metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.34D6, C24.62D10, C120.58C22, C60.171C23, (C8×D5)⋊8S3, C15⋊Q8.3C4, C8⋊S3⋊7D5, C5⋊5(C8○D12), C15⋊8(C8○D4), C3⋊C8.23D10, D6.1(C4×D5), C8.24(S3×D5), (D5×C24)⋊12C2, (C4×D5).84D6, C5⋊D12.3C4, C5⋊2C8.44D6, C15⋊D4.3C4, C3⋊D20.3C4, C40⋊S3⋊11C2, D15⋊2C8⋊9C2, D30.18(C2×C4), D10.11(C4×S3), (C4×S3).32D10, Dic3.1(C4×D5), C3⋊1(D20.2C4), C30.37(C22×C4), Dic5.14(C4×S3), C20.32D6⋊11C2, (S3×C20).32C22, C20.168(C22×S3), C15⋊3C8.27C22, Dic15.19(C2×C4), D6.D10.1C2, (C4×D15).38C22, C12.168(C22×D5), (D5×C12).100C22, C2.9(C4×S3×D5), C6.6(C2×C4×D5), (S3×C5⋊2C8)⋊9C2, C10.37(S3×C2×C4), C4.141(C2×S3×D5), (C5×C8⋊S3)⋊7C2, (C6×D5).33(C2×C4), (C5×C3⋊C8).23C22, (S3×C10).18(C2×C4), (C3×C5⋊2C8).42C22, (C3×Dic5).38(C2×C4), (C5×Dic3).19(C2×C4), SmallGroup(480,342)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.34D6
G = < a,b,c | a40=b6=1, c2=a20, bab-1=a9, cac-1=a29, 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, C4.Dic3, C2×C24, 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, 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, D20.2C4, S3×C5⋊2C8, D15⋊2C8, C20.32D6, D5×C24, C5×C8⋊S3, C40⋊S3, D6.D10, C40.34D6
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, D20.2C4, C4×S3×D5, C40.34D6
►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 47 153)(2 56 154 10 48 122)(3 65 155 19 49 131)(4 74 156 28 50 140)(5 43 157 37 51 149)(6 52 158)(7 61 159 15 53 127)(8 70 160 24 54 136)(9 79 121 33 55 145)(11 57 123)(12 66 124 20 58 132)(13 75 125 29 59 141)(14 44 126 38 60 150)(16 62 128)(17 71 129 25 63 137)(18 80 130 34 64 146)(21 67 133)(22 76 134 30 68 142)(23 45 135 39 69 151)(26 72 138)(27 41 139 35 73 147)(31 77 143)(32 46 144 40 78 152)(36 42 148)(81 196 207 101 176 227)(82 165 208 110 177 236)(83 174 209 119 178 205)(84 183 210 88 179 214)(85 192 211 97 180 223)(86 161 212 106 181 232)(87 170 213 115 182 201)(89 188 215 93 184 219)(90 197 216 102 185 228)(91 166 217 111 186 237)(92 175 218 120 187 206)(94 193 220 98 189 224)(95 162 221 107 190 233)(96 171 222 116 191 202)(99 198 225 103 194 229)(100 167 226 112 195 238)(104 163 230 108 199 234)(105 172 231 117 200 203)(109 168 235 113 164 239)(114 173 240 118 169 204)
(1 111 21 91)(2 100 22 120)(3 89 23 109)(4 118 24 98)(5 107 25 87)(6 96 26 116)(7 85 27 105)(8 114 28 94)(9 103 29 83)(10 92 30 112)(11 81 31 101)(12 110 32 90)(13 99 33 119)(14 88 34 108)(15 117 35 97)(16 106 36 86)(17 95 37 115)(18 84 38 104)(19 113 39 93)(20 102 40 82)(41 223 61 203)(42 212 62 232)(43 201 63 221)(44 230 64 210)(45 219 65 239)(46 208 66 228)(47 237 67 217)(48 226 68 206)(49 215 69 235)(50 204 70 224)(51 233 71 213)(52 222 72 202)(53 211 73 231)(54 240 74 220)(55 229 75 209)(56 218 76 238)(57 207 77 227)(58 236 78 216)(59 225 79 205)(60 214 80 234)(121 198 141 178)(122 187 142 167)(123 176 143 196)(124 165 144 185)(125 194 145 174)(126 183 146 163)(127 172 147 192)(128 161 148 181)(129 190 149 170)(130 179 150 199)(131 168 151 188)(132 197 152 177)(133 186 153 166)(134 175 154 195)(135 164 155 184)(136 193 156 173)(137 182 157 162)(138 171 158 191)(139 200 159 180)(140 189 160 169)
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), (1,47,153)(2,56,154,10,48,122)(3,65,155,19,49,131)(4,74,156,28,50,140)(5,43,157,37,51,149)(6,52,158)(7,61,159,15,53,127)(8,70,160,24,54,136)(9,79,121,33,55,145)(11,57,123)(12,66,124,20,58,132)(13,75,125,29,59,141)(14,44,126,38,60,150)(16,62,128)(17,71,129,25,63,137)(18,80,130,34,64,146)(21,67,133)(22,76,134,30,68,142)(23,45,135,39,69,151)(26,72,138)(27,41,139,35,73,147)(31,77,143)(32,46,144,40,78,152)(36,42,148)(81,196,207,101,176,227)(82,165,208,110,177,236)(83,174,209,119,178,205)(84,183,210,88,179,214)(85,192,211,97,180,223)(86,161,212,106,181,232)(87,170,213,115,182,201)(89,188,215,93,184,219)(90,197,216,102,185,228)(91,166,217,111,186,237)(92,175,218,120,187,206)(94,193,220,98,189,224)(95,162,221,107,190,233)(96,171,222,116,191,202)(99,198,225,103,194,229)(100,167,226,112,195,238)(104,163,230,108,199,234)(105,172,231,117,200,203)(109,168,235,113,164,239)(114,173,240,118,169,204), (1,111,21,91)(2,100,22,120)(3,89,23,109)(4,118,24,98)(5,107,25,87)(6,96,26,116)(7,85,27,105)(8,114,28,94)(9,103,29,83)(10,92,30,112)(11,81,31,101)(12,110,32,90)(13,99,33,119)(14,88,34,108)(15,117,35,97)(16,106,36,86)(17,95,37,115)(18,84,38,104)(19,113,39,93)(20,102,40,82)(41,223,61,203)(42,212,62,232)(43,201,63,221)(44,230,64,210)(45,219,65,239)(46,208,66,228)(47,237,67,217)(48,226,68,206)(49,215,69,235)(50,204,70,224)(51,233,71,213)(52,222,72,202)(53,211,73,231)(54,240,74,220)(55,229,75,209)(56,218,76,238)(57,207,77,227)(58,236,78,216)(59,225,79,205)(60,214,80,234)(121,198,141,178)(122,187,142,167)(123,176,143,196)(124,165,144,185)(125,194,145,174)(126,183,146,163)(127,172,147,192)(128,161,148,181)(129,190,149,170)(130,179,150,199)(131,168,151,188)(132,197,152,177)(133,186,153,166)(134,175,154,195)(135,164,155,184)(136,193,156,173)(137,182,157,162)(138,171,158,191)(139,200,159,180)(140,189,160,169)>;
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), (1,47,153)(2,56,154,10,48,122)(3,65,155,19,49,131)(4,74,156,28,50,140)(5,43,157,37,51,149)(6,52,158)(7,61,159,15,53,127)(8,70,160,24,54,136)(9,79,121,33,55,145)(11,57,123)(12,66,124,20,58,132)(13,75,125,29,59,141)(14,44,126,38,60,150)(16,62,128)(17,71,129,25,63,137)(18,80,130,34,64,146)(21,67,133)(22,76,134,30,68,142)(23,45,135,39,69,151)(26,72,138)(27,41,139,35,73,147)(31,77,143)(32,46,144,40,78,152)(36,42,148)(81,196,207,101,176,227)(82,165,208,110,177,236)(83,174,209,119,178,205)(84,183,210,88,179,214)(85,192,211,97,180,223)(86,161,212,106,181,232)(87,170,213,115,182,201)(89,188,215,93,184,219)(90,197,216,102,185,228)(91,166,217,111,186,237)(92,175,218,120,187,206)(94,193,220,98,189,224)(95,162,221,107,190,233)(96,171,222,116,191,202)(99,198,225,103,194,229)(100,167,226,112,195,238)(104,163,230,108,199,234)(105,172,231,117,200,203)(109,168,235,113,164,239)(114,173,240,118,169,204), (1,111,21,91)(2,100,22,120)(3,89,23,109)(4,118,24,98)(5,107,25,87)(6,96,26,116)(7,85,27,105)(8,114,28,94)(9,103,29,83)(10,92,30,112)(11,81,31,101)(12,110,32,90)(13,99,33,119)(14,88,34,108)(15,117,35,97)(16,106,36,86)(17,95,37,115)(18,84,38,104)(19,113,39,93)(20,102,40,82)(41,223,61,203)(42,212,62,232)(43,201,63,221)(44,230,64,210)(45,219,65,239)(46,208,66,228)(47,237,67,217)(48,226,68,206)(49,215,69,235)(50,204,70,224)(51,233,71,213)(52,222,72,202)(53,211,73,231)(54,240,74,220)(55,229,75,209)(56,218,76,238)(57,207,77,227)(58,236,78,216)(59,225,79,205)(60,214,80,234)(121,198,141,178)(122,187,142,167)(123,176,143,196)(124,165,144,185)(125,194,145,174)(126,183,146,163)(127,172,147,192)(128,161,148,181)(129,190,149,170)(130,179,150,199)(131,168,151,188)(132,197,152,177)(133,186,153,166)(134,175,154,195)(135,164,155,184)(136,193,156,173)(137,182,157,162)(138,171,158,191)(139,200,159,180)(140,189,160,169) );
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)], [(1,47,153),(2,56,154,10,48,122),(3,65,155,19,49,131),(4,74,156,28,50,140),(5,43,157,37,51,149),(6,52,158),(7,61,159,15,53,127),(8,70,160,24,54,136),(9,79,121,33,55,145),(11,57,123),(12,66,124,20,58,132),(13,75,125,29,59,141),(14,44,126,38,60,150),(16,62,128),(17,71,129,25,63,137),(18,80,130,34,64,146),(21,67,133),(22,76,134,30,68,142),(23,45,135,39,69,151),(26,72,138),(27,41,139,35,73,147),(31,77,143),(32,46,144,40,78,152),(36,42,148),(81,196,207,101,176,227),(82,165,208,110,177,236),(83,174,209,119,178,205),(84,183,210,88,179,214),(85,192,211,97,180,223),(86,161,212,106,181,232),(87,170,213,115,182,201),(89,188,215,93,184,219),(90,197,216,102,185,228),(91,166,217,111,186,237),(92,175,218,120,187,206),(94,193,220,98,189,224),(95,162,221,107,190,233),(96,171,222,116,191,202),(99,198,225,103,194,229),(100,167,226,112,195,238),(104,163,230,108,199,234),(105,172,231,117,200,203),(109,168,235,113,164,239),(114,173,240,118,169,204)], [(1,111,21,91),(2,100,22,120),(3,89,23,109),(4,118,24,98),(5,107,25,87),(6,96,26,116),(7,85,27,105),(8,114,28,94),(9,103,29,83),(10,92,30,112),(11,81,31,101),(12,110,32,90),(13,99,33,119),(14,88,34,108),(15,117,35,97),(16,106,36,86),(17,95,37,115),(18,84,38,104),(19,113,39,93),(20,102,40,82),(41,223,61,203),(42,212,62,232),(43,201,63,221),(44,230,64,210),(45,219,65,239),(46,208,66,228),(47,237,67,217),(48,226,68,206),(49,215,69,235),(50,204,70,224),(51,233,71,213),(52,222,72,202),(53,211,73,231),(54,240,74,220),(55,229,75,209),(56,218,76,238),(57,207,77,227),(58,236,78,216),(59,225,79,205),(60,214,80,234),(121,198,141,178),(122,187,142,167),(123,176,143,196),(124,165,144,185),(125,194,145,174),(126,183,146,163),(127,172,147,192),(128,161,148,181),(129,190,149,170),(130,179,150,199),(131,168,151,188),(132,197,152,177),(133,186,153,166),(134,175,154,195),(135,164,155,184),(136,193,156,173),(137,182,157,162),(138,171,158,191),(139,200,159,180),(140,189,160,169)]])
72 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 | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 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 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 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 | 10 | 10 | 2 | 2 | 5 | 5 | 5 | 5 | 6 | 6 | 30 | 30 | 2 | 2 | 12 | 12 | 2 | 2 | 10 | 10 | 4 | 4 | 2 | 2 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
72 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 | 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 | C8○D12 | S3×D5 | C2×S3×D5 | D20.2C4 | C4×S3×D5 | C40.34D6 |
| kernel | C40.34D6 | S3×C5⋊2C8 | D15⋊2C8 | C20.32D6 | D5×C24 | C5×C8⋊S3 | C40⋊S3 | 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 | C5 | C8 | 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 | 8 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C40.34D6 ►in GL4(𝔽241) generated by
| 195 | 46 | 0 | 0 |
| 195 | 177 | 0 | 0 |
| 0 | 0 | 233 | 0 |
| 0 | 0 | 0 | 8 |
| 189 | 52 | 0 | 0 |
| 240 | 52 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 16 |
| 52 | 189 | 0 | 0 |
| 1 | 189 | 0 | 0 |
| 0 | 0 | 0 | 225 |
| 0 | 0 | 226 | 0 |
G:=sub<GL(4,GF(241))| [195,195,0,0,46,177,0,0,0,0,233,0,0,0,0,8],[189,240,0,0,52,52,0,0,0,0,15,0,0,0,0,16],[52,1,0,0,189,189,0,0,0,0,0,226,0,0,225,0] >;
C40.34D6 in GAP, Magma, Sage, TeX
C_{40}._{34}D_6 % in TeX
G:=Group("C40.34D6"); // GroupNames label
G:=SmallGroup(480,342);
// by ID
G=gap.SmallGroup(480,342);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,219,58,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^40=b^6=1,c^2=a^20,b*a*b^-1=a^9,c*a*c^-1=a^29,c*b*c^-1=a^20*b^-1>;
// generators/relations