metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D120⋊8C2, Q16⋊3D15, C8.10D30, C40.24D6, D30.19D4, C24.24D10, Q8.10D30, C60.73C23, C120.15C22, Dic15.51D4, D60.24C22, (C8×D15)⋊3C2, (C5×Q16)⋊3S3, (C3×Q16)⋊3D5, C15⋊34(C4○D8), (C15×Q16)⋊3C2, C2.24(D4×D15), C6.117(D4×D5), (C5×Q8).34D6, C5⋊4(D24⋊C2), C3⋊4(Q8.D10), C30.324(C2×D4), C10.119(S3×D4), (C3×Q8).17D10, Q8⋊3D15⋊10C2, Q8⋊2D15⋊12C2, C4.10(C22×D15), C20.111(C22×S3), C15⋊3C8.34C22, (C4×D15).47C22, C12.111(C22×D5), (Q8×C15).26C22, SmallGroup(480,884)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D120⋊8C2
G = < a,b,c | a120=b2=c2=1, bab=a-1, cac=a89, cbc=a28b >
Subgroups: 884 in 124 conjugacy classes, 41 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C8, C8, C2×C4, D4, Q8, D5, C10, Dic3, C12, C12, D6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, C20, D10, C3⋊C8, C24, C4×S3, D12, C3×Q8, D15, C30, C4○D8, C5⋊2C8, C40, C4×D5, D20, C5×Q8, S3×C8, D24, Q8⋊2S3, C3×Q16, Q8⋊3S3, Dic15, C60, C60, D30, D30, C8×D5, D40, Q8⋊D5, C5×Q16, Q8⋊2D5, D24⋊C2, C15⋊3C8, C120, C4×D15, C4×D15, D60, D60, Q8×C15, Q8.D10, C8×D15, D120, Q8⋊2D15, C15×Q16, Q8⋊3D15, D120⋊8C2
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, D15, C4○D8, C22×D5, S3×D4, D30, D4×D5, D24⋊C2, C22×D15, Q8.D10, D4×D15, D120⋊8C2
►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)
(1 153)(2 152)(3 151)(4 150)(5 149)(6 148)(7 147)(8 146)(9 145)(10 144)(11 143)(12 142)(13 141)(14 140)(15 139)(16 138)(17 137)(18 136)(19 135)(20 134)(21 133)(22 132)(23 131)(24 130)(25 129)(26 128)(27 127)(28 126)(29 125)(30 124)(31 123)(32 122)(33 121)(34 240)(35 239)(36 238)(37 237)(38 236)(39 235)(40 234)(41 233)(42 232)(43 231)(44 230)(45 229)(46 228)(47 227)(48 226)(49 225)(50 224)(51 223)(52 222)(53 221)(54 220)(55 219)(56 218)(57 217)(58 216)(59 215)(60 214)(61 213)(62 212)(63 211)(64 210)(65 209)(66 208)(67 207)(68 206)(69 205)(70 204)(71 203)(72 202)(73 201)(74 200)(75 199)(76 198)(77 197)(78 196)(79 195)(80 194)(81 193)(82 192)(83 191)(84 190)(85 189)(86 188)(87 187)(88 186)(89 185)(90 184)(91 183)(92 182)(93 181)(94 180)(95 179)(96 178)(97 177)(98 176)(99 175)(100 174)(101 173)(102 172)(103 171)(104 170)(105 169)(106 168)(107 167)(108 166)(109 165)(110 164)(111 163)(112 162)(113 161)(114 160)(115 159)(116 158)(117 157)(118 156)(119 155)(120 154)
(2 90)(3 59)(4 28)(5 117)(6 86)(7 55)(8 24)(9 113)(10 82)(11 51)(12 20)(13 109)(14 78)(15 47)(17 105)(18 74)(19 43)(21 101)(22 70)(23 39)(25 97)(26 66)(27 35)(29 93)(30 62)(32 120)(33 89)(34 58)(36 116)(37 85)(38 54)(40 112)(41 81)(42 50)(44 108)(45 77)(48 104)(49 73)(52 100)(53 69)(56 96)(57 65)(60 92)(63 119)(64 88)(67 115)(68 84)(71 111)(72 80)(75 107)(79 103)(83 99)(87 95)(94 118)(98 114)(102 110)(121 157)(122 126)(123 215)(124 184)(125 153)(127 211)(128 180)(129 149)(130 238)(131 207)(132 176)(133 145)(134 234)(135 203)(136 172)(137 141)(138 230)(139 199)(140 168)(142 226)(143 195)(144 164)(146 222)(147 191)(148 160)(150 218)(151 187)(152 156)(154 214)(155 183)(158 210)(159 179)(161 237)(162 206)(163 175)(165 233)(166 202)(167 171)(169 229)(170 198)(173 225)(174 194)(177 221)(178 190)(181 217)(182 186)(185 213)(188 240)(189 209)(192 236)(193 205)(196 232)(197 201)(200 228)(204 224)(208 220)(212 216)(219 239)(223 235)(227 231)
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), (1,153)(2,152)(3,151)(4,150)(5,149)(6,148)(7,147)(8,146)(9,145)(10,144)(11,143)(12,142)(13,141)(14,140)(15,139)(16,138)(17,137)(18,136)(19,135)(20,134)(21,133)(22,132)(23,131)(24,130)(25,129)(26,128)(27,127)(28,126)(29,125)(30,124)(31,123)(32,122)(33,121)(34,240)(35,239)(36,238)(37,237)(38,236)(39,235)(40,234)(41,233)(42,232)(43,231)(44,230)(45,229)(46,228)(47,227)(48,226)(49,225)(50,224)(51,223)(52,222)(53,221)(54,220)(55,219)(56,218)(57,217)(58,216)(59,215)(60,214)(61,213)(62,212)(63,211)(64,210)(65,209)(66,208)(67,207)(68,206)(69,205)(70,204)(71,203)(72,202)(73,201)(74,200)(75,199)(76,198)(77,197)(78,196)(79,195)(80,194)(81,193)(82,192)(83,191)(84,190)(85,189)(86,188)(87,187)(88,186)(89,185)(90,184)(91,183)(92,182)(93,181)(94,180)(95,179)(96,178)(97,177)(98,176)(99,175)(100,174)(101,173)(102,172)(103,171)(104,170)(105,169)(106,168)(107,167)(108,166)(109,165)(110,164)(111,163)(112,162)(113,161)(114,160)(115,159)(116,158)(117,157)(118,156)(119,155)(120,154), (2,90)(3,59)(4,28)(5,117)(6,86)(7,55)(8,24)(9,113)(10,82)(11,51)(12,20)(13,109)(14,78)(15,47)(17,105)(18,74)(19,43)(21,101)(22,70)(23,39)(25,97)(26,66)(27,35)(29,93)(30,62)(32,120)(33,89)(34,58)(36,116)(37,85)(38,54)(40,112)(41,81)(42,50)(44,108)(45,77)(48,104)(49,73)(52,100)(53,69)(56,96)(57,65)(60,92)(63,119)(64,88)(67,115)(68,84)(71,111)(72,80)(75,107)(79,103)(83,99)(87,95)(94,118)(98,114)(102,110)(121,157)(122,126)(123,215)(124,184)(125,153)(127,211)(128,180)(129,149)(130,238)(131,207)(132,176)(133,145)(134,234)(135,203)(136,172)(137,141)(138,230)(139,199)(140,168)(142,226)(143,195)(144,164)(146,222)(147,191)(148,160)(150,218)(151,187)(152,156)(154,214)(155,183)(158,210)(159,179)(161,237)(162,206)(163,175)(165,233)(166,202)(167,171)(169,229)(170,198)(173,225)(174,194)(177,221)(178,190)(181,217)(182,186)(185,213)(188,240)(189,209)(192,236)(193,205)(196,232)(197,201)(200,228)(204,224)(208,220)(212,216)(219,239)(223,235)(227,231)>;
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), (1,153)(2,152)(3,151)(4,150)(5,149)(6,148)(7,147)(8,146)(9,145)(10,144)(11,143)(12,142)(13,141)(14,140)(15,139)(16,138)(17,137)(18,136)(19,135)(20,134)(21,133)(22,132)(23,131)(24,130)(25,129)(26,128)(27,127)(28,126)(29,125)(30,124)(31,123)(32,122)(33,121)(34,240)(35,239)(36,238)(37,237)(38,236)(39,235)(40,234)(41,233)(42,232)(43,231)(44,230)(45,229)(46,228)(47,227)(48,226)(49,225)(50,224)(51,223)(52,222)(53,221)(54,220)(55,219)(56,218)(57,217)(58,216)(59,215)(60,214)(61,213)(62,212)(63,211)(64,210)(65,209)(66,208)(67,207)(68,206)(69,205)(70,204)(71,203)(72,202)(73,201)(74,200)(75,199)(76,198)(77,197)(78,196)(79,195)(80,194)(81,193)(82,192)(83,191)(84,190)(85,189)(86,188)(87,187)(88,186)(89,185)(90,184)(91,183)(92,182)(93,181)(94,180)(95,179)(96,178)(97,177)(98,176)(99,175)(100,174)(101,173)(102,172)(103,171)(104,170)(105,169)(106,168)(107,167)(108,166)(109,165)(110,164)(111,163)(112,162)(113,161)(114,160)(115,159)(116,158)(117,157)(118,156)(119,155)(120,154), (2,90)(3,59)(4,28)(5,117)(6,86)(7,55)(8,24)(9,113)(10,82)(11,51)(12,20)(13,109)(14,78)(15,47)(17,105)(18,74)(19,43)(21,101)(22,70)(23,39)(25,97)(26,66)(27,35)(29,93)(30,62)(32,120)(33,89)(34,58)(36,116)(37,85)(38,54)(40,112)(41,81)(42,50)(44,108)(45,77)(48,104)(49,73)(52,100)(53,69)(56,96)(57,65)(60,92)(63,119)(64,88)(67,115)(68,84)(71,111)(72,80)(75,107)(79,103)(83,99)(87,95)(94,118)(98,114)(102,110)(121,157)(122,126)(123,215)(124,184)(125,153)(127,211)(128,180)(129,149)(130,238)(131,207)(132,176)(133,145)(134,234)(135,203)(136,172)(137,141)(138,230)(139,199)(140,168)(142,226)(143,195)(144,164)(146,222)(147,191)(148,160)(150,218)(151,187)(152,156)(154,214)(155,183)(158,210)(159,179)(161,237)(162,206)(163,175)(165,233)(166,202)(167,171)(169,229)(170,198)(173,225)(174,194)(177,221)(178,190)(181,217)(182,186)(185,213)(188,240)(189,209)(192,236)(193,205)(196,232)(197,201)(200,228)(204,224)(208,220)(212,216)(219,239)(223,235)(227,231) );
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)], [(1,153),(2,152),(3,151),(4,150),(5,149),(6,148),(7,147),(8,146),(9,145),(10,144),(11,143),(12,142),(13,141),(14,140),(15,139),(16,138),(17,137),(18,136),(19,135),(20,134),(21,133),(22,132),(23,131),(24,130),(25,129),(26,128),(27,127),(28,126),(29,125),(30,124),(31,123),(32,122),(33,121),(34,240),(35,239),(36,238),(37,237),(38,236),(39,235),(40,234),(41,233),(42,232),(43,231),(44,230),(45,229),(46,228),(47,227),(48,226),(49,225),(50,224),(51,223),(52,222),(53,221),(54,220),(55,219),(56,218),(57,217),(58,216),(59,215),(60,214),(61,213),(62,212),(63,211),(64,210),(65,209),(66,208),(67,207),(68,206),(69,205),(70,204),(71,203),(72,202),(73,201),(74,200),(75,199),(76,198),(77,197),(78,196),(79,195),(80,194),(81,193),(82,192),(83,191),(84,190),(85,189),(86,188),(87,187),(88,186),(89,185),(90,184),(91,183),(92,182),(93,181),(94,180),(95,179),(96,178),(97,177),(98,176),(99,175),(100,174),(101,173),(102,172),(103,171),(104,170),(105,169),(106,168),(107,167),(108,166),(109,165),(110,164),(111,163),(112,162),(113,161),(114,160),(115,159),(116,158),(117,157),(118,156),(119,155),(120,154)], [(2,90),(3,59),(4,28),(5,117),(6,86),(7,55),(8,24),(9,113),(10,82),(11,51),(12,20),(13,109),(14,78),(15,47),(17,105),(18,74),(19,43),(21,101),(22,70),(23,39),(25,97),(26,66),(27,35),(29,93),(30,62),(32,120),(33,89),(34,58),(36,116),(37,85),(38,54),(40,112),(41,81),(42,50),(44,108),(45,77),(48,104),(49,73),(52,100),(53,69),(56,96),(57,65),(60,92),(63,119),(64,88),(67,115),(68,84),(71,111),(72,80),(75,107),(79,103),(83,99),(87,95),(94,118),(98,114),(102,110),(121,157),(122,126),(123,215),(124,184),(125,153),(127,211),(128,180),(129,149),(130,238),(131,207),(132,176),(133,145),(134,234),(135,203),(136,172),(137,141),(138,230),(139,199),(140,168),(142,226),(143,195),(144,164),(146,222),(147,191),(148,160),(150,218),(151,187),(152,156),(154,214),(155,183),(158,210),(159,179),(161,237),(162,206),(163,175),(165,233),(166,202),(167,171),(169,229),(170,198),(173,225),(174,194),(177,221),(178,190),(181,217),(182,186),(185,213),(188,240),(189,209),(192,236),(193,205),(196,232),(197,201),(200,228),(204,224),(208,220),(212,216),(219,239),(223,235),(227,231)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6 | 8A | 8B | 8C | 8D | 10A | 10B | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 30A | 30B | 30C | 30D | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 60E | ··· | 60L | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 30 | 60 | 60 | 2 | 2 | 4 | 4 | 15 | 15 | 2 | 2 | 2 | 2 | 2 | 30 | 30 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 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 | D10 | D10 | D15 | C4○D8 | D30 | D30 | S3×D4 | D4×D5 | D24⋊C2 | Q8.D10 | D4×D15 | D120⋊8C2 |
| kernel | D120⋊8C2 | C8×D15 | D120 | Q8⋊2D15 | C15×Q16 | Q8⋊3D15 | C5×Q16 | Dic15 | D30 | C3×Q16 | C40 | C5×Q8 | C24 | C3×Q8 | Q16 | C15 | C8 | Q8 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 1 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of D120⋊8C2 ►in GL4(𝔽241) generated by
| 211 | 63 | 0 | 0 |
| 178 | 68 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 6 | 211 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 138 | 90 |
| 0 | 0 | 91 | 103 |
| 178 | 30 | 0 | 0 |
| 173 | 63 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 13 | 240 |
G:=sub<GL(4,GF(241))| [211,178,0,0,63,68,0,0,0,0,8,6,0,0,0,211],[0,1,0,0,1,0,0,0,0,0,138,91,0,0,90,103],[178,173,0,0,30,63,0,0,0,0,1,13,0,0,0,240] >;
D120⋊8C2 in GAP, Magma, Sage, TeX
D_{120}\rtimes_8C_2 % in TeX
G:=Group("D120:8C2"); // GroupNames label
G:=SmallGroup(480,884);
// by ID
G=gap.SmallGroup(480,884);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,422,135,100,346,185,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^120=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^89,c*b*c=a^28*b>;
// generators/relations