metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60⋊8D4, C12⋊7D20, D10⋊3D12, (C6×D5)⋊11D4, C4⋊Dic3⋊9D5, C6.23(D4×D5), (C2×D60)⋊26C2, C4⋊2(C3⋊D20), C15⋊7(C4⋊D4), C3⋊3(C4⋊D20), C5⋊1(C12⋊7D4), C20⋊2(C3⋊D4), C2.25(D5×D12), C30.60(C2×D4), C6.61(C2×D20), C10.24(C2×D12), (C2×C20).130D6, D30⋊4C4⋊18C2, C30.85(C4○D4), (C2×C12).306D10, (C22×D5).92D6, C10.15(C4○D12), (C2×C60).150C22, (C2×C30).140C23, C6.17(Q8⋊2D5), (C2×Dic5).184D6, (C2×Dic3).44D10, C2.17(C12.28D10), (C10×Dic3).87C22, (C6×Dic5).211C22, (C22×D15).47C22, (C2×C4×D5)⋊2S3, (D5×C2×C12)⋊3C2, (C2×C3⋊D20)⋊3C2, (C5×C4⋊Dic3)⋊6C2, (C2×C4).163(S3×D5), C2.19(C2×C3⋊D20), C10.16(C2×C3⋊D4), C22.192(C2×S3×D5), (D5×C2×C6).108C22, (C2×C6).152(C22×D5), (C2×C10).152(C22×S3), SmallGroup(480,526)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊7D20
G = < a,b,c | a12=b20=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 1276 in 188 conjugacy classes, 56 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, C12, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C3×D5, D15, C30, C4⋊D4, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C4⋊Dic3, D6⋊C4, C2×D12, C2×C3⋊D4, C22×C12, C5×Dic3, C3×Dic5, C60, C6×D5, C6×D5, D30, C2×C30, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C12⋊7D4, C3⋊D20, D5×C12, C6×Dic5, C10×Dic3, D60, C2×C60, D5×C2×C6, C22×D15, C4⋊D20, D30⋊4C4, C5×C4⋊Dic3, C2×C3⋊D20, D5×C2×C12, C2×D60, C12⋊7D20
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, D12, C3⋊D4, C22×S3, C4⋊D4, D20, C22×D5, C2×D12, C4○D12, C2×C3⋊D4, S3×D5, C2×D20, D4×D5, Q8⋊2D5, C12⋊7D4, C3⋊D20, C2×S3×D5, C4⋊D20, C12.28D10, D5×D12, C2×C3⋊D20, C12⋊7D20
►On 240 points
Generators in S240
(1 191 232 159 133 30 204 69 101 83 169 50)(2 51 170 84 102 70 205 31 134 160 233 192)(3 193 234 141 135 32 206 71 103 85 171 52)(4 53 172 86 104 72 207 33 136 142 235 194)(5 195 236 143 137 34 208 73 105 87 173 54)(6 55 174 88 106 74 209 35 138 144 237 196)(7 197 238 145 139 36 210 75 107 89 175 56)(8 57 176 90 108 76 211 37 140 146 239 198)(9 199 240 147 121 38 212 77 109 91 177 58)(10 59 178 92 110 78 213 39 122 148 221 200)(11 181 222 149 123 40 214 79 111 93 179 60)(12 41 180 94 112 80 215 21 124 150 223 182)(13 183 224 151 125 22 216 61 113 95 161 42)(14 43 162 96 114 62 217 23 126 152 225 184)(15 185 226 153 127 24 218 63 115 97 163 44)(16 45 164 98 116 64 219 25 128 154 227 186)(17 187 228 155 129 26 220 65 117 99 165 46)(18 47 166 100 118 66 201 27 130 156 229 188)(19 189 230 157 131 28 202 67 119 81 167 48)(20 49 168 82 120 68 203 29 132 158 231 190)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 30)(39 40)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(59 60)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 80)(78 79)(81 84)(82 83)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)(101 120)(102 119)(103 118)(104 117)(105 116)(106 115)(107 114)(108 113)(109 112)(110 111)(121 124)(122 123)(125 140)(126 139)(127 138)(128 137)(129 136)(130 135)(131 134)(132 133)(141 156)(142 155)(143 154)(144 153)(145 152)(146 151)(147 150)(148 149)(157 160)(158 159)(161 176)(162 175)(163 174)(164 173)(165 172)(166 171)(167 170)(168 169)(177 180)(178 179)(181 200)(182 199)(183 198)(184 197)(185 196)(186 195)(187 194)(188 193)(189 192)(190 191)(201 206)(202 205)(203 204)(207 220)(208 219)(209 218)(210 217)(211 216)(212 215)(213 214)(221 222)(223 240)(224 239)(225 238)(226 237)(227 236)(228 235)(229 234)(230 233)(231 232)
G:=sub<Sym(240)| (1,191,232,159,133,30,204,69,101,83,169,50)(2,51,170,84,102,70,205,31,134,160,233,192)(3,193,234,141,135,32,206,71,103,85,171,52)(4,53,172,86,104,72,207,33,136,142,235,194)(5,195,236,143,137,34,208,73,105,87,173,54)(6,55,174,88,106,74,209,35,138,144,237,196)(7,197,238,145,139,36,210,75,107,89,175,56)(8,57,176,90,108,76,211,37,140,146,239,198)(9,199,240,147,121,38,212,77,109,91,177,58)(10,59,178,92,110,78,213,39,122,148,221,200)(11,181,222,149,123,40,214,79,111,93,179,60)(12,41,180,94,112,80,215,21,124,150,223,182)(13,183,224,151,125,22,216,61,113,95,161,42)(14,43,162,96,114,62,217,23,126,152,225,184)(15,185,226,153,127,24,218,63,115,97,163,44)(16,45,164,98,116,64,219,25,128,154,227,186)(17,187,228,155,129,26,220,65,117,99,165,46)(18,47,166,100,118,66,201,27,130,156,229,188)(19,189,230,157,131,28,202,67,119,81,167,48)(20,49,168,82,120,68,203,29,132,158,231,190), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(39,40)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79)(81,84)(82,83)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(101,120)(102,119)(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111)(121,124)(122,123)(125,140)(126,139)(127,138)(128,137)(129,136)(130,135)(131,134)(132,133)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149)(157,160)(158,159)(161,176)(162,175)(163,174)(164,173)(165,172)(166,171)(167,170)(168,169)(177,180)(178,179)(181,200)(182,199)(183,198)(184,197)(185,196)(186,195)(187,194)(188,193)(189,192)(190,191)(201,206)(202,205)(203,204)(207,220)(208,219)(209,218)(210,217)(211,216)(212,215)(213,214)(221,222)(223,240)(224,239)(225,238)(226,237)(227,236)(228,235)(229,234)(230,233)(231,232)>;
G:=Group( (1,191,232,159,133,30,204,69,101,83,169,50)(2,51,170,84,102,70,205,31,134,160,233,192)(3,193,234,141,135,32,206,71,103,85,171,52)(4,53,172,86,104,72,207,33,136,142,235,194)(5,195,236,143,137,34,208,73,105,87,173,54)(6,55,174,88,106,74,209,35,138,144,237,196)(7,197,238,145,139,36,210,75,107,89,175,56)(8,57,176,90,108,76,211,37,140,146,239,198)(9,199,240,147,121,38,212,77,109,91,177,58)(10,59,178,92,110,78,213,39,122,148,221,200)(11,181,222,149,123,40,214,79,111,93,179,60)(12,41,180,94,112,80,215,21,124,150,223,182)(13,183,224,151,125,22,216,61,113,95,161,42)(14,43,162,96,114,62,217,23,126,152,225,184)(15,185,226,153,127,24,218,63,115,97,163,44)(16,45,164,98,116,64,219,25,128,154,227,186)(17,187,228,155,129,26,220,65,117,99,165,46)(18,47,166,100,118,66,201,27,130,156,229,188)(19,189,230,157,131,28,202,67,119,81,167,48)(20,49,168,82,120,68,203,29,132,158,231,190), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(39,40)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79)(81,84)(82,83)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(101,120)(102,119)(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111)(121,124)(122,123)(125,140)(126,139)(127,138)(128,137)(129,136)(130,135)(131,134)(132,133)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149)(157,160)(158,159)(161,176)(162,175)(163,174)(164,173)(165,172)(166,171)(167,170)(168,169)(177,180)(178,179)(181,200)(182,199)(183,198)(184,197)(185,196)(186,195)(187,194)(188,193)(189,192)(190,191)(201,206)(202,205)(203,204)(207,220)(208,219)(209,218)(210,217)(211,216)(212,215)(213,214)(221,222)(223,240)(224,239)(225,238)(226,237)(227,236)(228,235)(229,234)(230,233)(231,232) );
G=PermutationGroup([[(1,191,232,159,133,30,204,69,101,83,169,50),(2,51,170,84,102,70,205,31,134,160,233,192),(3,193,234,141,135,32,206,71,103,85,171,52),(4,53,172,86,104,72,207,33,136,142,235,194),(5,195,236,143,137,34,208,73,105,87,173,54),(6,55,174,88,106,74,209,35,138,144,237,196),(7,197,238,145,139,36,210,75,107,89,175,56),(8,57,176,90,108,76,211,37,140,146,239,198),(9,199,240,147,121,38,212,77,109,91,177,58),(10,59,178,92,110,78,213,39,122,148,221,200),(11,181,222,149,123,40,214,79,111,93,179,60),(12,41,180,94,112,80,215,21,124,150,223,182),(13,183,224,151,125,22,216,61,113,95,161,42),(14,43,162,96,114,62,217,23,126,152,225,184),(15,185,226,153,127,24,218,63,115,97,163,44),(16,45,164,98,116,64,219,25,128,154,227,186),(17,187,228,155,129,26,220,65,117,99,165,46),(18,47,166,100,118,66,201,27,130,156,229,188),(19,189,230,157,131,28,202,67,119,81,167,48),(20,49,168,82,120,68,203,29,132,158,231,190)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,38),(22,37),(23,36),(24,35),(25,34),(26,33),(27,32),(28,31),(29,30),(39,40),(41,58),(42,57),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50),(59,60),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,70),(68,69),(77,80),(78,79),(81,84),(82,83),(85,100),(86,99),(87,98),(88,97),(89,96),(90,95),(91,94),(92,93),(101,120),(102,119),(103,118),(104,117),(105,116),(106,115),(107,114),(108,113),(109,112),(110,111),(121,124),(122,123),(125,140),(126,139),(127,138),(128,137),(129,136),(130,135),(131,134),(132,133),(141,156),(142,155),(143,154),(144,153),(145,152),(146,151),(147,150),(148,149),(157,160),(158,159),(161,176),(162,175),(163,174),(164,173),(165,172),(166,171),(167,170),(168,169),(177,180),(178,179),(181,200),(182,199),(183,198),(184,197),(185,196),(186,195),(187,194),(188,193),(189,192),(190,191),(201,206),(202,205),(203,204),(207,220),(208,219),(209,218),(210,217),(211,216),(212,215),(213,214),(221,222),(223,240),(224,239),(225,238),(226,237),(227,236),(228,235),(229,234),(230,233),(231,232)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 60 | 60 | 2 | 2 | 2 | 10 | 10 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | C3⋊D4 | D12 | D20 | C4○D12 | S3×D5 | D4×D5 | Q8⋊2D5 | C3⋊D20 | C2×S3×D5 | C12.28D10 | D5×D12 |
| kernel | C12⋊7D20 | D30⋊4C4 | C5×C4⋊Dic3 | C2×C3⋊D20 | D5×C2×C12 | C2×D60 | C2×C4×D5 | C60 | C6×D5 | C4⋊Dic3 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | C20 | D10 | C12 | C10 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 8 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C12⋊7D20 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 22 |
| 0 | 0 | 0 | 29 |
| 7 | 32 | 0 | 0 |
| 29 | 2 | 0 | 0 |
| 0 | 0 | 31 | 5 |
| 0 | 0 | 15 | 30 |
| 7 | 32 | 0 | 0 |
| 29 | 54 | 0 | 0 |
| 0 | 0 | 30 | 3 |
| 0 | 0 | 46 | 31 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,22,29],[7,29,0,0,32,2,0,0,0,0,31,15,0,0,5,30],[7,29,0,0,32,54,0,0,0,0,30,46,0,0,3,31] >;
C12⋊7D20 in GAP, Magma, Sage, TeX
C_{12}\rtimes_7D_{20} % in TeX
G:=Group("C12:7D20"); // GroupNames label
G:=SmallGroup(480,526);
// by ID
G=gap.SmallGroup(480,526);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,64,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^12=b^20=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations