metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊6D4, (C5×D4)⋊6D4, (C2×D8)⋊6D5, C4.60(D4×D5), (C10×D8)⋊14C2, C20⋊2D4⋊5C2, C5⋊5(D4⋊D4), D4⋊3(C5⋊D4), (C2×C8).35D10, (C2×D4).65D10, C20.167(C2×D4), D10⋊1C8⋊28C2, C10.56C22≀C2, C10.35(C4○D8), D4⋊Dic5⋊30C2, (C22×D5).43D4, C22.258(D4×D5), C2.30(D8⋊D5), C2.19(D8⋊3D5), C20.44D4⋊29C2, C10.51(C8⋊C22), (C2×C20).435C23, (C2×C40).249C22, (C2×Dic5).237D4, (D4×C10).84C22, C2.24(C23⋊D10), C4⋊Dic5.166C22, (C2×Dic10).126C22, C4.37(C2×C5⋊D4), (C2×D4⋊2D5)⋊2C2, (C2×D4.D5)⋊19C2, (C2×C4×D5).49C22, (C2×C10).348(C2×D4), (C2×C4).525(C22×D5), (C2×C5⋊2C8).149C22, SmallGroup(320,785)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=cac-1=a-1, dad=a9, cbc-1=a5b, dbd=a10b, dcd=c-1 >
Subgroups: 686 in 162 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, D4⋊D4, C2×C5⋊2C8, C4⋊Dic5, D4.D5, C23.D5, C2×C40, C5×D8, C2×Dic10, C2×C4×D5, D4⋊2D5, C22×Dic5, C2×C5⋊D4, D4×C10, C20.44D4, D10⋊1C8, D4⋊Dic5, C2×D4.D5, C20⋊2D4, C10×D8, C2×D4⋊2D5, Dic10⋊D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C4○D8, C8⋊C22, C5⋊D4, C22×D5, D4⋊D4, D4×D5, C2×C5⋊D4, D8⋊D5, D8⋊3D5, C23⋊D10, Dic10⋊D4
►On 160 points
Generators in S160
(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)
(1 160 11 150)(2 159 12 149)(3 158 13 148)(4 157 14 147)(5 156 15 146)(6 155 16 145)(7 154 17 144)(8 153 18 143)(9 152 19 142)(10 151 20 141)(21 43 31 53)(22 42 32 52)(23 41 33 51)(24 60 34 50)(25 59 35 49)(26 58 36 48)(27 57 37 47)(28 56 38 46)(29 55 39 45)(30 54 40 44)(61 103 71 113)(62 102 72 112)(63 101 73 111)(64 120 74 110)(65 119 75 109)(66 118 76 108)(67 117 77 107)(68 116 78 106)(69 115 79 105)(70 114 80 104)(81 132 91 122)(82 131 92 121)(83 130 93 140)(84 129 94 139)(85 128 95 138)(86 127 96 137)(87 126 97 136)(88 125 98 135)(89 124 99 134)(90 123 100 133)
(1 106 60 91)(2 105 41 90)(3 104 42 89)(4 103 43 88)(5 102 44 87)(6 101 45 86)(7 120 46 85)(8 119 47 84)(9 118 48 83)(10 117 49 82)(11 116 50 81)(12 115 51 100)(13 114 52 99)(14 113 53 98)(15 112 54 97)(16 111 55 96)(17 110 56 95)(18 109 57 94)(19 108 58 93)(20 107 59 92)(21 130 147 76)(22 129 148 75)(23 128 149 74)(24 127 150 73)(25 126 151 72)(26 125 152 71)(27 124 153 70)(28 123 154 69)(29 122 155 68)(30 121 156 67)(31 140 157 66)(32 139 158 65)(33 138 159 64)(34 137 160 63)(35 136 141 62)(36 135 142 61)(37 134 143 80)(38 133 144 79)(39 132 145 78)(40 131 146 77)
(1 60)(2 49)(3 58)(4 47)(5 56)(6 45)(7 54)(8 43)(9 52)(10 41)(11 50)(12 59)(13 48)(14 57)(15 46)(16 55)(17 44)(18 53)(19 42)(20 51)(21 153)(22 142)(23 151)(24 160)(25 149)(26 158)(27 147)(28 156)(29 145)(30 154)(31 143)(32 152)(33 141)(34 150)(35 159)(36 148)(37 157)(38 146)(39 155)(40 144)(61 75)(62 64)(63 73)(65 71)(66 80)(67 69)(68 78)(70 76)(72 74)(77 79)(82 90)(83 99)(84 88)(85 97)(87 95)(89 93)(92 100)(94 98)(102 110)(103 119)(104 108)(105 117)(107 115)(109 113)(112 120)(114 118)(121 123)(122 132)(124 130)(125 139)(126 128)(127 137)(129 135)(131 133)(134 140)(136 138)
G:=sub<Sym(160)| (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), (1,160,11,150)(2,159,12,149)(3,158,13,148)(4,157,14,147)(5,156,15,146)(6,155,16,145)(7,154,17,144)(8,153,18,143)(9,152,19,142)(10,151,20,141)(21,43,31,53)(22,42,32,52)(23,41,33,51)(24,60,34,50)(25,59,35,49)(26,58,36,48)(27,57,37,47)(28,56,38,46)(29,55,39,45)(30,54,40,44)(61,103,71,113)(62,102,72,112)(63,101,73,111)(64,120,74,110)(65,119,75,109)(66,118,76,108)(67,117,77,107)(68,116,78,106)(69,115,79,105)(70,114,80,104)(81,132,91,122)(82,131,92,121)(83,130,93,140)(84,129,94,139)(85,128,95,138)(86,127,96,137)(87,126,97,136)(88,125,98,135)(89,124,99,134)(90,123,100,133), (1,106,60,91)(2,105,41,90)(3,104,42,89)(4,103,43,88)(5,102,44,87)(6,101,45,86)(7,120,46,85)(8,119,47,84)(9,118,48,83)(10,117,49,82)(11,116,50,81)(12,115,51,100)(13,114,52,99)(14,113,53,98)(15,112,54,97)(16,111,55,96)(17,110,56,95)(18,109,57,94)(19,108,58,93)(20,107,59,92)(21,130,147,76)(22,129,148,75)(23,128,149,74)(24,127,150,73)(25,126,151,72)(26,125,152,71)(27,124,153,70)(28,123,154,69)(29,122,155,68)(30,121,156,67)(31,140,157,66)(32,139,158,65)(33,138,159,64)(34,137,160,63)(35,136,141,62)(36,135,142,61)(37,134,143,80)(38,133,144,79)(39,132,145,78)(40,131,146,77), (1,60)(2,49)(3,58)(4,47)(5,56)(6,45)(7,54)(8,43)(9,52)(10,41)(11,50)(12,59)(13,48)(14,57)(15,46)(16,55)(17,44)(18,53)(19,42)(20,51)(21,153)(22,142)(23,151)(24,160)(25,149)(26,158)(27,147)(28,156)(29,145)(30,154)(31,143)(32,152)(33,141)(34,150)(35,159)(36,148)(37,157)(38,146)(39,155)(40,144)(61,75)(62,64)(63,73)(65,71)(66,80)(67,69)(68,78)(70,76)(72,74)(77,79)(82,90)(83,99)(84,88)(85,97)(87,95)(89,93)(92,100)(94,98)(102,110)(103,119)(104,108)(105,117)(107,115)(109,113)(112,120)(114,118)(121,123)(122,132)(124,130)(125,139)(126,128)(127,137)(129,135)(131,133)(134,140)(136,138)>;
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), (1,160,11,150)(2,159,12,149)(3,158,13,148)(4,157,14,147)(5,156,15,146)(6,155,16,145)(7,154,17,144)(8,153,18,143)(9,152,19,142)(10,151,20,141)(21,43,31,53)(22,42,32,52)(23,41,33,51)(24,60,34,50)(25,59,35,49)(26,58,36,48)(27,57,37,47)(28,56,38,46)(29,55,39,45)(30,54,40,44)(61,103,71,113)(62,102,72,112)(63,101,73,111)(64,120,74,110)(65,119,75,109)(66,118,76,108)(67,117,77,107)(68,116,78,106)(69,115,79,105)(70,114,80,104)(81,132,91,122)(82,131,92,121)(83,130,93,140)(84,129,94,139)(85,128,95,138)(86,127,96,137)(87,126,97,136)(88,125,98,135)(89,124,99,134)(90,123,100,133), (1,106,60,91)(2,105,41,90)(3,104,42,89)(4,103,43,88)(5,102,44,87)(6,101,45,86)(7,120,46,85)(8,119,47,84)(9,118,48,83)(10,117,49,82)(11,116,50,81)(12,115,51,100)(13,114,52,99)(14,113,53,98)(15,112,54,97)(16,111,55,96)(17,110,56,95)(18,109,57,94)(19,108,58,93)(20,107,59,92)(21,130,147,76)(22,129,148,75)(23,128,149,74)(24,127,150,73)(25,126,151,72)(26,125,152,71)(27,124,153,70)(28,123,154,69)(29,122,155,68)(30,121,156,67)(31,140,157,66)(32,139,158,65)(33,138,159,64)(34,137,160,63)(35,136,141,62)(36,135,142,61)(37,134,143,80)(38,133,144,79)(39,132,145,78)(40,131,146,77), (1,60)(2,49)(3,58)(4,47)(5,56)(6,45)(7,54)(8,43)(9,52)(10,41)(11,50)(12,59)(13,48)(14,57)(15,46)(16,55)(17,44)(18,53)(19,42)(20,51)(21,153)(22,142)(23,151)(24,160)(25,149)(26,158)(27,147)(28,156)(29,145)(30,154)(31,143)(32,152)(33,141)(34,150)(35,159)(36,148)(37,157)(38,146)(39,155)(40,144)(61,75)(62,64)(63,73)(65,71)(66,80)(67,69)(68,78)(70,76)(72,74)(77,79)(82,90)(83,99)(84,88)(85,97)(87,95)(89,93)(92,100)(94,98)(102,110)(103,119)(104,108)(105,117)(107,115)(109,113)(112,120)(114,118)(121,123)(122,132)(124,130)(125,139)(126,128)(127,137)(129,135)(131,133)(134,140)(136,138) );
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)], [(1,160,11,150),(2,159,12,149),(3,158,13,148),(4,157,14,147),(5,156,15,146),(6,155,16,145),(7,154,17,144),(8,153,18,143),(9,152,19,142),(10,151,20,141),(21,43,31,53),(22,42,32,52),(23,41,33,51),(24,60,34,50),(25,59,35,49),(26,58,36,48),(27,57,37,47),(28,56,38,46),(29,55,39,45),(30,54,40,44),(61,103,71,113),(62,102,72,112),(63,101,73,111),(64,120,74,110),(65,119,75,109),(66,118,76,108),(67,117,77,107),(68,116,78,106),(69,115,79,105),(70,114,80,104),(81,132,91,122),(82,131,92,121),(83,130,93,140),(84,129,94,139),(85,128,95,138),(86,127,96,137),(87,126,97,136),(88,125,98,135),(89,124,99,134),(90,123,100,133)], [(1,106,60,91),(2,105,41,90),(3,104,42,89),(4,103,43,88),(5,102,44,87),(6,101,45,86),(7,120,46,85),(8,119,47,84),(9,118,48,83),(10,117,49,82),(11,116,50,81),(12,115,51,100),(13,114,52,99),(14,113,53,98),(15,112,54,97),(16,111,55,96),(17,110,56,95),(18,109,57,94),(19,108,58,93),(20,107,59,92),(21,130,147,76),(22,129,148,75),(23,128,149,74),(24,127,150,73),(25,126,151,72),(26,125,152,71),(27,124,153,70),(28,123,154,69),(29,122,155,68),(30,121,156,67),(31,140,157,66),(32,139,158,65),(33,138,159,64),(34,137,160,63),(35,136,141,62),(36,135,142,61),(37,134,143,80),(38,133,144,79),(39,132,145,78),(40,131,146,77)], [(1,60),(2,49),(3,58),(4,47),(5,56),(6,45),(7,54),(8,43),(9,52),(10,41),(11,50),(12,59),(13,48),(14,57),(15,46),(16,55),(17,44),(18,53),(19,42),(20,51),(21,153),(22,142),(23,151),(24,160),(25,149),(26,158),(27,147),(28,156),(29,145),(30,154),(31,143),(32,152),(33,141),(34,150),(35,159),(36,148),(37,157),(38,146),(39,155),(40,144),(61,75),(62,64),(63,73),(65,71),(66,80),(67,69),(68,78),(70,76),(72,74),(77,79),(82,90),(83,99),(84,88),(85,97),(87,95),(89,93),(92,100),(94,98),(102,110),(103,119),(104,108),(105,117),(107,115),(109,113),(112,120),(114,118),(121,123),(122,132),(124,130),(125,139),(126,128),(127,137),(129,135),(131,133),(134,140),(136,138)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 20 | 2 | 2 | 10 | 10 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | D10 | D10 | C4○D8 | C5⋊D4 | C8⋊C22 | D4×D5 | D4×D5 | D8⋊D5 | D8⋊3D5 |
| kernel | Dic10⋊D4 | C20.44D4 | D10⋊1C8 | D4⋊Dic5 | C2×D4.D5 | C20⋊2D4 | C10×D8 | C2×D4⋊2D5 | Dic10 | C2×Dic5 | C5×D4 | C22×D5 | C2×D8 | C2×C8 | C2×D4 | C10 | D4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of Dic10⋊D4 ►in GL6(𝔽41)
| 7 | 1 | 0 | 0 | 0 | 0 |
| 33 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 40 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 7 | 6 | 0 | 0 | 0 | 0 |
| 33 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 26 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 38 | 26 |
| 0 | 0 | 0 | 0 | 28 | 3 |
| 34 | 35 | 0 | 0 | 0 | 0 |
| 8 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 9 | 0 | 0 |
| 0 | 0 | 5 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 2 | 9 |
| 34 | 35 | 0 | 0 | 0 | 0 |
| 8 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 26 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 9 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [7,33,0,0,0,0,1,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,40,32],[7,33,0,0,0,0,6,34,0,0,0,0,0,0,40,26,0,0,0,0,0,1,0,0,0,0,0,0,38,28,0,0,0,0,26,3],[34,8,0,0,0,0,35,7,0,0,0,0,0,0,35,5,0,0,0,0,9,6,0,0,0,0,0,0,32,2,0,0,0,0,0,9],[34,8,0,0,0,0,35,7,0,0,0,0,0,0,40,26,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,9,40] >;
Dic10⋊D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes D_4 % in TeX
G:=Group("Dic10:D4"); // GroupNames label
G:=SmallGroup(320,785);
// by ID
G=gap.SmallGroup(320,785);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,254,219,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=c*a*c^-1=a^-1,d*a*d=a^9,c*b*c^-1=a^5*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations