metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.19D6, D120⋊12C2, Dic12⋊7D5, D10.7D12, C24.46D10, C120.23C22, C60.125C23, Dic5.26D12, Dic6.22D10, D60.34C22, (C8×D5)⋊3S3, (D5×C24)⋊3C2, C5⋊3(C4○D24), C15⋊3(C4○D8), C6.10(D4×D5), C8.22(S3×D5), (C4×D5).81D6, (C6×D5).45D4, C2.15(D5×D12), C30.21(C2×D4), C5⋊2C8.34D6, C3⋊1(Q8.D10), (C5×Dic12)⋊3C2, C10.10(C2×D12), C12.28D10⋊10C2, Dic6⋊D5⋊12C2, C20.78(C22×S3), (C3×Dic5).49D4, (D5×C12).95C22, C12.148(C22×D5), (C5×Dic6).26C22, C4.73(C2×S3×D5), (C3×C5⋊2C8).38C22, SmallGroup(480,347)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D120⋊C2
G = < a,b,c | a120=b2=c2=1, bab=a-1, cac=a49, cbc=a108b >
Subgroups: 828 in 124 conjugacy classes, 40 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, Q8, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, C20, D10, D10, C24, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D5, D15, C30, C4○D8, C5⋊2C8, C40, C4×D5, C4×D5, D20, C5×Q8, C24⋊C2, D24, Dic12, C2×C24, C4○D12, C5×Dic3, C3×Dic5, C60, C6×D5, D30, C8×D5, D40, Q8⋊D5, C5×Q16, Q8⋊2D5, C4○D24, C3×C5⋊2C8, C120, D30.C2, C3⋊D20, D5×C12, C5×Dic6, D60, Q8.D10, Dic6⋊D5, D5×C24, C5×Dic12, D120, C12.28D10, D120⋊C2
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, C22×S3, C4○D8, C22×D5, C2×D12, S3×D5, D4×D5, C4○D24, C2×S3×D5, Q8.D10, D5×D12, D120⋊C2
►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)
(1 185)(2 184)(3 183)(4 182)(5 181)(6 180)(7 179)(8 178)(9 177)(10 176)(11 175)(12 174)(13 173)(14 172)(15 171)(16 170)(17 169)(18 168)(19 167)(20 166)(21 165)(22 164)(23 163)(24 162)(25 161)(26 160)(27 159)(28 158)(29 157)(30 156)(31 155)(32 154)(33 153)(34 152)(35 151)(36 150)(37 149)(38 148)(39 147)(40 146)(41 145)(42 144)(43 143)(44 142)(45 141)(46 140)(47 139)(48 138)(49 137)(50 136)(51 135)(52 134)(53 133)(54 132)(55 131)(56 130)(57 129)(58 128)(59 127)(60 126)(61 125)(62 124)(63 123)(64 122)(65 121)(66 240)(67 239)(68 238)(69 237)(70 236)(71 235)(72 234)(73 233)(74 232)(75 231)(76 230)(77 229)(78 228)(79 227)(80 226)(81 225)(82 224)(83 223)(84 222)(85 221)(86 220)(87 219)(88 218)(89 217)(90 216)(91 215)(92 214)(93 213)(94 212)(95 211)(96 210)(97 209)(98 208)(99 207)(100 206)(101 205)(102 204)(103 203)(104 202)(105 201)(106 200)(107 199)(108 198)(109 197)(110 196)(111 195)(112 194)(113 193)(114 192)(115 191)(116 190)(117 189)(118 188)(119 187)(120 186)
(2 50)(3 99)(4 28)(5 77)(7 55)(8 104)(9 33)(10 82)(12 60)(13 109)(14 38)(15 87)(17 65)(18 114)(19 43)(20 92)(22 70)(23 119)(24 48)(25 97)(27 75)(29 53)(30 102)(32 80)(34 58)(35 107)(37 85)(39 63)(40 112)(42 90)(44 68)(45 117)(47 95)(49 73)(52 100)(54 78)(57 105)(59 83)(62 110)(64 88)(67 115)(69 93)(72 120)(74 98)(79 103)(84 108)(89 113)(94 118)(121 181)(122 230)(123 159)(124 208)(125 137)(126 186)(127 235)(128 164)(129 213)(130 142)(131 191)(132 240)(133 169)(134 218)(135 147)(136 196)(138 174)(139 223)(140 152)(141 201)(143 179)(144 228)(145 157)(146 206)(148 184)(149 233)(150 162)(151 211)(153 189)(154 238)(155 167)(156 216)(158 194)(160 172)(161 221)(163 199)(165 177)(166 226)(168 204)(170 182)(171 231)(173 209)(175 187)(176 236)(178 214)(180 192)(183 219)(185 197)(188 224)(190 202)(193 229)(195 207)(198 234)(200 212)(203 239)(205 217)(210 222)(215 227)(220 232)(225 237)
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), (1,185)(2,184)(3,183)(4,182)(5,181)(6,180)(7,179)(8,178)(9,177)(10,176)(11,175)(12,174)(13,173)(14,172)(15,171)(16,170)(17,169)(18,168)(19,167)(20,166)(21,165)(22,164)(23,163)(24,162)(25,161)(26,160)(27,159)(28,158)(29,157)(30,156)(31,155)(32,154)(33,153)(34,152)(35,151)(36,150)(37,149)(38,148)(39,147)(40,146)(41,145)(42,144)(43,143)(44,142)(45,141)(46,140)(47,139)(48,138)(49,137)(50,136)(51,135)(52,134)(53,133)(54,132)(55,131)(56,130)(57,129)(58,128)(59,127)(60,126)(61,125)(62,124)(63,123)(64,122)(65,121)(66,240)(67,239)(68,238)(69,237)(70,236)(71,235)(72,234)(73,233)(74,232)(75,231)(76,230)(77,229)(78,228)(79,227)(80,226)(81,225)(82,224)(83,223)(84,222)(85,221)(86,220)(87,219)(88,218)(89,217)(90,216)(91,215)(92,214)(93,213)(94,212)(95,211)(96,210)(97,209)(98,208)(99,207)(100,206)(101,205)(102,204)(103,203)(104,202)(105,201)(106,200)(107,199)(108,198)(109,197)(110,196)(111,195)(112,194)(113,193)(114,192)(115,191)(116,190)(117,189)(118,188)(119,187)(120,186), (2,50)(3,99)(4,28)(5,77)(7,55)(8,104)(9,33)(10,82)(12,60)(13,109)(14,38)(15,87)(17,65)(18,114)(19,43)(20,92)(22,70)(23,119)(24,48)(25,97)(27,75)(29,53)(30,102)(32,80)(34,58)(35,107)(37,85)(39,63)(40,112)(42,90)(44,68)(45,117)(47,95)(49,73)(52,100)(54,78)(57,105)(59,83)(62,110)(64,88)(67,115)(69,93)(72,120)(74,98)(79,103)(84,108)(89,113)(94,118)(121,181)(122,230)(123,159)(124,208)(125,137)(126,186)(127,235)(128,164)(129,213)(130,142)(131,191)(132,240)(133,169)(134,218)(135,147)(136,196)(138,174)(139,223)(140,152)(141,201)(143,179)(144,228)(145,157)(146,206)(148,184)(149,233)(150,162)(151,211)(153,189)(154,238)(155,167)(156,216)(158,194)(160,172)(161,221)(163,199)(165,177)(166,226)(168,204)(170,182)(171,231)(173,209)(175,187)(176,236)(178,214)(180,192)(183,219)(185,197)(188,224)(190,202)(193,229)(195,207)(198,234)(200,212)(203,239)(205,217)(210,222)(215,227)(220,232)(225,237)>;
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), (1,185)(2,184)(3,183)(4,182)(5,181)(6,180)(7,179)(8,178)(9,177)(10,176)(11,175)(12,174)(13,173)(14,172)(15,171)(16,170)(17,169)(18,168)(19,167)(20,166)(21,165)(22,164)(23,163)(24,162)(25,161)(26,160)(27,159)(28,158)(29,157)(30,156)(31,155)(32,154)(33,153)(34,152)(35,151)(36,150)(37,149)(38,148)(39,147)(40,146)(41,145)(42,144)(43,143)(44,142)(45,141)(46,140)(47,139)(48,138)(49,137)(50,136)(51,135)(52,134)(53,133)(54,132)(55,131)(56,130)(57,129)(58,128)(59,127)(60,126)(61,125)(62,124)(63,123)(64,122)(65,121)(66,240)(67,239)(68,238)(69,237)(70,236)(71,235)(72,234)(73,233)(74,232)(75,231)(76,230)(77,229)(78,228)(79,227)(80,226)(81,225)(82,224)(83,223)(84,222)(85,221)(86,220)(87,219)(88,218)(89,217)(90,216)(91,215)(92,214)(93,213)(94,212)(95,211)(96,210)(97,209)(98,208)(99,207)(100,206)(101,205)(102,204)(103,203)(104,202)(105,201)(106,200)(107,199)(108,198)(109,197)(110,196)(111,195)(112,194)(113,193)(114,192)(115,191)(116,190)(117,189)(118,188)(119,187)(120,186), (2,50)(3,99)(4,28)(5,77)(7,55)(8,104)(9,33)(10,82)(12,60)(13,109)(14,38)(15,87)(17,65)(18,114)(19,43)(20,92)(22,70)(23,119)(24,48)(25,97)(27,75)(29,53)(30,102)(32,80)(34,58)(35,107)(37,85)(39,63)(40,112)(42,90)(44,68)(45,117)(47,95)(49,73)(52,100)(54,78)(57,105)(59,83)(62,110)(64,88)(67,115)(69,93)(72,120)(74,98)(79,103)(84,108)(89,113)(94,118)(121,181)(122,230)(123,159)(124,208)(125,137)(126,186)(127,235)(128,164)(129,213)(130,142)(131,191)(132,240)(133,169)(134,218)(135,147)(136,196)(138,174)(139,223)(140,152)(141,201)(143,179)(144,228)(145,157)(146,206)(148,184)(149,233)(150,162)(151,211)(153,189)(154,238)(155,167)(156,216)(158,194)(160,172)(161,221)(163,199)(165,177)(166,226)(168,204)(170,182)(171,231)(173,209)(175,187)(176,236)(178,214)(180,192)(183,219)(185,197)(188,224)(190,202)(193,229)(195,207)(198,234)(200,212)(203,239)(205,217)(210,222)(215,227)(220,232)(225,237) );
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)], [(1,185),(2,184),(3,183),(4,182),(5,181),(6,180),(7,179),(8,178),(9,177),(10,176),(11,175),(12,174),(13,173),(14,172),(15,171),(16,170),(17,169),(18,168),(19,167),(20,166),(21,165),(22,164),(23,163),(24,162),(25,161),(26,160),(27,159),(28,158),(29,157),(30,156),(31,155),(32,154),(33,153),(34,152),(35,151),(36,150),(37,149),(38,148),(39,147),(40,146),(41,145),(42,144),(43,143),(44,142),(45,141),(46,140),(47,139),(48,138),(49,137),(50,136),(51,135),(52,134),(53,133),(54,132),(55,131),(56,130),(57,129),(58,128),(59,127),(60,126),(61,125),(62,124),(63,123),(64,122),(65,121),(66,240),(67,239),(68,238),(69,237),(70,236),(71,235),(72,234),(73,233),(74,232),(75,231),(76,230),(77,229),(78,228),(79,227),(80,226),(81,225),(82,224),(83,223),(84,222),(85,221),(86,220),(87,219),(88,218),(89,217),(90,216),(91,215),(92,214),(93,213),(94,212),(95,211),(96,210),(97,209),(98,208),(99,207),(100,206),(101,205),(102,204),(103,203),(104,202),(105,201),(106,200),(107,199),(108,198),(109,197),(110,196),(111,195),(112,194),(113,193),(114,192),(115,191),(116,190),(117,189),(118,188),(119,187),(120,186)], [(2,50),(3,99),(4,28),(5,77),(7,55),(8,104),(9,33),(10,82),(12,60),(13,109),(14,38),(15,87),(17,65),(18,114),(19,43),(20,92),(22,70),(23,119),(24,48),(25,97),(27,75),(29,53),(30,102),(32,80),(34,58),(35,107),(37,85),(39,63),(40,112),(42,90),(44,68),(45,117),(47,95),(49,73),(52,100),(54,78),(57,105),(59,83),(62,110),(64,88),(67,115),(69,93),(72,120),(74,98),(79,103),(84,108),(89,113),(94,118),(121,181),(122,230),(123,159),(124,208),(125,137),(126,186),(127,235),(128,164),(129,213),(130,142),(131,191),(132,240),(133,169),(134,218),(135,147),(136,196),(138,174),(139,223),(140,152),(141,201),(143,179),(144,228),(145,157),(146,206),(148,184),(149,233),(150,162),(151,211),(153,189),(154,238),(155,167),(156,216),(158,194),(160,172),(161,221),(163,199),(165,177),(166,226),(168,204),(170,182),(171,231),(173,209),(175,187),(176,236),(178,214),(180,192),(183,219),(185,197),(188,224),(190,202),(193,229),(195,207),(198,234),(200,212),(203,239),(205,217),(210,222),(215,227),(220,232),(225,237)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | 10B | 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 | 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 | 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 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 10 | 60 | 60 | 2 | 2 | 5 | 5 | 12 | 12 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 10 | 10 | 4 | 4 | 4 | 4 | 24 | 24 | 24 | 24 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 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 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D12 | D12 | C4○D8 | C4○D24 | S3×D5 | D4×D5 | C2×S3×D5 | Q8.D10 | D5×D12 | D120⋊C2 |
| kernel | D120⋊C2 | Dic6⋊D5 | D5×C24 | C5×Dic12 | D120 | C12.28D10 | C8×D5 | C3×Dic5 | C6×D5 | Dic12 | C5⋊2C8 | C40 | C4×D5 | C24 | Dic6 | Dic5 | D10 | C15 | C5 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 8 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of D120⋊C2 ►in GL4(𝔽241) generated by
| 128 | 0 | 0 | 0 |
| 23 | 209 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 51 |
| 150 | 172 | 0 | 0 |
| 120 | 91 | 0 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 0 | 240 | 0 |
| 1 | 0 | 0 | 0 |
| 172 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 51 | 240 |
G:=sub<GL(4,GF(241))| [128,23,0,0,0,209,0,0,0,0,0,240,0,0,1,51],[150,120,0,0,172,91,0,0,0,0,0,240,0,0,240,0],[1,172,0,0,0,240,0,0,0,0,1,51,0,0,0,240] >;
D120⋊C2 in GAP, Magma, Sage, TeX
D_{120}\rtimes C_2 % in TeX
G:=Group("D120:C2"); // GroupNames label
G:=SmallGroup(480,347);
// by ID
G=gap.SmallGroup(480,347);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,135,142,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^120=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^49,c*b*c=a^108*b>;
// generators/relations