30.48(C22⋊C4), C2.1(C6.D20), C6.27(D10⋊C4), C2.6(D10⋊Dic3), C22.12(C3⋊D20), C10.16(C6.D4), (C2×C3⋊C8)⋊1D5, (C10×C3⋊C8)⋊4C2, (C2×C4).82(S3×D5), (C2×C10).23(C3⋊D4), SmallGroup(480,41)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.D40
G = < a,b,c | a6=b40=1, c2=a3, bab-1=cac-1=a-1, cbc-1=a3b-1 >
Subgroups: 572 in 100 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, C12, C2×C6, C2×C6, C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20, D10, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C3×D4, C22×C6, C3×D5, C30, D4⋊C4, C40, D20, D20, C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8, C4⋊Dic3, C6×D4, Dic15, C60, C6×D5, C2×C30, C4⋊Dic5, C2×C40, C2×D20, D4⋊Dic3, C5×C3⋊C8, C3×D20, C3×D20, C2×Dic15, C2×C60, D5×C2×C6, D20⋊5C4, C10×C3⋊C8, C60⋊5C4, C6×D20, C6.D40
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, D8, SD16, D10, C2×Dic3, C3⋊D4, D4⋊C4, C4×D5, D20, C5⋊D4, D4⋊S3, D4.S3, C6.D4, S3×D5, C40⋊C2, D40, D10⋊C4, D4⋊Dic3, D5×Dic3, C15⋊D4, C3⋊D20, D20⋊5C4, C3⋊D40, C6.D20, D10⋊Dic3, C6.D40
►On 240 points
Generators in S240
(1 47 85 148 175 236)(2 237 176 149 86 48)(3 49 87 150 177 238)(4 239 178 151 88 50)(5 51 89 152 179 240)(6 201 180 153 90 52)(7 53 91 154 181 202)(8 203 182 155 92 54)(9 55 93 156 183 204)(10 205 184 157 94 56)(11 57 95 158 185 206)(12 207 186 159 96 58)(13 59 97 160 187 208)(14 209 188 121 98 60)(15 61 99 122 189 210)(16 211 190 123 100 62)(17 63 101 124 191 212)(18 213 192 125 102 64)(19 65 103 126 193 214)(20 215 194 127 104 66)(21 67 105 128 195 216)(22 217 196 129 106 68)(23 69 107 130 197 218)(24 219 198 131 108 70)(25 71 109 132 199 220)(26 221 200 133 110 72)(27 73 111 134 161 222)(28 223 162 135 112 74)(29 75 113 136 163 224)(30 225 164 137 114 76)(31 77 115 138 165 226)(32 227 166 139 116 78)(33 79 117 140 167 228)(34 229 168 141 118 80)(35 41 119 142 169 230)(36 231 170 143 120 42)(37 43 81 144 171 232)(38 233 172 145 82 44)(39 45 83 146 173 234)(40 235 174 147 84 46)
(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)
(1 147 148 40)(2 39 149 146)(3 145 150 38)(4 37 151 144)(5 143 152 36)(6 35 153 142)(7 141 154 34)(8 33 155 140)(9 139 156 32)(10 31 157 138)(11 137 158 30)(12 29 159 136)(13 135 160 28)(14 27 121 134)(15 133 122 26)(16 25 123 132)(17 131 124 24)(18 23 125 130)(19 129 126 22)(20 21 127 128)(41 180 169 52)(42 51 170 179)(43 178 171 50)(44 49 172 177)(45 176 173 48)(46 47 174 175)(53 168 181 80)(54 79 182 167)(55 166 183 78)(56 77 184 165)(57 164 185 76)(58 75 186 163)(59 162 187 74)(60 73 188 161)(61 200 189 72)(62 71 190 199)(63 198 191 70)(64 69 192 197)(65 196 193 68)(66 67 194 195)(81 239 232 88)(82 87 233 238)(83 237 234 86)(84 85 235 236)(89 231 240 120)(90 119 201 230)(91 229 202 118)(92 117 203 228)(93 227 204 116)(94 115 205 226)(95 225 206 114)(96 113 207 224)(97 223 208 112)(98 111 209 222)(99 221 210 110)(100 109 211 220)(101 219 212 108)(102 107 213 218)(103 217 214 106)(104 105 215 216)
G:=sub<Sym(240)| (1,47,85,148,175,236)(2,237,176,149,86,48)(3,49,87,150,177,238)(4,239,178,151,88,50)(5,51,89,152,179,240)(6,201,180,153,90,52)(7,53,91,154,181,202)(8,203,182,155,92,54)(9,55,93,156,183,204)(10,205,184,157,94,56)(11,57,95,158,185,206)(12,207,186,159,96,58)(13,59,97,160,187,208)(14,209,188,121,98,60)(15,61,99,122,189,210)(16,211,190,123,100,62)(17,63,101,124,191,212)(18,213,192,125,102,64)(19,65,103,126,193,214)(20,215,194,127,104,66)(21,67,105,128,195,216)(22,217,196,129,106,68)(23,69,107,130,197,218)(24,219,198,131,108,70)(25,71,109,132,199,220)(26,221,200,133,110,72)(27,73,111,134,161,222)(28,223,162,135,112,74)(29,75,113,136,163,224)(30,225,164,137,114,76)(31,77,115,138,165,226)(32,227,166,139,116,78)(33,79,117,140,167,228)(34,229,168,141,118,80)(35,41,119,142,169,230)(36,231,170,143,120,42)(37,43,81,144,171,232)(38,233,172,145,82,44)(39,45,83,146,173,234)(40,235,174,147,84,46), (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), (1,147,148,40)(2,39,149,146)(3,145,150,38)(4,37,151,144)(5,143,152,36)(6,35,153,142)(7,141,154,34)(8,33,155,140)(9,139,156,32)(10,31,157,138)(11,137,158,30)(12,29,159,136)(13,135,160,28)(14,27,121,134)(15,133,122,26)(16,25,123,132)(17,131,124,24)(18,23,125,130)(19,129,126,22)(20,21,127,128)(41,180,169,52)(42,51,170,179)(43,178,171,50)(44,49,172,177)(45,176,173,48)(46,47,174,175)(53,168,181,80)(54,79,182,167)(55,166,183,78)(56,77,184,165)(57,164,185,76)(58,75,186,163)(59,162,187,74)(60,73,188,161)(61,200,189,72)(62,71,190,199)(63,198,191,70)(64,69,192,197)(65,196,193,68)(66,67,194,195)(81,239,232,88)(82,87,233,238)(83,237,234,86)(84,85,235,236)(89,231,240,120)(90,119,201,230)(91,229,202,118)(92,117,203,228)(93,227,204,116)(94,115,205,226)(95,225,206,114)(96,113,207,224)(97,223,208,112)(98,111,209,222)(99,221,210,110)(100,109,211,220)(101,219,212,108)(102,107,213,218)(103,217,214,106)(104,105,215,216)>;
G:=Group( (1,47,85,148,175,236)(2,237,176,149,86,48)(3,49,87,150,177,238)(4,239,178,151,88,50)(5,51,89,152,179,240)(6,201,180,153,90,52)(7,53,91,154,181,202)(8,203,182,155,92,54)(9,55,93,156,183,204)(10,205,184,157,94,56)(11,57,95,158,185,206)(12,207,186,159,96,58)(13,59,97,160,187,208)(14,209,188,121,98,60)(15,61,99,122,189,210)(16,211,190,123,100,62)(17,63,101,124,191,212)(18,213,192,125,102,64)(19,65,103,126,193,214)(20,215,194,127,104,66)(21,67,105,128,195,216)(22,217,196,129,106,68)(23,69,107,130,197,218)(24,219,198,131,108,70)(25,71,109,132,199,220)(26,221,200,133,110,72)(27,73,111,134,161,222)(28,223,162,135,112,74)(29,75,113,136,163,224)(30,225,164,137,114,76)(31,77,115,138,165,226)(32,227,166,139,116,78)(33,79,117,140,167,228)(34,229,168,141,118,80)(35,41,119,142,169,230)(36,231,170,143,120,42)(37,43,81,144,171,232)(38,233,172,145,82,44)(39,45,83,146,173,234)(40,235,174,147,84,46), (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), (1,147,148,40)(2,39,149,146)(3,145,150,38)(4,37,151,144)(5,143,152,36)(6,35,153,142)(7,141,154,34)(8,33,155,140)(9,139,156,32)(10,31,157,138)(11,137,158,30)(12,29,159,136)(13,135,160,28)(14,27,121,134)(15,133,122,26)(16,25,123,132)(17,131,124,24)(18,23,125,130)(19,129,126,22)(20,21,127,128)(41,180,169,52)(42,51,170,179)(43,178,171,50)(44,49,172,177)(45,176,173,48)(46,47,174,175)(53,168,181,80)(54,79,182,167)(55,166,183,78)(56,77,184,165)(57,164,185,76)(58,75,186,163)(59,162,187,74)(60,73,188,161)(61,200,189,72)(62,71,190,199)(63,198,191,70)(64,69,192,197)(65,196,193,68)(66,67,194,195)(81,239,232,88)(82,87,233,238)(83,237,234,86)(84,85,235,236)(89,231,240,120)(90,119,201,230)(91,229,202,118)(92,117,203,228)(93,227,204,116)(94,115,205,226)(95,225,206,114)(96,113,207,224)(97,223,208,112)(98,111,209,222)(99,221,210,110)(100,109,211,220)(101,219,212,108)(102,107,213,218)(103,217,214,106)(104,105,215,216) );
G=PermutationGroup([[(1,47,85,148,175,236),(2,237,176,149,86,48),(3,49,87,150,177,238),(4,239,178,151,88,50),(5,51,89,152,179,240),(6,201,180,153,90,52),(7,53,91,154,181,202),(8,203,182,155,92,54),(9,55,93,156,183,204),(10,205,184,157,94,56),(11,57,95,158,185,206),(12,207,186,159,96,58),(13,59,97,160,187,208),(14,209,188,121,98,60),(15,61,99,122,189,210),(16,211,190,123,100,62),(17,63,101,124,191,212),(18,213,192,125,102,64),(19,65,103,126,193,214),(20,215,194,127,104,66),(21,67,105,128,195,216),(22,217,196,129,106,68),(23,69,107,130,197,218),(24,219,198,131,108,70),(25,71,109,132,199,220),(26,221,200,133,110,72),(27,73,111,134,161,222),(28,223,162,135,112,74),(29,75,113,136,163,224),(30,225,164,137,114,76),(31,77,115,138,165,226),(32,227,166,139,116,78),(33,79,117,140,167,228),(34,229,168,141,118,80),(35,41,119,142,169,230),(36,231,170,143,120,42),(37,43,81,144,171,232),(38,233,172,145,82,44),(39,45,83,146,173,234),(40,235,174,147,84,46)], [(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)], [(1,147,148,40),(2,39,149,146),(3,145,150,38),(4,37,151,144),(5,143,152,36),(6,35,153,142),(7,141,154,34),(8,33,155,140),(9,139,156,32),(10,31,157,138),(11,137,158,30),(12,29,159,136),(13,135,160,28),(14,27,121,134),(15,133,122,26),(16,25,123,132),(17,131,124,24),(18,23,125,130),(19,129,126,22),(20,21,127,128),(41,180,169,52),(42,51,170,179),(43,178,171,50),(44,49,172,177),(45,176,173,48),(46,47,174,175),(53,168,181,80),(54,79,182,167),(55,166,183,78),(56,77,184,165),(57,164,185,76),(58,75,186,163),(59,162,187,74),(60,73,188,161),(61,200,189,72),(62,71,190,199),(63,198,191,70),(64,69,192,197),(65,196,193,68),(66,67,194,195),(81,239,232,88),(82,87,233,238),(83,237,234,86),(84,85,235,236),(89,231,240,120),(90,119,201,230),(91,229,202,118),(92,117,203,228),(93,227,204,116),(94,115,205,226),(95,225,206,114),(96,113,207,224),(97,223,208,112),(98,111,209,222),(99,221,210,110),(100,109,211,220),(101,219,212,108),(102,107,213,218),(103,217,214,106),(104,105,215,216)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30F | 40A | ··· | 40P | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 2 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | + | - | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D5 | Dic3 | D6 | D8 | SD16 | D10 | C3⋊D4 | C3⋊D4 | C4×D5 | C5⋊D4 | D20 | C40⋊C2 | D40 | D4⋊S3 | D4.S3 | S3×D5 | D5×Dic3 | C15⋊D4 | C3⋊D20 | C3⋊D40 | C6.D20 |
| kernel | C6.D40 | C10×C3⋊C8 | C60⋊5C4 | C6×D20 | C3×D20 | C2×D20 | C60 | C2×C30 | C2×C3⋊C8 | D20 | C2×C20 | C30 | C30 | C2×C12 | C20 | C2×C10 | C12 | C12 | C2×C6 | C6 | C6 | C10 | C10 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C6.D40 ►in GL4(𝔽241) generated by
| 0 | 1 | 0 | 0 |
| 240 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 84 | 202 | 0 | 0 |
| 45 | 157 | 0 | 0 |
| 0 | 0 | 194 | 227 |
| 0 | 0 | 14 | 199 |
| 84 | 202 | 0 | 0 |
| 45 | 157 | 0 | 0 |
| 0 | 0 | 194 | 227 |
| 0 | 0 | 20 | 47 |
G:=sub<GL(4,GF(241))| [0,240,0,0,1,1,0,0,0,0,1,0,0,0,0,1],[84,45,0,0,202,157,0,0,0,0,194,14,0,0,227,199],[84,45,0,0,202,157,0,0,0,0,194,20,0,0,227,47] >;
C6.D40 in GAP, Magma, Sage, TeX
C_6.D_{40} % in TeX
G:=Group("C6.D40"); // GroupNames label
G:=SmallGroup(480,41);
// by ID
G=gap.SmallGroup(480,41);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,85,92,422,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^6=b^40=1,c^2=a^3,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^3*b^-1>;
// generators/relations