metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊4Q16, Q8.6D20, Dic10.12D4, C4.93(D4×D5), C4.6(C2×D20), (C5×Q8).1D4, C2.9(D5×Q16), C4⋊C4.25D10, Q8⋊C4⋊4D5, (C2×C8).16D10, (C2×Dic20)⋊6C2, C20.122(C2×D4), C10.17(C2×Q16), C5⋊2(C22⋊Q16), C10.24C22≀C2, D10⋊1C8.2C2, C10.Q16⋊12C2, (C2×C40).16C22, D10⋊2Q8.2C2, (C2×Q8).108D10, (C2×Dic5).39D4, C22.198(D4×D5), (C2×C20).248C23, (C22×D5).115D4, (Q8×C10).31C22, C2.27(C22⋊D20), C2.17(SD16⋊D5), C10.35(C8.C22), (C2×Dic10).75C22, (C2×Q8×D5).4C2, (C2×C5⋊Q16)⋊3C2, (C5×Q8⋊C4)⋊4C2, (C2×C4×D5).25C22, (C2×C10).261(C2×D4), (C5×C4⋊C4).49C22, (C2×C5⋊2C8).39C22, (C2×C4).355(C22×D5), SmallGroup(320,435)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊4Q16
G = < a,b,c,d | a10=b2=c8=1, d2=c4, bab=cac-1=dad-1=a-1, cbc-1=dbd-1=a3b, dcd-1=c-1 >
Subgroups: 606 in 148 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C23, D5, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C2×Q8, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, C22⋊C8, Q8⋊C4, Q8⋊C4, C22⋊Q8, C2×Q16, C22×Q8, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, C22⋊Q16, Dic20, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, C5⋊Q16, C5×C4⋊C4, C2×C40, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, Q8×D5, Q8×C10, C10.Q16, D10⋊1C8, C5×Q8⋊C4, D10⋊2Q8, C2×Dic20, C2×C5⋊Q16, C2×Q8×D5, D10⋊4Q16
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, D10, C22≀C2, C2×Q16, C8.C22, D20, C22×D5, C22⋊Q16, C2×D20, D4×D5, C22⋊D20, SD16⋊D5, D5×Q16, D10⋊4Q16
►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 32)(2 31)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 41)(12 50)(13 49)(14 48)(15 47)(16 46)(17 45)(18 44)(19 43)(20 42)(21 141)(22 150)(23 149)(24 148)(25 147)(26 146)(27 145)(28 144)(29 143)(30 142)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 74)(58 73)(59 72)(60 71)(61 85)(62 84)(63 83)(64 82)(65 81)(66 90)(67 89)(68 88)(69 87)(70 86)(91 115)(92 114)(93 113)(94 112)(95 111)(96 120)(97 119)(98 118)(99 117)(100 116)(101 130)(102 129)(103 128)(104 127)(105 126)(106 125)(107 124)(108 123)(109 122)(110 121)(131 160)(132 159)(133 158)(134 157)(135 156)(136 155)(137 154)(138 153)(139 152)(140 151)
(1 21 13 139 33 147 50 158)(2 30 14 138 34 146 41 157)(3 29 15 137 35 145 42 156)(4 28 16 136 36 144 43 155)(5 27 17 135 37 143 44 154)(6 26 18 134 38 142 45 153)(7 25 19 133 39 141 46 152)(8 24 20 132 40 150 47 151)(9 23 11 131 31 149 48 160)(10 22 12 140 32 148 49 159)(51 128 87 111 80 110 69 92)(52 127 88 120 71 109 70 91)(53 126 89 119 72 108 61 100)(54 125 90 118 73 107 62 99)(55 124 81 117 74 106 63 98)(56 123 82 116 75 105 64 97)(57 122 83 115 76 104 65 96)(58 121 84 114 77 103 66 95)(59 130 85 113 78 102 67 94)(60 129 86 112 79 101 68 93)
(1 59 33 78)(2 58 34 77)(3 57 35 76)(4 56 36 75)(5 55 37 74)(6 54 38 73)(7 53 39 72)(8 52 40 71)(9 51 31 80)(10 60 32 79)(11 69 48 87)(12 68 49 86)(13 67 50 85)(14 66 41 84)(15 65 42 83)(16 64 43 82)(17 63 44 81)(18 62 45 90)(19 61 46 89)(20 70 47 88)(21 94 147 113)(22 93 148 112)(23 92 149 111)(24 91 150 120)(25 100 141 119)(26 99 142 118)(27 98 143 117)(28 97 144 116)(29 96 145 115)(30 95 146 114)(101 159 129 140)(102 158 130 139)(103 157 121 138)(104 156 122 137)(105 155 123 136)(106 154 124 135)(107 153 125 134)(108 152 126 133)(109 151 127 132)(110 160 128 131)
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), (1,32)(2,31)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,41)(12,50)(13,49)(14,48)(15,47)(16,46)(17,45)(18,44)(19,43)(20,42)(21,141)(22,150)(23,149)(24,148)(25,147)(26,146)(27,145)(28,144)(29,143)(30,142)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,85)(62,84)(63,83)(64,82)(65,81)(66,90)(67,89)(68,88)(69,87)(70,86)(91,115)(92,114)(93,113)(94,112)(95,111)(96,120)(97,119)(98,118)(99,117)(100,116)(101,130)(102,129)(103,128)(104,127)(105,126)(106,125)(107,124)(108,123)(109,122)(110,121)(131,160)(132,159)(133,158)(134,157)(135,156)(136,155)(137,154)(138,153)(139,152)(140,151), (1,21,13,139,33,147,50,158)(2,30,14,138,34,146,41,157)(3,29,15,137,35,145,42,156)(4,28,16,136,36,144,43,155)(5,27,17,135,37,143,44,154)(6,26,18,134,38,142,45,153)(7,25,19,133,39,141,46,152)(8,24,20,132,40,150,47,151)(9,23,11,131,31,149,48,160)(10,22,12,140,32,148,49,159)(51,128,87,111,80,110,69,92)(52,127,88,120,71,109,70,91)(53,126,89,119,72,108,61,100)(54,125,90,118,73,107,62,99)(55,124,81,117,74,106,63,98)(56,123,82,116,75,105,64,97)(57,122,83,115,76,104,65,96)(58,121,84,114,77,103,66,95)(59,130,85,113,78,102,67,94)(60,129,86,112,79,101,68,93), (1,59,33,78)(2,58,34,77)(3,57,35,76)(4,56,36,75)(5,55,37,74)(6,54,38,73)(7,53,39,72)(8,52,40,71)(9,51,31,80)(10,60,32,79)(11,69,48,87)(12,68,49,86)(13,67,50,85)(14,66,41,84)(15,65,42,83)(16,64,43,82)(17,63,44,81)(18,62,45,90)(19,61,46,89)(20,70,47,88)(21,94,147,113)(22,93,148,112)(23,92,149,111)(24,91,150,120)(25,100,141,119)(26,99,142,118)(27,98,143,117)(28,97,144,116)(29,96,145,115)(30,95,146,114)(101,159,129,140)(102,158,130,139)(103,157,121,138)(104,156,122,137)(105,155,123,136)(106,154,124,135)(107,153,125,134)(108,152,126,133)(109,151,127,132)(110,160,128,131)>;
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), (1,32)(2,31)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,41)(12,50)(13,49)(14,48)(15,47)(16,46)(17,45)(18,44)(19,43)(20,42)(21,141)(22,150)(23,149)(24,148)(25,147)(26,146)(27,145)(28,144)(29,143)(30,142)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,85)(62,84)(63,83)(64,82)(65,81)(66,90)(67,89)(68,88)(69,87)(70,86)(91,115)(92,114)(93,113)(94,112)(95,111)(96,120)(97,119)(98,118)(99,117)(100,116)(101,130)(102,129)(103,128)(104,127)(105,126)(106,125)(107,124)(108,123)(109,122)(110,121)(131,160)(132,159)(133,158)(134,157)(135,156)(136,155)(137,154)(138,153)(139,152)(140,151), (1,21,13,139,33,147,50,158)(2,30,14,138,34,146,41,157)(3,29,15,137,35,145,42,156)(4,28,16,136,36,144,43,155)(5,27,17,135,37,143,44,154)(6,26,18,134,38,142,45,153)(7,25,19,133,39,141,46,152)(8,24,20,132,40,150,47,151)(9,23,11,131,31,149,48,160)(10,22,12,140,32,148,49,159)(51,128,87,111,80,110,69,92)(52,127,88,120,71,109,70,91)(53,126,89,119,72,108,61,100)(54,125,90,118,73,107,62,99)(55,124,81,117,74,106,63,98)(56,123,82,116,75,105,64,97)(57,122,83,115,76,104,65,96)(58,121,84,114,77,103,66,95)(59,130,85,113,78,102,67,94)(60,129,86,112,79,101,68,93), (1,59,33,78)(2,58,34,77)(3,57,35,76)(4,56,36,75)(5,55,37,74)(6,54,38,73)(7,53,39,72)(8,52,40,71)(9,51,31,80)(10,60,32,79)(11,69,48,87)(12,68,49,86)(13,67,50,85)(14,66,41,84)(15,65,42,83)(16,64,43,82)(17,63,44,81)(18,62,45,90)(19,61,46,89)(20,70,47,88)(21,94,147,113)(22,93,148,112)(23,92,149,111)(24,91,150,120)(25,100,141,119)(26,99,142,118)(27,98,143,117)(28,97,144,116)(29,96,145,115)(30,95,146,114)(101,159,129,140)(102,158,130,139)(103,157,121,138)(104,156,122,137)(105,155,123,136)(106,154,124,135)(107,153,125,134)(108,152,126,133)(109,151,127,132)(110,160,128,131) );
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)], [(1,32),(2,31),(3,40),(4,39),(5,38),(6,37),(7,36),(8,35),(9,34),(10,33),(11,41),(12,50),(13,49),(14,48),(15,47),(16,46),(17,45),(18,44),(19,43),(20,42),(21,141),(22,150),(23,149),(24,148),(25,147),(26,146),(27,145),(28,144),(29,143),(30,142),(51,80),(52,79),(53,78),(54,77),(55,76),(56,75),(57,74),(58,73),(59,72),(60,71),(61,85),(62,84),(63,83),(64,82),(65,81),(66,90),(67,89),(68,88),(69,87),(70,86),(91,115),(92,114),(93,113),(94,112),(95,111),(96,120),(97,119),(98,118),(99,117),(100,116),(101,130),(102,129),(103,128),(104,127),(105,126),(106,125),(107,124),(108,123),(109,122),(110,121),(131,160),(132,159),(133,158),(134,157),(135,156),(136,155),(137,154),(138,153),(139,152),(140,151)], [(1,21,13,139,33,147,50,158),(2,30,14,138,34,146,41,157),(3,29,15,137,35,145,42,156),(4,28,16,136,36,144,43,155),(5,27,17,135,37,143,44,154),(6,26,18,134,38,142,45,153),(7,25,19,133,39,141,46,152),(8,24,20,132,40,150,47,151),(9,23,11,131,31,149,48,160),(10,22,12,140,32,148,49,159),(51,128,87,111,80,110,69,92),(52,127,88,120,71,109,70,91),(53,126,89,119,72,108,61,100),(54,125,90,118,73,107,62,99),(55,124,81,117,74,106,63,98),(56,123,82,116,75,105,64,97),(57,122,83,115,76,104,65,96),(58,121,84,114,77,103,66,95),(59,130,85,113,78,102,67,94),(60,129,86,112,79,101,68,93)], [(1,59,33,78),(2,58,34,77),(3,57,35,76),(4,56,36,75),(5,55,37,74),(6,54,38,73),(7,53,39,72),(8,52,40,71),(9,51,31,80),(10,60,32,79),(11,69,48,87),(12,68,49,86),(13,67,50,85),(14,66,41,84),(15,65,42,83),(16,64,43,82),(17,63,44,81),(18,62,45,90),(19,61,46,89),(20,70,47,88),(21,94,147,113),(22,93,148,112),(23,92,149,111),(24,91,150,120),(25,100,141,119),(26,99,142,118),(27,98,143,117),(28,97,144,116),(29,96,145,115),(30,95,146,114),(101,159,129,140),(102,158,130,139),(103,157,121,138),(104,156,122,137),(105,155,123,136),(106,154,124,135),(107,153,125,134),(108,152,126,133),(109,151,127,132),(110,160,128,131)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 4 | 4 | 8 | 20 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 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 | Q16 | D10 | D10 | D10 | D20 | C8.C22 | D4×D5 | D4×D5 | SD16⋊D5 | D5×Q16 |
| kernel | D10⋊4Q16 | C10.Q16 | D10⋊1C8 | C5×Q8⋊C4 | D10⋊2Q8 | C2×Dic20 | C2×C5⋊Q16 | C2×Q8×D5 | Dic10 | C2×Dic5 | C5×Q8 | C22×D5 | Q8⋊C4 | D10 | C4⋊C4 | C2×C8 | C2×Q8 | Q8 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D10⋊4Q16 ►in GL6(𝔽41)
| 35 | 35 | 0 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 35 | 35 | 0 | 0 | 0 | 0 |
| 40 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 27 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 37 | 0 | 0 |
| 0 | 0 | 22 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 0 | 35 | 24 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 37 | 0 | 0 |
| 0 | 0 | 1 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 3 |
| 0 | 0 | 0 | 0 | 21 | 10 |
G:=sub<GL(6,GF(41))| [35,6,0,0,0,0,35,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[35,40,0,0,0,0,35,6,0,0,0,0,0,0,40,27,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,28,22,0,0,0,0,37,13,0,0,0,0,0,0,0,35,0,0,0,0,7,24],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,28,1,0,0,0,0,37,13,0,0,0,0,0,0,31,21,0,0,0,0,3,10] >;
D10⋊4Q16 in GAP, Magma, Sage, TeX
D_{10}\rtimes_4Q_{16} % in TeX
G:=Group("D10:4Q16"); // GroupNames label
G:=SmallGroup(320,435);
// by ID
G=gap.SmallGroup(320,435);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,226,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=c^4,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations