direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×C40⋊C2, C24⋊34D10, C30⋊8SD16, C60.174D4, C12.44D20, C120⋊42C22, C60.260C23, C8⋊8(C6×D5), (C2×C40)⋊7C6, C40⋊9(C2×C6), C5⋊1(C6×SD16), (C2×C24)⋊13D5, C10.7(C6×D4), C4.6(C3×D20), (C2×C120)⋊19C2, D20.6(C2×C6), (C2×D20).5C6, C10⋊1(C3×SD16), C6.80(C2×D20), (C2×C6).53D20, C2.11(C6×D20), C20.29(C3×D4), C15⋊17(C2×SD16), Dic10⋊3(C2×C6), (C2×Dic10)⋊5C6, (C6×D20).16C2, (C2×C30).112D4, C30.281(C2×D4), (C6×Dic10)⋊21C2, (C2×C12).429D10, C20.27(C22×C6), C22.12(C3×D20), (C2×C60).508C22, (C3×D20).45C22, C12.233(C22×D5), (C3×Dic10)⋊30C22, (C2×C8)⋊5(C3×D5), C4.26(D5×C2×C6), (C2×C4).80(C6×D5), (C2×C20).91(C2×C6), (C2×C10).16(C3×D4), SmallGroup(480,695)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C40⋊C2
G = < a,b,c | a6=b40=c2=1, ab=ba, ac=ca, cbc=b19 >
Subgroups: 560 in 136 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, Dic5, C20, D10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C3×D5, C30, C30, C2×SD16, C40, Dic10, Dic10, D20, D20, C2×Dic5, C2×C20, C22×D5, C2×C24, C3×SD16, C6×D4, C6×Q8, C3×Dic5, C60, C6×D5, C2×C30, C40⋊C2, C2×C40, C2×Dic10, C2×D20, C6×SD16, C120, C3×Dic10, C3×Dic10, C3×D20, C3×D20, C6×Dic5, C2×C60, D5×C2×C6, C2×C40⋊C2, C3×C40⋊C2, C2×C120, C6×Dic10, C6×D20, C6×C40⋊C2
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, SD16, C2×D4, D10, C3×D4, C22×C6, C3×D5, C2×SD16, D20, C22×D5, C3×SD16, C6×D4, C6×D5, C40⋊C2, C2×D20, C6×SD16, C3×D20, D5×C2×C6, C2×C40⋊C2, C3×C40⋊C2, C6×D20, C6×C40⋊C2
►On 240 points
Generators in S240
(1 162 58 124 208 106)(2 163 59 125 209 107)(3 164 60 126 210 108)(4 165 61 127 211 109)(5 166 62 128 212 110)(6 167 63 129 213 111)(7 168 64 130 214 112)(8 169 65 131 215 113)(9 170 66 132 216 114)(10 171 67 133 217 115)(11 172 68 134 218 116)(12 173 69 135 219 117)(13 174 70 136 220 118)(14 175 71 137 221 119)(15 176 72 138 222 120)(16 177 73 139 223 81)(17 178 74 140 224 82)(18 179 75 141 225 83)(19 180 76 142 226 84)(20 181 77 143 227 85)(21 182 78 144 228 86)(22 183 79 145 229 87)(23 184 80 146 230 88)(24 185 41 147 231 89)(25 186 42 148 232 90)(26 187 43 149 233 91)(27 188 44 150 234 92)(28 189 45 151 235 93)(29 190 46 152 236 94)(30 191 47 153 237 95)(31 192 48 154 238 96)(32 193 49 155 239 97)(33 194 50 156 240 98)(34 195 51 157 201 99)(35 196 52 158 202 100)(36 197 53 159 203 101)(37 198 54 160 204 102)(38 199 55 121 205 103)(39 200 56 122 206 104)(40 161 57 123 207 105)
(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)
(1 124)(2 143)(3 122)(4 141)(5 160)(6 139)(7 158)(8 137)(9 156)(10 135)(11 154)(12 133)(13 152)(14 131)(15 150)(16 129)(17 148)(18 127)(19 146)(20 125)(21 144)(22 123)(23 142)(24 121)(25 140)(26 159)(27 138)(28 157)(29 136)(30 155)(31 134)(32 153)(33 132)(34 151)(35 130)(36 149)(37 128)(38 147)(39 126)(40 145)(41 103)(42 82)(43 101)(44 120)(45 99)(46 118)(47 97)(48 116)(49 95)(50 114)(51 93)(52 112)(53 91)(54 110)(55 89)(56 108)(57 87)(58 106)(59 85)(60 104)(61 83)(62 102)(63 81)(64 100)(65 119)(66 98)(67 117)(68 96)(69 115)(70 94)(71 113)(72 92)(73 111)(74 90)(75 109)(76 88)(77 107)(78 86)(79 105)(80 84)(161 229)(162 208)(163 227)(164 206)(165 225)(166 204)(167 223)(168 202)(169 221)(170 240)(171 219)(172 238)(173 217)(174 236)(175 215)(176 234)(177 213)(178 232)(179 211)(180 230)(181 209)(182 228)(183 207)(184 226)(185 205)(186 224)(187 203)(188 222)(189 201)(190 220)(191 239)(192 218)(193 237)(194 216)(195 235)(196 214)(197 233)(198 212)(199 231)(200 210)
G:=sub<Sym(240)| (1,162,58,124,208,106)(2,163,59,125,209,107)(3,164,60,126,210,108)(4,165,61,127,211,109)(5,166,62,128,212,110)(6,167,63,129,213,111)(7,168,64,130,214,112)(8,169,65,131,215,113)(9,170,66,132,216,114)(10,171,67,133,217,115)(11,172,68,134,218,116)(12,173,69,135,219,117)(13,174,70,136,220,118)(14,175,71,137,221,119)(15,176,72,138,222,120)(16,177,73,139,223,81)(17,178,74,140,224,82)(18,179,75,141,225,83)(19,180,76,142,226,84)(20,181,77,143,227,85)(21,182,78,144,228,86)(22,183,79,145,229,87)(23,184,80,146,230,88)(24,185,41,147,231,89)(25,186,42,148,232,90)(26,187,43,149,233,91)(27,188,44,150,234,92)(28,189,45,151,235,93)(29,190,46,152,236,94)(30,191,47,153,237,95)(31,192,48,154,238,96)(32,193,49,155,239,97)(33,194,50,156,240,98)(34,195,51,157,201,99)(35,196,52,158,202,100)(36,197,53,159,203,101)(37,198,54,160,204,102)(38,199,55,121,205,103)(39,200,56,122,206,104)(40,161,57,123,207,105), (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), (1,124)(2,143)(3,122)(4,141)(5,160)(6,139)(7,158)(8,137)(9,156)(10,135)(11,154)(12,133)(13,152)(14,131)(15,150)(16,129)(17,148)(18,127)(19,146)(20,125)(21,144)(22,123)(23,142)(24,121)(25,140)(26,159)(27,138)(28,157)(29,136)(30,155)(31,134)(32,153)(33,132)(34,151)(35,130)(36,149)(37,128)(38,147)(39,126)(40,145)(41,103)(42,82)(43,101)(44,120)(45,99)(46,118)(47,97)(48,116)(49,95)(50,114)(51,93)(52,112)(53,91)(54,110)(55,89)(56,108)(57,87)(58,106)(59,85)(60,104)(61,83)(62,102)(63,81)(64,100)(65,119)(66,98)(67,117)(68,96)(69,115)(70,94)(71,113)(72,92)(73,111)(74,90)(75,109)(76,88)(77,107)(78,86)(79,105)(80,84)(161,229)(162,208)(163,227)(164,206)(165,225)(166,204)(167,223)(168,202)(169,221)(170,240)(171,219)(172,238)(173,217)(174,236)(175,215)(176,234)(177,213)(178,232)(179,211)(180,230)(181,209)(182,228)(183,207)(184,226)(185,205)(186,224)(187,203)(188,222)(189,201)(190,220)(191,239)(192,218)(193,237)(194,216)(195,235)(196,214)(197,233)(198,212)(199,231)(200,210)>;
G:=Group( (1,162,58,124,208,106)(2,163,59,125,209,107)(3,164,60,126,210,108)(4,165,61,127,211,109)(5,166,62,128,212,110)(6,167,63,129,213,111)(7,168,64,130,214,112)(8,169,65,131,215,113)(9,170,66,132,216,114)(10,171,67,133,217,115)(11,172,68,134,218,116)(12,173,69,135,219,117)(13,174,70,136,220,118)(14,175,71,137,221,119)(15,176,72,138,222,120)(16,177,73,139,223,81)(17,178,74,140,224,82)(18,179,75,141,225,83)(19,180,76,142,226,84)(20,181,77,143,227,85)(21,182,78,144,228,86)(22,183,79,145,229,87)(23,184,80,146,230,88)(24,185,41,147,231,89)(25,186,42,148,232,90)(26,187,43,149,233,91)(27,188,44,150,234,92)(28,189,45,151,235,93)(29,190,46,152,236,94)(30,191,47,153,237,95)(31,192,48,154,238,96)(32,193,49,155,239,97)(33,194,50,156,240,98)(34,195,51,157,201,99)(35,196,52,158,202,100)(36,197,53,159,203,101)(37,198,54,160,204,102)(38,199,55,121,205,103)(39,200,56,122,206,104)(40,161,57,123,207,105), (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), (1,124)(2,143)(3,122)(4,141)(5,160)(6,139)(7,158)(8,137)(9,156)(10,135)(11,154)(12,133)(13,152)(14,131)(15,150)(16,129)(17,148)(18,127)(19,146)(20,125)(21,144)(22,123)(23,142)(24,121)(25,140)(26,159)(27,138)(28,157)(29,136)(30,155)(31,134)(32,153)(33,132)(34,151)(35,130)(36,149)(37,128)(38,147)(39,126)(40,145)(41,103)(42,82)(43,101)(44,120)(45,99)(46,118)(47,97)(48,116)(49,95)(50,114)(51,93)(52,112)(53,91)(54,110)(55,89)(56,108)(57,87)(58,106)(59,85)(60,104)(61,83)(62,102)(63,81)(64,100)(65,119)(66,98)(67,117)(68,96)(69,115)(70,94)(71,113)(72,92)(73,111)(74,90)(75,109)(76,88)(77,107)(78,86)(79,105)(80,84)(161,229)(162,208)(163,227)(164,206)(165,225)(166,204)(167,223)(168,202)(169,221)(170,240)(171,219)(172,238)(173,217)(174,236)(175,215)(176,234)(177,213)(178,232)(179,211)(180,230)(181,209)(182,228)(183,207)(184,226)(185,205)(186,224)(187,203)(188,222)(189,201)(190,220)(191,239)(192,218)(193,237)(194,216)(195,235)(196,214)(197,233)(198,212)(199,231)(200,210) );
G=PermutationGroup([[(1,162,58,124,208,106),(2,163,59,125,209,107),(3,164,60,126,210,108),(4,165,61,127,211,109),(5,166,62,128,212,110),(6,167,63,129,213,111),(7,168,64,130,214,112),(8,169,65,131,215,113),(9,170,66,132,216,114),(10,171,67,133,217,115),(11,172,68,134,218,116),(12,173,69,135,219,117),(13,174,70,136,220,118),(14,175,71,137,221,119),(15,176,72,138,222,120),(16,177,73,139,223,81),(17,178,74,140,224,82),(18,179,75,141,225,83),(19,180,76,142,226,84),(20,181,77,143,227,85),(21,182,78,144,228,86),(22,183,79,145,229,87),(23,184,80,146,230,88),(24,185,41,147,231,89),(25,186,42,148,232,90),(26,187,43,149,233,91),(27,188,44,150,234,92),(28,189,45,151,235,93),(29,190,46,152,236,94),(30,191,47,153,237,95),(31,192,48,154,238,96),(32,193,49,155,239,97),(33,194,50,156,240,98),(34,195,51,157,201,99),(35,196,52,158,202,100),(36,197,53,159,203,101),(37,198,54,160,204,102),(38,199,55,121,205,103),(39,200,56,122,206,104),(40,161,57,123,207,105)], [(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)], [(1,124),(2,143),(3,122),(4,141),(5,160),(6,139),(7,158),(8,137),(9,156),(10,135),(11,154),(12,133),(13,152),(14,131),(15,150),(16,129),(17,148),(18,127),(19,146),(20,125),(21,144),(22,123),(23,142),(24,121),(25,140),(26,159),(27,138),(28,157),(29,136),(30,155),(31,134),(32,153),(33,132),(34,151),(35,130),(36,149),(37,128),(38,147),(39,126),(40,145),(41,103),(42,82),(43,101),(44,120),(45,99),(46,118),(47,97),(48,116),(49,95),(50,114),(51,93),(52,112),(53,91),(54,110),(55,89),(56,108),(57,87),(58,106),(59,85),(60,104),(61,83),(62,102),(63,81),(64,100),(65,119),(66,98),(67,117),(68,96),(69,115),(70,94),(71,113),(72,92),(73,111),(74,90),(75,109),(76,88),(77,107),(78,86),(79,105),(80,84),(161,229),(162,208),(163,227),(164,206),(165,225),(166,204),(167,223),(168,202),(169,221),(170,240),(171,219),(172,238),(173,217),(174,236),(175,215),(176,234),(177,213),(178,232),(179,211),(180,230),(181,209),(182,228),(183,207),(184,226),(185,205),(186,224),(187,203),(188,222),(189,201),(190,220),(191,239),(192,218),(193,237),(194,216),(195,235),(196,214),(197,233),(198,212),(199,231),(200,210)]])
138 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 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 | 1 | 1 | 2 | 2 | 20 | 20 | 2 | 2 | 1 | ··· | 1 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 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 | 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 | C2 | C3 | C6 | C6 | C6 | C6 | D4 | D4 | D5 | SD16 | D10 | D10 | C3×D4 | C3×D4 | C3×D5 | D20 | D20 | C3×SD16 | C6×D5 | C6×D5 | C40⋊C2 | C3×D20 | C3×D20 | C3×C40⋊C2 |
| kernel | C6×C40⋊C2 | C3×C40⋊C2 | C2×C120 | C6×Dic10 | C6×D20 | C2×C40⋊C2 | C40⋊C2 | C2×C40 | C2×Dic10 | 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 | 1 | 1 | 2 | 8 | 2 | 2 | 2 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 4 | 16 | 8 | 8 | 32 |
Matrix representation of C6×C40⋊C2 ►in GL4(𝔽241) generated by
| 226 | 0 | 0 | 0 |
| 0 | 226 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 189 | 0 | 0 |
| 51 | 189 | 0 | 0 |
| 0 | 0 | 204 | 68 |
| 0 | 0 | 173 | 125 |
| 52 | 188 | 0 | 0 |
| 51 | 189 | 0 | 0 |
| 0 | 0 | 240 | 52 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(241))| [226,0,0,0,0,226,0,0,0,0,16,0,0,0,0,16],[0,51,0,0,189,189,0,0,0,0,204,173,0,0,68,125],[52,51,0,0,188,189,0,0,0,0,240,0,0,0,52,1] >;
C6×C40⋊C2 in GAP, Magma, Sage, TeX
C_6\times C_{40}\rtimes C_2 % in TeX
G:=Group("C6xC40:C2"); // GroupNames label
G:=SmallGroup(480,695);
// by ID
G=gap.SmallGroup(480,695);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,590,142,2524,102,18822]);
// 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^19>;
// generators/relations