direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×D40, C30⋊5D8, C24⋊31D10, C60.175D4, C12.45D20, C120⋊38C22, C60.261C23, C5⋊1(C6×D8), C8⋊7(C6×D5), C40⋊8(C2×C6), (C2×C40)⋊5C6, (C2×C24)⋊8D5, C10⋊1(C3×D8), C15⋊11(C2×D8), D20⋊3(C2×C6), (C2×D20)⋊5C6, C10.8(C6×D4), C4.7(C3×D20), (C2×C120)⋊12C2, (C6×D20)⋊21C2, (C2×C6).54D20, C6.81(C2×D20), C20.30(C3×D4), C2.12(C6×D20), (C2×C30).113D4, C30.282(C2×D4), (C2×C12).430D10, (C3×D20)⋊33C22, C20.28(C22×C6), C22.13(C3×D20), (C2×C60).509C22, C12.234(C22×D5), (C2×C8)⋊3(C3×D5), C4.27(D5×C2×C6), (C2×C4).81(C6×D5), (C2×C20).92(C2×C6), (C2×C10).17(C3×D4), SmallGroup(480,696)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×D40
G = < a,b,c | a6=b40=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 752 in 152 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, C6, C6, C6, C8, C2×C4, D4, C23, D5, C10, C10, C12, C2×C6, C2×C6, C15, C2×C8, D8, C2×D4, C20, D10, C2×C10, C24, C2×C12, C3×D4, C22×C6, C3×D5, C30, C30, C2×D8, C40, D20, D20, C2×C20, C22×D5, C2×C24, C3×D8, C6×D4, C60, C6×D5, C2×C30, D40, C2×C40, C2×D20, C6×D8, C120, C3×D20, C3×D20, C2×C60, D5×C2×C6, C2×D40, C3×D40, C2×C120, C6×D20, C6×D40
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, D8, C2×D4, D10, C3×D4, C22×C6, C3×D5, C2×D8, D20, C22×D5, C3×D8, C6×D4, C6×D5, D40, C2×D20, C6×D8, C3×D20, D5×C2×C6, C2×D40, C3×D40, C6×D20, C6×D40
►On 240 points
Generators in S240
(1 225 164 73 134 94)(2 226 165 74 135 95)(3 227 166 75 136 96)(4 228 167 76 137 97)(5 229 168 77 138 98)(6 230 169 78 139 99)(7 231 170 79 140 100)(8 232 171 80 141 101)(9 233 172 41 142 102)(10 234 173 42 143 103)(11 235 174 43 144 104)(12 236 175 44 145 105)(13 237 176 45 146 106)(14 238 177 46 147 107)(15 239 178 47 148 108)(16 240 179 48 149 109)(17 201 180 49 150 110)(18 202 181 50 151 111)(19 203 182 51 152 112)(20 204 183 52 153 113)(21 205 184 53 154 114)(22 206 185 54 155 115)(23 207 186 55 156 116)(24 208 187 56 157 117)(25 209 188 57 158 118)(26 210 189 58 159 119)(27 211 190 59 160 120)(28 212 191 60 121 81)(29 213 192 61 122 82)(30 214 193 62 123 83)(31 215 194 63 124 84)(32 216 195 64 125 85)(33 217 196 65 126 86)(34 218 197 66 127 87)(35 219 198 67 128 88)(36 220 199 68 129 89)(37 221 200 69 130 90)(38 222 161 70 131 91)(39 223 162 71 132 92)(40 224 163 72 133 93)
(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 72)(2 71)(3 70)(4 69)(5 68)(6 67)(7 66)(8 65)(9 64)(10 63)(11 62)(12 61)(13 60)(14 59)(15 58)(16 57)(17 56)(18 55)(19 54)(20 53)(21 52)(22 51)(23 50)(24 49)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 80)(34 79)(35 78)(36 77)(37 76)(38 75)(39 74)(40 73)(81 176)(82 175)(83 174)(84 173)(85 172)(86 171)(87 170)(88 169)(89 168)(90 167)(91 166)(92 165)(93 164)(94 163)(95 162)(96 161)(97 200)(98 199)(99 198)(100 197)(101 196)(102 195)(103 194)(104 193)(105 192)(106 191)(107 190)(108 189)(109 188)(110 187)(111 186)(112 185)(113 184)(114 183)(115 182)(116 181)(117 180)(118 179)(119 178)(120 177)(121 237)(122 236)(123 235)(124 234)(125 233)(126 232)(127 231)(128 230)(129 229)(130 228)(131 227)(132 226)(133 225)(134 224)(135 223)(136 222)(137 221)(138 220)(139 219)(140 218)(141 217)(142 216)(143 215)(144 214)(145 213)(146 212)(147 211)(148 210)(149 209)(150 208)(151 207)(152 206)(153 205)(154 204)(155 203)(156 202)(157 201)(158 240)(159 239)(160 238)
G:=sub<Sym(240)| (1,225,164,73,134,94)(2,226,165,74,135,95)(3,227,166,75,136,96)(4,228,167,76,137,97)(5,229,168,77,138,98)(6,230,169,78,139,99)(7,231,170,79,140,100)(8,232,171,80,141,101)(9,233,172,41,142,102)(10,234,173,42,143,103)(11,235,174,43,144,104)(12,236,175,44,145,105)(13,237,176,45,146,106)(14,238,177,46,147,107)(15,239,178,47,148,108)(16,240,179,48,149,109)(17,201,180,49,150,110)(18,202,181,50,151,111)(19,203,182,51,152,112)(20,204,183,52,153,113)(21,205,184,53,154,114)(22,206,185,54,155,115)(23,207,186,55,156,116)(24,208,187,56,157,117)(25,209,188,57,158,118)(26,210,189,58,159,119)(27,211,190,59,160,120)(28,212,191,60,121,81)(29,213,192,61,122,82)(30,214,193,62,123,83)(31,215,194,63,124,84)(32,216,195,64,125,85)(33,217,196,65,126,86)(34,218,197,66,127,87)(35,219,198,67,128,88)(36,220,199,68,129,89)(37,221,200,69,130,90)(38,222,161,70,131,91)(39,223,162,71,132,92)(40,224,163,72,133,93), (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), (1,72)(2,71)(3,70)(4,69)(5,68)(6,67)(7,66)(8,65)(9,64)(10,63)(11,62)(12,61)(13,60)(14,59)(15,58)(16,57)(17,56)(18,55)(19,54)(20,53)(21,52)(22,51)(23,50)(24,49)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(81,176)(82,175)(83,174)(84,173)(85,172)(86,171)(87,170)(88,169)(89,168)(90,167)(91,166)(92,165)(93,164)(94,163)(95,162)(96,161)(97,200)(98,199)(99,198)(100,197)(101,196)(102,195)(103,194)(104,193)(105,192)(106,191)(107,190)(108,189)(109,188)(110,187)(111,186)(112,185)(113,184)(114,183)(115,182)(116,181)(117,180)(118,179)(119,178)(120,177)(121,237)(122,236)(123,235)(124,234)(125,233)(126,232)(127,231)(128,230)(129,229)(130,228)(131,227)(132,226)(133,225)(134,224)(135,223)(136,222)(137,221)(138,220)(139,219)(140,218)(141,217)(142,216)(143,215)(144,214)(145,213)(146,212)(147,211)(148,210)(149,209)(150,208)(151,207)(152,206)(153,205)(154,204)(155,203)(156,202)(157,201)(158,240)(159,239)(160,238)>;
G:=Group( (1,225,164,73,134,94)(2,226,165,74,135,95)(3,227,166,75,136,96)(4,228,167,76,137,97)(5,229,168,77,138,98)(6,230,169,78,139,99)(7,231,170,79,140,100)(8,232,171,80,141,101)(9,233,172,41,142,102)(10,234,173,42,143,103)(11,235,174,43,144,104)(12,236,175,44,145,105)(13,237,176,45,146,106)(14,238,177,46,147,107)(15,239,178,47,148,108)(16,240,179,48,149,109)(17,201,180,49,150,110)(18,202,181,50,151,111)(19,203,182,51,152,112)(20,204,183,52,153,113)(21,205,184,53,154,114)(22,206,185,54,155,115)(23,207,186,55,156,116)(24,208,187,56,157,117)(25,209,188,57,158,118)(26,210,189,58,159,119)(27,211,190,59,160,120)(28,212,191,60,121,81)(29,213,192,61,122,82)(30,214,193,62,123,83)(31,215,194,63,124,84)(32,216,195,64,125,85)(33,217,196,65,126,86)(34,218,197,66,127,87)(35,219,198,67,128,88)(36,220,199,68,129,89)(37,221,200,69,130,90)(38,222,161,70,131,91)(39,223,162,71,132,92)(40,224,163,72,133,93), (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), (1,72)(2,71)(3,70)(4,69)(5,68)(6,67)(7,66)(8,65)(9,64)(10,63)(11,62)(12,61)(13,60)(14,59)(15,58)(16,57)(17,56)(18,55)(19,54)(20,53)(21,52)(22,51)(23,50)(24,49)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(81,176)(82,175)(83,174)(84,173)(85,172)(86,171)(87,170)(88,169)(89,168)(90,167)(91,166)(92,165)(93,164)(94,163)(95,162)(96,161)(97,200)(98,199)(99,198)(100,197)(101,196)(102,195)(103,194)(104,193)(105,192)(106,191)(107,190)(108,189)(109,188)(110,187)(111,186)(112,185)(113,184)(114,183)(115,182)(116,181)(117,180)(118,179)(119,178)(120,177)(121,237)(122,236)(123,235)(124,234)(125,233)(126,232)(127,231)(128,230)(129,229)(130,228)(131,227)(132,226)(133,225)(134,224)(135,223)(136,222)(137,221)(138,220)(139,219)(140,218)(141,217)(142,216)(143,215)(144,214)(145,213)(146,212)(147,211)(148,210)(149,209)(150,208)(151,207)(152,206)(153,205)(154,204)(155,203)(156,202)(157,201)(158,240)(159,239)(160,238) );
G=PermutationGroup([[(1,225,164,73,134,94),(2,226,165,74,135,95),(3,227,166,75,136,96),(4,228,167,76,137,97),(5,229,168,77,138,98),(6,230,169,78,139,99),(7,231,170,79,140,100),(8,232,171,80,141,101),(9,233,172,41,142,102),(10,234,173,42,143,103),(11,235,174,43,144,104),(12,236,175,44,145,105),(13,237,176,45,146,106),(14,238,177,46,147,107),(15,239,178,47,148,108),(16,240,179,48,149,109),(17,201,180,49,150,110),(18,202,181,50,151,111),(19,203,182,51,152,112),(20,204,183,52,153,113),(21,205,184,53,154,114),(22,206,185,54,155,115),(23,207,186,55,156,116),(24,208,187,56,157,117),(25,209,188,57,158,118),(26,210,189,58,159,119),(27,211,190,59,160,120),(28,212,191,60,121,81),(29,213,192,61,122,82),(30,214,193,62,123,83),(31,215,194,63,124,84),(32,216,195,64,125,85),(33,217,196,65,126,86),(34,218,197,66,127,87),(35,219,198,67,128,88),(36,220,199,68,129,89),(37,221,200,69,130,90),(38,222,161,70,131,91),(39,223,162,71,132,92),(40,224,163,72,133,93)], [(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)], [(1,72),(2,71),(3,70),(4,69),(5,68),(6,67),(7,66),(8,65),(9,64),(10,63),(11,62),(12,61),(13,60),(14,59),(15,58),(16,57),(17,56),(18,55),(19,54),(20,53),(21,52),(22,51),(23,50),(24,49),(25,48),(26,47),(27,46),(28,45),(29,44),(30,43),(31,42),(32,41),(33,80),(34,79),(35,78),(36,77),(37,76),(38,75),(39,74),(40,73),(81,176),(82,175),(83,174),(84,173),(85,172),(86,171),(87,170),(88,169),(89,168),(90,167),(91,166),(92,165),(93,164),(94,163),(95,162),(96,161),(97,200),(98,199),(99,198),(100,197),(101,196),(102,195),(103,194),(104,193),(105,192),(106,191),(107,190),(108,189),(109,188),(110,187),(111,186),(112,185),(113,184),(114,183),(115,182),(116,181),(117,180),(118,179),(119,178),(120,177),(121,237),(122,236),(123,235),(124,234),(125,233),(126,232),(127,231),(128,230),(129,229),(130,228),(131,227),(132,226),(133,225),(134,224),(135,223),(136,222),(137,221),(138,220),(139,219),(140,218),(141,217),(142,216),(143,215),(144,214),(145,213),(146,212),(147,211),(148,210),(149,209),(150,208),(151,207),(152,206),(153,205),(154,204),(155,203),(156,202),(157,201),(158,240),(159,239),(160,238)]])
138 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 5A | 5B | 6A | ··· | 6F | 6G | ··· | 6N | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 20 | 20 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 20 | ··· | 20 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
138 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D4 | D5 | D8 | D10 | D10 | C3×D4 | C3×D4 | C3×D5 | D20 | D20 | C3×D8 | C6×D5 | C6×D5 | D40 | C3×D20 | C3×D20 | C3×D40 |
| kernel | C6×D40 | C3×D40 | C2×C120 | C6×D20 | C2×D40 | D40 | C2×C40 | C2×D20 | C60 | C2×C30 | C2×C24 | C30 | C24 | C2×C12 | C20 | C2×C10 | C2×C8 | C12 | C2×C6 | C10 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 2 | 8 | 2 | 4 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 4 | 16 | 8 | 8 | 32 |
Matrix representation of C6×D40 ►in GL5(𝔽241)
| 240 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 219 | 55 | 0 | 0 |
| 0 | 92 | 0 | 0 | 0 |
| 0 | 0 | 0 | 199 | 227 |
| 0 | 0 | 0 | 14 | 194 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 219 | 55 | 0 | 0 |
| 0 | 92 | 22 | 0 | 0 |
| 0 | 0 | 0 | 221 | 29 |
| 0 | 0 | 0 | 194 | 20 |
G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,15,0,0,0,0,0,15,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,219,92,0,0,0,55,0,0,0,0,0,0,199,14,0,0,0,227,194],[1,0,0,0,0,0,219,92,0,0,0,55,22,0,0,0,0,0,221,194,0,0,0,29,20] >;
C6×D40 in GAP, Magma, Sage, TeX
C_6\times D_{40} % in TeX
G:=Group("C6xD40"); // GroupNames label
G:=SmallGroup(480,696);
// by ID
G=gap.SmallGroup(480,696);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,590,394,2524,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^6=b^40=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations