metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.54D6, C24.61D10, C60.170C23, C120.57C22, (S3×C8)⋊6D5, (C8×D5)⋊6S3, (S3×C40)⋊8C2, C15⋊Q8.6C4, C15⋊7(C8○D4), C5⋊4(C8○D12), C3⋊C8.33D10, D6.3(C4×D5), C8.40(S3×D5), (D5×C24)⋊11C2, (C8×D15)⋊14C2, (C4×D5).83D6, C3⋊D20.6C4, C15⋊D4.6C4, C5⋊2C8.37D6, C5⋊D12.6C4, D30.28(C2×C4), (C4×S3).43D10, D10.10(C4×S3), Dic3.5(C4×D5), C3⋊1(D20.3C4), D6.Dic5⋊15C2, C30.36(C22×C4), Dic5.13(C4×S3), D30.5C4⋊15C2, C20.32D6⋊15C2, (S3×C20).51C22, C20.167(C22×S3), C15⋊3C8.46C22, Dic15.35(C2×C4), D6.D10.4C2, (C4×D15).62C22, (D5×C12).99C22, C12.167(C22×D5), C2.8(C4×S3×D5), C6.5(C2×C4×D5), C10.36(S3×C2×C4), C4.140(C2×S3×D5), (C6×D5).32(C2×C4), (C5×C3⋊C8).37C22, (S3×C10).26(C2×C4), (C3×C5⋊2C8).41C22, (C5×Dic3).31(C2×C4), (C3×Dic5).37(C2×C4), SmallGroup(480,341)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.54D6
G = < a,b,c | a40=b6=1, c2=a20, bab-1=cac-1=a9, cbc-1=a20b-1 >
Subgroups: 540 in 124 conjugacy classes, 50 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, Q8, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8, M4(2), C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C3⋊C8, C3⋊C8, C24, C24, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12, C5×S3, C3×D5, D15, C30, C8○D4, C5⋊2C8, C5⋊2C8, C40, C40, Dic10, C4×D5, C4×D5, D20, C5⋊D4, C2×C20, S3×C8, S3×C8, C8⋊S3, C4.Dic3, C2×C24, C4○D12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D30, C8×D5, C8×D5, C8⋊D5, C4.Dic5, C2×C40, C4○D20, C8○D12, C5×C3⋊C8, C3×C5⋊2C8, C15⋊3C8, C120, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, D5×C12, S3×C20, C4×D15, D20.3C4, C20.32D6, D6.Dic5, D30.5C4, D5×C24, S3×C40, C8×D15, D6.D10, C40.54D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, D10, C4×S3, C22×S3, C8○D4, C4×D5, C22×D5, S3×C2×C4, S3×D5, C2×C4×D5, C8○D12, C2×S3×D5, D20.3C4, C4×S3×D5, C40.54D6
►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)
(1 150 237)(2 159 238 10 151 206)(3 128 239 19 152 215)(4 137 240 28 153 224)(5 146 201 37 154 233)(6 155 202)(7 124 203 15 156 211)(8 133 204 24 157 220)(9 142 205 33 158 229)(11 160 207)(12 129 208 20 121 216)(13 138 209 29 122 225)(14 147 210 38 123 234)(16 125 212)(17 134 213 25 126 221)(18 143 214 34 127 230)(21 130 217)(22 139 218 30 131 226)(23 148 219 39 132 235)(26 135 222)(27 144 223 35 136 231)(31 140 227)(32 149 228 40 141 236)(36 145 232)(41 161 109 45 197 113)(42 170 110 54 198 82)(43 179 111 63 199 91)(44 188 112 72 200 100)(46 166 114 50 162 118)(47 175 115 59 163 87)(48 184 116 68 164 96)(49 193 117 77 165 105)(51 171 119 55 167 83)(52 180 120 64 168 92)(53 189 81 73 169 101)(56 176 84 60 172 88)(57 185 85 69 173 97)(58 194 86 78 174 106)(61 181 89 65 177 93)(62 190 90 74 178 102)(66 186 94 70 182 98)(67 195 95 79 183 107)(71 191 99 75 187 103)(76 196 104 80 192 108)
(1 101 21 81)(2 110 22 90)(3 119 23 99)(4 88 24 108)(5 97 25 117)(6 106 26 86)(7 115 27 95)(8 84 28 104)(9 93 29 113)(10 102 30 82)(11 111 31 91)(12 120 32 100)(13 89 33 109)(14 98 34 118)(15 107 35 87)(16 116 36 96)(17 85 37 105)(18 94 38 114)(19 103 39 83)(20 112 40 92)(41 209 61 229)(42 218 62 238)(43 227 63 207)(44 236 64 216)(45 205 65 225)(46 214 66 234)(47 223 67 203)(48 232 68 212)(49 201 69 221)(50 210 70 230)(51 219 71 239)(52 228 72 208)(53 237 73 217)(54 206 74 226)(55 215 75 235)(56 224 76 204)(57 233 77 213)(58 202 78 222)(59 211 79 231)(60 220 80 240)(121 168 141 188)(122 177 142 197)(123 186 143 166)(124 195 144 175)(125 164 145 184)(126 173 146 193)(127 182 147 162)(128 191 148 171)(129 200 149 180)(130 169 150 189)(131 178 151 198)(132 187 152 167)(133 196 153 176)(134 165 154 185)(135 174 155 194)(136 183 156 163)(137 192 157 172)(138 161 158 181)(139 170 159 190)(140 179 160 199)
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), (1,150,237)(2,159,238,10,151,206)(3,128,239,19,152,215)(4,137,240,28,153,224)(5,146,201,37,154,233)(6,155,202)(7,124,203,15,156,211)(8,133,204,24,157,220)(9,142,205,33,158,229)(11,160,207)(12,129,208,20,121,216)(13,138,209,29,122,225)(14,147,210,38,123,234)(16,125,212)(17,134,213,25,126,221)(18,143,214,34,127,230)(21,130,217)(22,139,218,30,131,226)(23,148,219,39,132,235)(26,135,222)(27,144,223,35,136,231)(31,140,227)(32,149,228,40,141,236)(36,145,232)(41,161,109,45,197,113)(42,170,110,54,198,82)(43,179,111,63,199,91)(44,188,112,72,200,100)(46,166,114,50,162,118)(47,175,115,59,163,87)(48,184,116,68,164,96)(49,193,117,77,165,105)(51,171,119,55,167,83)(52,180,120,64,168,92)(53,189,81,73,169,101)(56,176,84,60,172,88)(57,185,85,69,173,97)(58,194,86,78,174,106)(61,181,89,65,177,93)(62,190,90,74,178,102)(66,186,94,70,182,98)(67,195,95,79,183,107)(71,191,99,75,187,103)(76,196,104,80,192,108), (1,101,21,81)(2,110,22,90)(3,119,23,99)(4,88,24,108)(5,97,25,117)(6,106,26,86)(7,115,27,95)(8,84,28,104)(9,93,29,113)(10,102,30,82)(11,111,31,91)(12,120,32,100)(13,89,33,109)(14,98,34,118)(15,107,35,87)(16,116,36,96)(17,85,37,105)(18,94,38,114)(19,103,39,83)(20,112,40,92)(41,209,61,229)(42,218,62,238)(43,227,63,207)(44,236,64,216)(45,205,65,225)(46,214,66,234)(47,223,67,203)(48,232,68,212)(49,201,69,221)(50,210,70,230)(51,219,71,239)(52,228,72,208)(53,237,73,217)(54,206,74,226)(55,215,75,235)(56,224,76,204)(57,233,77,213)(58,202,78,222)(59,211,79,231)(60,220,80,240)(121,168,141,188)(122,177,142,197)(123,186,143,166)(124,195,144,175)(125,164,145,184)(126,173,146,193)(127,182,147,162)(128,191,148,171)(129,200,149,180)(130,169,150,189)(131,178,151,198)(132,187,152,167)(133,196,153,176)(134,165,154,185)(135,174,155,194)(136,183,156,163)(137,192,157,172)(138,161,158,181)(139,170,159,190)(140,179,160,199)>;
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), (1,150,237)(2,159,238,10,151,206)(3,128,239,19,152,215)(4,137,240,28,153,224)(5,146,201,37,154,233)(6,155,202)(7,124,203,15,156,211)(8,133,204,24,157,220)(9,142,205,33,158,229)(11,160,207)(12,129,208,20,121,216)(13,138,209,29,122,225)(14,147,210,38,123,234)(16,125,212)(17,134,213,25,126,221)(18,143,214,34,127,230)(21,130,217)(22,139,218,30,131,226)(23,148,219,39,132,235)(26,135,222)(27,144,223,35,136,231)(31,140,227)(32,149,228,40,141,236)(36,145,232)(41,161,109,45,197,113)(42,170,110,54,198,82)(43,179,111,63,199,91)(44,188,112,72,200,100)(46,166,114,50,162,118)(47,175,115,59,163,87)(48,184,116,68,164,96)(49,193,117,77,165,105)(51,171,119,55,167,83)(52,180,120,64,168,92)(53,189,81,73,169,101)(56,176,84,60,172,88)(57,185,85,69,173,97)(58,194,86,78,174,106)(61,181,89,65,177,93)(62,190,90,74,178,102)(66,186,94,70,182,98)(67,195,95,79,183,107)(71,191,99,75,187,103)(76,196,104,80,192,108), (1,101,21,81)(2,110,22,90)(3,119,23,99)(4,88,24,108)(5,97,25,117)(6,106,26,86)(7,115,27,95)(8,84,28,104)(9,93,29,113)(10,102,30,82)(11,111,31,91)(12,120,32,100)(13,89,33,109)(14,98,34,118)(15,107,35,87)(16,116,36,96)(17,85,37,105)(18,94,38,114)(19,103,39,83)(20,112,40,92)(41,209,61,229)(42,218,62,238)(43,227,63,207)(44,236,64,216)(45,205,65,225)(46,214,66,234)(47,223,67,203)(48,232,68,212)(49,201,69,221)(50,210,70,230)(51,219,71,239)(52,228,72,208)(53,237,73,217)(54,206,74,226)(55,215,75,235)(56,224,76,204)(57,233,77,213)(58,202,78,222)(59,211,79,231)(60,220,80,240)(121,168,141,188)(122,177,142,197)(123,186,143,166)(124,195,144,175)(125,164,145,184)(126,173,146,193)(127,182,147,162)(128,191,148,171)(129,200,149,180)(130,169,150,189)(131,178,151,198)(132,187,152,167)(133,196,153,176)(134,165,154,185)(135,174,155,194)(136,183,156,163)(137,192,157,172)(138,161,158,181)(139,170,159,190)(140,179,160,199) );
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)], [(1,150,237),(2,159,238,10,151,206),(3,128,239,19,152,215),(4,137,240,28,153,224),(5,146,201,37,154,233),(6,155,202),(7,124,203,15,156,211),(8,133,204,24,157,220),(9,142,205,33,158,229),(11,160,207),(12,129,208,20,121,216),(13,138,209,29,122,225),(14,147,210,38,123,234),(16,125,212),(17,134,213,25,126,221),(18,143,214,34,127,230),(21,130,217),(22,139,218,30,131,226),(23,148,219,39,132,235),(26,135,222),(27,144,223,35,136,231),(31,140,227),(32,149,228,40,141,236),(36,145,232),(41,161,109,45,197,113),(42,170,110,54,198,82),(43,179,111,63,199,91),(44,188,112,72,200,100),(46,166,114,50,162,118),(47,175,115,59,163,87),(48,184,116,68,164,96),(49,193,117,77,165,105),(51,171,119,55,167,83),(52,180,120,64,168,92),(53,189,81,73,169,101),(56,176,84,60,172,88),(57,185,85,69,173,97),(58,194,86,78,174,106),(61,181,89,65,177,93),(62,190,90,74,178,102),(66,186,94,70,182,98),(67,195,95,79,183,107),(71,191,99,75,187,103),(76,196,104,80,192,108)], [(1,101,21,81),(2,110,22,90),(3,119,23,99),(4,88,24,108),(5,97,25,117),(6,106,26,86),(7,115,27,95),(8,84,28,104),(9,93,29,113),(10,102,30,82),(11,111,31,91),(12,120,32,100),(13,89,33,109),(14,98,34,118),(15,107,35,87),(16,116,36,96),(17,85,37,105),(18,94,38,114),(19,103,39,83),(20,112,40,92),(41,209,61,229),(42,218,62,238),(43,227,63,207),(44,236,64,216),(45,205,65,225),(46,214,66,234),(47,223,67,203),(48,232,68,212),(49,201,69,221),(50,210,70,230),(51,219,71,239),(52,228,72,208),(53,237,73,217),(54,206,74,226),(55,215,75,235),(56,224,76,204),(57,233,77,213),(58,202,78,222),(59,211,79,231),(60,220,80,240),(121,168,141,188),(122,177,142,197),(123,186,143,166),(124,195,144,175),(125,164,145,184),(126,173,146,193),(127,182,147,162),(128,191,148,171),(129,200,149,180),(130,169,150,189),(131,178,151,198),(132,187,152,167),(133,196,153,176),(134,165,154,185),(135,174,155,194),(136,183,156,163),(137,192,157,172),(138,161,158,181),(139,170,159,190),(140,179,160,199)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 30A | 30B | 40A | ··· | 40H | 40I | ··· | 40P | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 6 | 10 | 30 | 2 | 1 | 1 | 6 | 10 | 30 | 2 | 2 | 2 | 10 | 10 | 1 | 1 | 1 | 1 | 6 | 6 | 10 | 10 | 30 | 30 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 10 | 10 | 4 | 4 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C4×S3 | C4×S3 | C8○D4 | C4×D5 | C4×D5 | C8○D12 | D20.3C4 | S3×D5 | C2×S3×D5 | C4×S3×D5 | C40.54D6 |
| kernel | C40.54D6 | C20.32D6 | D6.Dic5 | D30.5C4 | D5×C24 | S3×C40 | C8×D15 | D6.D10 | C15⋊D4 | C3⋊D20 | C5⋊D12 | C15⋊Q8 | C8×D5 | S3×C8 | C5⋊2C8 | C40 | C4×D5 | C3⋊C8 | C24 | C4×S3 | Dic5 | D10 | C15 | Dic3 | D6 | C5 | C3 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 16 | 2 | 2 | 4 | 8 |
Matrix representation of C40.54D6 ►in GL4(𝔽241) generated by
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 0 | 84 |
| 0 | 0 | 127 | 157 |
| 225 | 233 | 0 | 0 |
| 0 | 226 | 0 | 0 |
| 0 | 0 | 51 | 51 |
| 0 | 0 | 1 | 190 |
| 15 | 233 | 0 | 0 |
| 209 | 226 | 0 | 0 |
| 0 | 0 | 69 | 20 |
| 0 | 0 | 27 | 172 |
G:=sub<GL(4,GF(241))| [30,0,0,0,0,30,0,0,0,0,0,127,0,0,84,157],[225,0,0,0,233,226,0,0,0,0,51,1,0,0,51,190],[15,209,0,0,233,226,0,0,0,0,69,27,0,0,20,172] >;
C40.54D6 in GAP, Magma, Sage, TeX
C_{40}._{54}D_6 % in TeX
G:=Group("C40.54D6"); // GroupNames label
G:=SmallGroup(480,341);
// by ID
G=gap.SmallGroup(480,341);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,58,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^40=b^6=1,c^2=a^20,b*a*b^-1=c*a*c^-1=a^9,c*b*c^-1=a^20*b^-1>;
// generators/relations