metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10.16D4, C4.65(D4×D5), (C5×D4).10D4, D10⋊3Q8⋊5C2, (C2×SD16)⋊14D5, (C2×C8).149D10, C20.177(C2×D4), D4.9(C5⋊D4), C5⋊5(D4.7D4), (C2×Q8).55D10, D10⋊1C8⋊35C2, C10.60C22≀C2, (C10×SD16)⋊23C2, (C2×D4).147D10, C10.65(C4○D8), D4⋊Dic5⋊35C2, (C22×D5).46D4, C22.270(D4×D5), C20.44D4⋊36C2, (C2×C40).296C22, (C2×C20).450C23, (C2×Dic5).241D4, (D4×C10).99C22, (Q8×C10).79C22, C2.28(C23⋊D10), C2.30(SD16⋊D5), C10.50(C8.C22), C4⋊Dic5.177C22, C2.31(SD16⋊3D5), (C2×Dic10).132C22, C4.45(C2×C5⋊D4), (C2×C5⋊Q16)⋊19C2, (C2×C4×D5).54C22, (C2×D4⋊2D5).6C2, (C2×C10).362(C2×D4), (C2×C4).539(C22×D5), (C2×C5⋊2C8).159C22, SmallGroup(320,800)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10.16D4
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=a10c-1 >
Subgroups: 606 in 152 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, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C22×C10, D4.7D4, C2×C5⋊2C8, C10.D4, C4⋊Dic5, D10⋊C4, C5⋊Q16, C2×C40, C5×SD16, C2×Dic10, C2×C4×D5, D4⋊2D5, C22×Dic5, C2×C5⋊D4, D4×C10, Q8×C10, C20.44D4, D10⋊1C8, D4⋊Dic5, C2×C5⋊Q16, D10⋊3Q8, C10×SD16, C2×D4⋊2D5, Dic10.16D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C4○D8, C8.C22, C5⋊D4, C22×D5, D4.7D4, D4×D5, C2×C5⋊D4, SD16⋊D5, SD16⋊3D5, C23⋊D10, Dic10.16D4
►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 87 11 97)(2 86 12 96)(3 85 13 95)(4 84 14 94)(5 83 15 93)(6 82 16 92)(7 81 17 91)(8 100 18 90)(9 99 19 89)(10 98 20 88)(21 44 31 54)(22 43 32 53)(23 42 33 52)(24 41 34 51)(25 60 35 50)(26 59 36 49)(27 58 37 48)(28 57 38 47)(29 56 39 46)(30 55 40 45)(61 145 71 155)(62 144 72 154)(63 143 73 153)(64 142 74 152)(65 141 75 151)(66 160 76 150)(67 159 77 149)(68 158 78 148)(69 157 79 147)(70 156 80 146)(101 135 111 125)(102 134 112 124)(103 133 113 123)(104 132 114 122)(105 131 115 121)(106 130 116 140)(107 129 117 139)(108 128 118 138)(109 127 119 137)(110 126 120 136)
(1 28 143 105)(2 27 144 104)(3 26 145 103)(4 25 146 102)(5 24 147 101)(6 23 148 120)(7 22 149 119)(8 21 150 118)(9 40 151 117)(10 39 152 116)(11 38 153 115)(12 37 154 114)(13 36 155 113)(14 35 156 112)(15 34 157 111)(16 33 158 110)(17 32 159 109)(18 31 160 108)(19 30 141 107)(20 29 142 106)(41 64 135 98)(42 63 136 97)(43 62 137 96)(44 61 138 95)(45 80 139 94)(46 79 140 93)(47 78 121 92)(48 77 122 91)(49 76 123 90)(50 75 124 89)(51 74 125 88)(52 73 126 87)(53 72 127 86)(54 71 128 85)(55 70 129 84)(56 69 130 83)(57 68 131 82)(58 67 132 81)(59 66 133 100)(60 65 134 99)
(1 153)(2 142)(3 151)(4 160)(5 149)(6 158)(7 147)(8 156)(9 145)(10 154)(11 143)(12 152)(13 141)(14 150)(15 159)(16 148)(17 157)(18 146)(19 155)(20 144)(21 25)(22 34)(24 32)(26 30)(27 39)(29 37)(31 35)(36 40)(41 43)(42 52)(44 50)(45 59)(46 48)(47 57)(49 55)(51 53)(54 60)(56 58)(61 99)(62 88)(63 97)(64 86)(65 95)(66 84)(67 93)(68 82)(69 91)(70 100)(71 89)(72 98)(73 87)(74 96)(75 85)(76 94)(77 83)(78 92)(79 81)(80 90)(101 109)(102 118)(103 107)(104 116)(106 114)(108 112)(111 119)(113 117)(121 131)(122 140)(123 129)(124 138)(125 127)(126 136)(128 134)(130 132)(133 139)(135 137)
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), (1,87,11,97)(2,86,12,96)(3,85,13,95)(4,84,14,94)(5,83,15,93)(6,82,16,92)(7,81,17,91)(8,100,18,90)(9,99,19,89)(10,98,20,88)(21,44,31,54)(22,43,32,53)(23,42,33,52)(24,41,34,51)(25,60,35,50)(26,59,36,49)(27,58,37,48)(28,57,38,47)(29,56,39,46)(30,55,40,45)(61,145,71,155)(62,144,72,154)(63,143,73,153)(64,142,74,152)(65,141,75,151)(66,160,76,150)(67,159,77,149)(68,158,78,148)(69,157,79,147)(70,156,80,146)(101,135,111,125)(102,134,112,124)(103,133,113,123)(104,132,114,122)(105,131,115,121)(106,130,116,140)(107,129,117,139)(108,128,118,138)(109,127,119,137)(110,126,120,136), (1,28,143,105)(2,27,144,104)(3,26,145,103)(4,25,146,102)(5,24,147,101)(6,23,148,120)(7,22,149,119)(8,21,150,118)(9,40,151,117)(10,39,152,116)(11,38,153,115)(12,37,154,114)(13,36,155,113)(14,35,156,112)(15,34,157,111)(16,33,158,110)(17,32,159,109)(18,31,160,108)(19,30,141,107)(20,29,142,106)(41,64,135,98)(42,63,136,97)(43,62,137,96)(44,61,138,95)(45,80,139,94)(46,79,140,93)(47,78,121,92)(48,77,122,91)(49,76,123,90)(50,75,124,89)(51,74,125,88)(52,73,126,87)(53,72,127,86)(54,71,128,85)(55,70,129,84)(56,69,130,83)(57,68,131,82)(58,67,132,81)(59,66,133,100)(60,65,134,99), (1,153)(2,142)(3,151)(4,160)(5,149)(6,158)(7,147)(8,156)(9,145)(10,154)(11,143)(12,152)(13,141)(14,150)(15,159)(16,148)(17,157)(18,146)(19,155)(20,144)(21,25)(22,34)(24,32)(26,30)(27,39)(29,37)(31,35)(36,40)(41,43)(42,52)(44,50)(45,59)(46,48)(47,57)(49,55)(51,53)(54,60)(56,58)(61,99)(62,88)(63,97)(64,86)(65,95)(66,84)(67,93)(68,82)(69,91)(70,100)(71,89)(72,98)(73,87)(74,96)(75,85)(76,94)(77,83)(78,92)(79,81)(80,90)(101,109)(102,118)(103,107)(104,116)(106,114)(108,112)(111,119)(113,117)(121,131)(122,140)(123,129)(124,138)(125,127)(126,136)(128,134)(130,132)(133,139)(135,137)>;
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), (1,87,11,97)(2,86,12,96)(3,85,13,95)(4,84,14,94)(5,83,15,93)(6,82,16,92)(7,81,17,91)(8,100,18,90)(9,99,19,89)(10,98,20,88)(21,44,31,54)(22,43,32,53)(23,42,33,52)(24,41,34,51)(25,60,35,50)(26,59,36,49)(27,58,37,48)(28,57,38,47)(29,56,39,46)(30,55,40,45)(61,145,71,155)(62,144,72,154)(63,143,73,153)(64,142,74,152)(65,141,75,151)(66,160,76,150)(67,159,77,149)(68,158,78,148)(69,157,79,147)(70,156,80,146)(101,135,111,125)(102,134,112,124)(103,133,113,123)(104,132,114,122)(105,131,115,121)(106,130,116,140)(107,129,117,139)(108,128,118,138)(109,127,119,137)(110,126,120,136), (1,28,143,105)(2,27,144,104)(3,26,145,103)(4,25,146,102)(5,24,147,101)(6,23,148,120)(7,22,149,119)(8,21,150,118)(9,40,151,117)(10,39,152,116)(11,38,153,115)(12,37,154,114)(13,36,155,113)(14,35,156,112)(15,34,157,111)(16,33,158,110)(17,32,159,109)(18,31,160,108)(19,30,141,107)(20,29,142,106)(41,64,135,98)(42,63,136,97)(43,62,137,96)(44,61,138,95)(45,80,139,94)(46,79,140,93)(47,78,121,92)(48,77,122,91)(49,76,123,90)(50,75,124,89)(51,74,125,88)(52,73,126,87)(53,72,127,86)(54,71,128,85)(55,70,129,84)(56,69,130,83)(57,68,131,82)(58,67,132,81)(59,66,133,100)(60,65,134,99), (1,153)(2,142)(3,151)(4,160)(5,149)(6,158)(7,147)(8,156)(9,145)(10,154)(11,143)(12,152)(13,141)(14,150)(15,159)(16,148)(17,157)(18,146)(19,155)(20,144)(21,25)(22,34)(24,32)(26,30)(27,39)(29,37)(31,35)(36,40)(41,43)(42,52)(44,50)(45,59)(46,48)(47,57)(49,55)(51,53)(54,60)(56,58)(61,99)(62,88)(63,97)(64,86)(65,95)(66,84)(67,93)(68,82)(69,91)(70,100)(71,89)(72,98)(73,87)(74,96)(75,85)(76,94)(77,83)(78,92)(79,81)(80,90)(101,109)(102,118)(103,107)(104,116)(106,114)(108,112)(111,119)(113,117)(121,131)(122,140)(123,129)(124,138)(125,127)(126,136)(128,134)(130,132)(133,139)(135,137) );
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)], [(1,87,11,97),(2,86,12,96),(3,85,13,95),(4,84,14,94),(5,83,15,93),(6,82,16,92),(7,81,17,91),(8,100,18,90),(9,99,19,89),(10,98,20,88),(21,44,31,54),(22,43,32,53),(23,42,33,52),(24,41,34,51),(25,60,35,50),(26,59,36,49),(27,58,37,48),(28,57,38,47),(29,56,39,46),(30,55,40,45),(61,145,71,155),(62,144,72,154),(63,143,73,153),(64,142,74,152),(65,141,75,151),(66,160,76,150),(67,159,77,149),(68,158,78,148),(69,157,79,147),(70,156,80,146),(101,135,111,125),(102,134,112,124),(103,133,113,123),(104,132,114,122),(105,131,115,121),(106,130,116,140),(107,129,117,139),(108,128,118,138),(109,127,119,137),(110,126,120,136)], [(1,28,143,105),(2,27,144,104),(3,26,145,103),(4,25,146,102),(5,24,147,101),(6,23,148,120),(7,22,149,119),(8,21,150,118),(9,40,151,117),(10,39,152,116),(11,38,153,115),(12,37,154,114),(13,36,155,113),(14,35,156,112),(15,34,157,111),(16,33,158,110),(17,32,159,109),(18,31,160,108),(19,30,141,107),(20,29,142,106),(41,64,135,98),(42,63,136,97),(43,62,137,96),(44,61,138,95),(45,80,139,94),(46,79,140,93),(47,78,121,92),(48,77,122,91),(49,76,123,90),(50,75,124,89),(51,74,125,88),(52,73,126,87),(53,72,127,86),(54,71,128,85),(55,70,129,84),(56,69,130,83),(57,68,131,82),(58,67,132,81),(59,66,133,100),(60,65,134,99)], [(1,153),(2,142),(3,151),(4,160),(5,149),(6,158),(7,147),(8,156),(9,145),(10,154),(11,143),(12,152),(13,141),(14,150),(15,159),(16,148),(17,157),(18,146),(19,155),(20,144),(21,25),(22,34),(24,32),(26,30),(27,39),(29,37),(31,35),(36,40),(41,43),(42,52),(44,50),(45,59),(46,48),(47,57),(49,55),(51,53),(54,60),(56,58),(61,99),(62,88),(63,97),(64,86),(65,95),(66,84),(67,93),(68,82),(69,91),(70,100),(71,89),(72,98),(73,87),(74,96),(75,85),(76,94),(77,83),(78,92),(79,81),(80,90),(101,109),(102,118),(103,107),(104,116),(106,114),(108,112),(111,119),(113,117),(121,131),(122,140),(123,129),(124,138),(125,127),(126,136),(128,134),(130,132),(133,139),(135,137)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 20 | 2 | 2 | 8 | 10 | 10 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 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 | D10 | C4○D8 | C5⋊D4 | C8.C22 | D4×D5 | D4×D5 | SD16⋊D5 | SD16⋊3D5 |
| kernel | Dic10.16D4 | C20.44D4 | D10⋊1C8 | D4⋊Dic5 | C2×C5⋊Q16 | D10⋊3Q8 | C10×SD16 | C2×D4⋊2D5 | Dic10 | C2×Dic5 | C5×D4 | C22×D5 | C2×SD16 | C2×C8 | C2×D4 | C2×Q8 | C10 | D4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of Dic10.16D4 ►in GL6(𝔽41)
| 40 | 1 | 0 | 0 | 0 | 0 |
| 33 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 34 | 1 | 0 | 0 | 0 | 0 |
| 34 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 34 | 1 | 0 | 0 | 0 | 0 |
| 34 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 9 | 0 | 0 |
| 0 | 0 | 17 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 34 | 1 | 0 | 0 | 0 | 0 |
| 34 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [40,33,0,0,0,0,1,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,9],[34,34,0,0,0,0,1,7,0,0,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,0,0,0,0,3,0,0,0,0,27,0],[34,34,0,0,0,0,1,7,0,0,0,0,0,0,25,17,0,0,0,0,9,16,0,0,0,0,0,0,0,32,0,0,0,0,9,0],[34,34,0,0,0,0,1,7,0,0,0,0,0,0,40,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
Dic10.16D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}._{16}D_4 % in TeX
G:=Group("Dic10.16D4"); // GroupNames label
G:=SmallGroup(320,800);
// by ID
G=gap.SmallGroup(320,800);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,254,219,184,851,438,102,12550]);
// 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=a^10*c^-1>;
// generators/relations