metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.35D6, C24.38D10, C120.60C22, C60.173C23, C15⋊Q8.5C4, C8⋊S3⋊6D5, C8⋊D5⋊6S3, C3⋊C8.24D10, D6.2(C4×D5), C8.33(S3×D5), (C8×D15)⋊15C2, C15⋊10(C8○D4), C5⋊5(D12.C4), D10.6(C4×S3), (C4×D5).57D6, C5⋊2C8.24D6, C5⋊D12.5C4, C3⋊D20.5C4, C15⋊D4.5C4, D30.29(C2×C4), (C4×S3).33D10, Dic3.2(C4×D5), Dic5.6(C4×S3), C3⋊2(D20.2C4), C30.39(C22×C4), D30.5C4⋊11C2, (S3×C20).33C22, C20.170(C22×S3), C15⋊3C8.47C22, Dic15.36(C2×C4), D6.D10.3C2, (D5×C12).57C22, (C4×D15).63C22, C12.170(C22×D5), C6.8(C2×C4×D5), (D5×C3⋊C8)⋊10C2, C2.11(C4×S3×D5), C10.39(S3×C2×C4), C4.143(C2×S3×D5), (C5×C8⋊S3)⋊8C2, (S3×C5⋊2C8)⋊10C2, (C6×D5).4(C2×C4), (C3×C8⋊D5)⋊10C2, (C5×C3⋊C8).24C22, (S3×C10).19(C2×C4), (C3×Dic5).4(C2×C4), (C3×C5⋊2C8).24C22, (C5×Dic3).20(C2×C4), SmallGroup(480,344)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.35D6
G = < a,b,c | a40=b6=1, c2=a20, bab-1=a29, 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, C8⋊S3, C8⋊S3, C2×C3⋊C8, C3×M4(2), C4○D12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D30, C8×D5, C8⋊D5, C8⋊D5, C2×C5⋊2C8, C5×M4(2), C4○D20, D12.C4, C5×C3⋊C8, C3×C5⋊2C8, C15⋊3C8, C120, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, D5×C12, S3×C20, C4×D15, D20.2C4, D5×C3⋊C8, S3×C5⋊2C8, D30.5C4, C3×C8⋊D5, C5×C8⋊S3, C8×D15, D6.D10, C40.35D6
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, D12.C4, C2×S3×D5, D20.2C4, C4×S3×D5, C40.35D6
►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)
(1 180 222)(2 169 223 30 181 211)(3 198 224 19 182 240)(4 187 225 8 183 229)(5 176 226 37 184 218)(6 165 227 26 185 207)(7 194 228 15 186 236)(9 172 230 33 188 214)(10 161 231 22 189 203)(11 190 232)(12 179 233 40 191 221)(13 168 234 29 192 210)(14 197 235 18 193 239)(16 175 237 36 195 217)(17 164 238 25 196 206)(20 171 201 32 199 213)(21 200 202)(23 178 204 39 162 220)(24 167 205 28 163 209)(27 174 208 35 166 216)(31 170 212)(34 177 215 38 173 219)(41 81 157 61 101 137)(42 110 158 50 102 126)(43 99 159 79 103 155)(44 88 160 68 104 144)(45 117 121 57 105 133)(46 106 122)(47 95 123 75 107 151)(48 84 124 64 108 140)(49 113 125 53 109 129)(51 91 127 71 111 147)(52 120 128 60 112 136)(54 98 130 78 114 154)(55 87 131 67 115 143)(56 116 132)(58 94 134 74 118 150)(59 83 135 63 119 139)(62 90 138 70 82 146)(65 97 141 77 85 153)(66 86 142)(69 93 145 73 89 149)(72 100 148 80 92 156)(76 96 152)
(1 91 21 111)(2 100 22 120)(3 109 23 89)(4 118 24 98)(5 87 25 107)(6 96 26 116)(7 105 27 85)(8 114 28 94)(9 83 29 103)(10 92 30 112)(11 101 31 81)(12 110 32 90)(13 119 33 99)(14 88 34 108)(15 97 35 117)(16 106 36 86)(17 115 37 95)(18 84 38 104)(19 93 39 113)(20 102 40 82)(41 170 61 190)(42 179 62 199)(43 188 63 168)(44 197 64 177)(45 166 65 186)(46 175 66 195)(47 184 67 164)(48 193 68 173)(49 162 69 182)(50 171 70 191)(51 180 71 200)(52 189 72 169)(53 198 73 178)(54 167 74 187)(55 176 75 196)(56 185 76 165)(57 194 77 174)(58 163 78 183)(59 172 79 192)(60 181 80 161)(121 208 141 228)(122 217 142 237)(123 226 143 206)(124 235 144 215)(125 204 145 224)(126 213 146 233)(127 222 147 202)(128 231 148 211)(129 240 149 220)(130 209 150 229)(131 218 151 238)(132 227 152 207)(133 236 153 216)(134 205 154 225)(135 214 155 234)(136 223 156 203)(137 232 157 212)(138 201 158 221)(139 210 159 230)(140 219 160 239)
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), (1,180,222)(2,169,223,30,181,211)(3,198,224,19,182,240)(4,187,225,8,183,229)(5,176,226,37,184,218)(6,165,227,26,185,207)(7,194,228,15,186,236)(9,172,230,33,188,214)(10,161,231,22,189,203)(11,190,232)(12,179,233,40,191,221)(13,168,234,29,192,210)(14,197,235,18,193,239)(16,175,237,36,195,217)(17,164,238,25,196,206)(20,171,201,32,199,213)(21,200,202)(23,178,204,39,162,220)(24,167,205,28,163,209)(27,174,208,35,166,216)(31,170,212)(34,177,215,38,173,219)(41,81,157,61,101,137)(42,110,158,50,102,126)(43,99,159,79,103,155)(44,88,160,68,104,144)(45,117,121,57,105,133)(46,106,122)(47,95,123,75,107,151)(48,84,124,64,108,140)(49,113,125,53,109,129)(51,91,127,71,111,147)(52,120,128,60,112,136)(54,98,130,78,114,154)(55,87,131,67,115,143)(56,116,132)(58,94,134,74,118,150)(59,83,135,63,119,139)(62,90,138,70,82,146)(65,97,141,77,85,153)(66,86,142)(69,93,145,73,89,149)(72,100,148,80,92,156)(76,96,152), (1,91,21,111)(2,100,22,120)(3,109,23,89)(4,118,24,98)(5,87,25,107)(6,96,26,116)(7,105,27,85)(8,114,28,94)(9,83,29,103)(10,92,30,112)(11,101,31,81)(12,110,32,90)(13,119,33,99)(14,88,34,108)(15,97,35,117)(16,106,36,86)(17,115,37,95)(18,84,38,104)(19,93,39,113)(20,102,40,82)(41,170,61,190)(42,179,62,199)(43,188,63,168)(44,197,64,177)(45,166,65,186)(46,175,66,195)(47,184,67,164)(48,193,68,173)(49,162,69,182)(50,171,70,191)(51,180,71,200)(52,189,72,169)(53,198,73,178)(54,167,74,187)(55,176,75,196)(56,185,76,165)(57,194,77,174)(58,163,78,183)(59,172,79,192)(60,181,80,161)(121,208,141,228)(122,217,142,237)(123,226,143,206)(124,235,144,215)(125,204,145,224)(126,213,146,233)(127,222,147,202)(128,231,148,211)(129,240,149,220)(130,209,150,229)(131,218,151,238)(132,227,152,207)(133,236,153,216)(134,205,154,225)(135,214,155,234)(136,223,156,203)(137,232,157,212)(138,201,158,221)(139,210,159,230)(140,219,160,239)>;
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), (1,180,222)(2,169,223,30,181,211)(3,198,224,19,182,240)(4,187,225,8,183,229)(5,176,226,37,184,218)(6,165,227,26,185,207)(7,194,228,15,186,236)(9,172,230,33,188,214)(10,161,231,22,189,203)(11,190,232)(12,179,233,40,191,221)(13,168,234,29,192,210)(14,197,235,18,193,239)(16,175,237,36,195,217)(17,164,238,25,196,206)(20,171,201,32,199,213)(21,200,202)(23,178,204,39,162,220)(24,167,205,28,163,209)(27,174,208,35,166,216)(31,170,212)(34,177,215,38,173,219)(41,81,157,61,101,137)(42,110,158,50,102,126)(43,99,159,79,103,155)(44,88,160,68,104,144)(45,117,121,57,105,133)(46,106,122)(47,95,123,75,107,151)(48,84,124,64,108,140)(49,113,125,53,109,129)(51,91,127,71,111,147)(52,120,128,60,112,136)(54,98,130,78,114,154)(55,87,131,67,115,143)(56,116,132)(58,94,134,74,118,150)(59,83,135,63,119,139)(62,90,138,70,82,146)(65,97,141,77,85,153)(66,86,142)(69,93,145,73,89,149)(72,100,148,80,92,156)(76,96,152), (1,91,21,111)(2,100,22,120)(3,109,23,89)(4,118,24,98)(5,87,25,107)(6,96,26,116)(7,105,27,85)(8,114,28,94)(9,83,29,103)(10,92,30,112)(11,101,31,81)(12,110,32,90)(13,119,33,99)(14,88,34,108)(15,97,35,117)(16,106,36,86)(17,115,37,95)(18,84,38,104)(19,93,39,113)(20,102,40,82)(41,170,61,190)(42,179,62,199)(43,188,63,168)(44,197,64,177)(45,166,65,186)(46,175,66,195)(47,184,67,164)(48,193,68,173)(49,162,69,182)(50,171,70,191)(51,180,71,200)(52,189,72,169)(53,198,73,178)(54,167,74,187)(55,176,75,196)(56,185,76,165)(57,194,77,174)(58,163,78,183)(59,172,79,192)(60,181,80,161)(121,208,141,228)(122,217,142,237)(123,226,143,206)(124,235,144,215)(125,204,145,224)(126,213,146,233)(127,222,147,202)(128,231,148,211)(129,240,149,220)(130,209,150,229)(131,218,151,238)(132,227,152,207)(133,236,153,216)(134,205,154,225)(135,214,155,234)(136,223,156,203)(137,232,157,212)(138,201,158,221)(139,210,159,230)(140,219,160,239) );
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)], [(1,180,222),(2,169,223,30,181,211),(3,198,224,19,182,240),(4,187,225,8,183,229),(5,176,226,37,184,218),(6,165,227,26,185,207),(7,194,228,15,186,236),(9,172,230,33,188,214),(10,161,231,22,189,203),(11,190,232),(12,179,233,40,191,221),(13,168,234,29,192,210),(14,197,235,18,193,239),(16,175,237,36,195,217),(17,164,238,25,196,206),(20,171,201,32,199,213),(21,200,202),(23,178,204,39,162,220),(24,167,205,28,163,209),(27,174,208,35,166,216),(31,170,212),(34,177,215,38,173,219),(41,81,157,61,101,137),(42,110,158,50,102,126),(43,99,159,79,103,155),(44,88,160,68,104,144),(45,117,121,57,105,133),(46,106,122),(47,95,123,75,107,151),(48,84,124,64,108,140),(49,113,125,53,109,129),(51,91,127,71,111,147),(52,120,128,60,112,136),(54,98,130,78,114,154),(55,87,131,67,115,143),(56,116,132),(58,94,134,74,118,150),(59,83,135,63,119,139),(62,90,138,70,82,146),(65,97,141,77,85,153),(66,86,142),(69,93,145,73,89,149),(72,100,148,80,92,156),(76,96,152)], [(1,91,21,111),(2,100,22,120),(3,109,23,89),(4,118,24,98),(5,87,25,107),(6,96,26,116),(7,105,27,85),(8,114,28,94),(9,83,29,103),(10,92,30,112),(11,101,31,81),(12,110,32,90),(13,119,33,99),(14,88,34,108),(15,97,35,117),(16,106,36,86),(17,115,37,95),(18,84,38,104),(19,93,39,113),(20,102,40,82),(41,170,61,190),(42,179,62,199),(43,188,63,168),(44,197,64,177),(45,166,65,186),(46,175,66,195),(47,184,67,164),(48,193,68,173),(49,162,69,182),(50,171,70,191),(51,180,71,200),(52,189,72,169),(53,198,73,178),(54,167,74,187),(55,176,75,196),(56,185,76,165),(57,194,77,174),(58,163,78,183),(59,172,79,192),(60,181,80,161),(121,208,141,228),(122,217,142,237),(123,226,143,206),(124,235,144,215),(125,204,145,224),(126,213,146,233),(127,222,147,202),(128,231,148,211),(129,240,149,220),(130,209,150,229),(131,218,151,238),(132,227,152,207),(133,236,153,216),(134,205,154,225),(135,214,155,234),(136,223,156,203),(137,232,157,212),(138,201,158,221),(139,210,159,230),(140,219,160,239)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 24C | 24D | 30A | 30B | 40A | 40B | 40C | 40D | 40E | 40F | 40G | 40H | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 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 | 20 | 2 | 2 | 6 | 6 | 10 | 10 | 15 | 15 | 15 | 15 | 2 | 2 | 12 | 12 | 2 | 2 | 20 | 4 | 4 | 2 | 2 | 2 | 2 | 12 | 12 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
66 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 | 4 | 4 | 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 | S3×D5 | D12.C4 | C2×S3×D5 | D20.2C4 | C4×S3×D5 | C40.35D6 |
| kernel | C40.35D6 | D5×C3⋊C8 | S3×C5⋊2C8 | D30.5C4 | C3×C8⋊D5 | C5×C8⋊S3 | C8×D15 | D6.D10 | C15⋊D4 | C3⋊D20 | C5⋊D12 | C15⋊Q8 | C8⋊D5 | C8⋊S3 | C5⋊2C8 | C40 | C4×D5 | C3⋊C8 | C24 | C4×S3 | Dic5 | D10 | C15 | Dic3 | D6 | C8 | C5 | C4 | C3 | 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 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C40.35D6 ►in GL6(𝔽241)
| 64 | 54 | 0 | 0 | 0 | 0 |
| 86 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 51 | 0 | 0 |
| 0 | 0 | 189 | 189 | 0 | 0 |
| 0 | 0 | 0 | 0 | 177 | 0 |
| 0 | 0 | 0 | 0 | 0 | 177 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 194 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 189 | 190 | 0 | 0 |
| 0 | 0 | 53 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 240 |
| 30 | 191 | 0 | 0 | 0 | 0 |
| 206 | 211 | 0 | 0 | 0 | 0 |
| 0 | 0 | 189 | 190 | 0 | 0 |
| 0 | 0 | 53 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 89 | 81 |
| 0 | 0 | 0 | 0 | 170 | 152 |
G:=sub<GL(6,GF(241))| [64,86,0,0,0,0,54,177,0,0,0,0,0,0,0,189,0,0,0,0,51,189,0,0,0,0,0,0,177,0,0,0,0,0,0,177],[1,194,0,0,0,0,0,240,0,0,0,0,0,0,189,53,0,0,0,0,190,52,0,0,0,0,0,0,0,1,0,0,0,0,240,240],[30,206,0,0,0,0,191,211,0,0,0,0,0,0,189,53,0,0,0,0,190,52,0,0,0,0,0,0,89,170,0,0,0,0,81,152] >;
C40.35D6 in GAP, Magma, Sage, TeX
C_{40}._{35}D_6 % in TeX
G:=Group("C40.35D6"); // GroupNames label
G:=SmallGroup(480,344);
// by ID
G=gap.SmallGroup(480,344);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,422,219,58,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^40=b^6=1,c^2=a^20,b*a*b^-1=a^29,c*a*c^-1=a^9,c*b*c^-1=a^20*b^-1>;
// generators/relations