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 [×3], C2 [×2], C4 [×2], C4 [×7], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4 [×11], Q8 [×2], Q8 [×10], C23, D5 [×2], C10 [×3], C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16 [×4], C22×C4 [×2], C2×Q8, C2×Q8 [×7], Dic5 [×4], C20 [×2], C20 [×3], D10 [×2], D10 [×2], C2×C10, C22⋊C8, Q8⋊C4, Q8⋊C4, C22⋊Q8, C2×Q16 [×2], C22×Q8, C5⋊2C8, C40, Dic10 [×2], Dic10 [×7], C4×D5 [×6], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C5×Q8, C22×D5, C22⋊Q16, Dic20 [×2], C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, C5⋊Q16 [×2], C5×C4⋊C4, C2×C40, C2×Dic10 [×2], C2×Dic10, C2×C4×D5, C2×C4×D5, Q8×D5 [×4], Q8×C10, C10.Q16, D10⋊1C8, C5×Q8⋊C4, D10⋊2Q8, C2×Dic20, C2×C5⋊Q16, C2×Q8×D5, D10⋊4Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, Q16 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×Q16, C8.C22, D20 [×2], C22×D5, C22⋊Q16, C2×D20, D4×D5 [×2], 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 42)(12 41)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 147)(22 146)(23 145)(24 144)(25 143)(26 142)(27 141)(28 150)(29 149)(30 148)(51 76)(52 75)(53 74)(54 73)(55 72)(56 71)(57 80)(58 79)(59 78)(60 77)(61 81)(62 90)(63 89)(64 88)(65 87)(66 86)(67 85)(68 84)(69 83)(70 82)(91 111)(92 120)(93 119)(94 118)(95 117)(96 116)(97 115)(98 114)(99 113)(100 112)(101 126)(102 125)(103 124)(104 123)(105 122)(106 121)(107 130)(108 129)(109 128)(110 127)(131 156)(132 155)(133 154)(134 153)(135 152)(136 151)(137 160)(138 159)(139 158)(140 157)
(1 21 14 135 33 143 50 158)(2 30 15 134 34 142 41 157)(3 29 16 133 35 141 42 156)(4 28 17 132 36 150 43 155)(5 27 18 131 37 149 44 154)(6 26 19 140 38 148 45 153)(7 25 20 139 39 147 46 152)(8 24 11 138 40 146 47 151)(9 23 12 137 31 145 48 160)(10 22 13 136 32 144 49 159)(51 124 81 117 74 102 69 94)(52 123 82 116 75 101 70 93)(53 122 83 115 76 110 61 92)(54 121 84 114 77 109 62 91)(55 130 85 113 78 108 63 100)(56 129 86 112 79 107 64 99)(57 128 87 111 80 106 65 98)(58 127 88 120 71 105 66 97)(59 126 89 119 72 104 67 96)(60 125 90 118 73 103 68 95)
(1 55 33 78)(2 54 34 77)(3 53 35 76)(4 52 36 75)(5 51 37 74)(6 60 38 73)(7 59 39 72)(8 58 40 71)(9 57 31 80)(10 56 32 79)(11 66 47 88)(12 65 48 87)(13 64 49 86)(14 63 50 85)(15 62 41 84)(16 61 42 83)(17 70 43 82)(18 69 44 81)(19 68 45 90)(20 67 46 89)(21 100 143 113)(22 99 144 112)(23 98 145 111)(24 97 146 120)(25 96 147 119)(26 95 148 118)(27 94 149 117)(28 93 150 116)(29 92 141 115)(30 91 142 114)(101 155 123 132)(102 154 124 131)(103 153 125 140)(104 152 126 139)(105 151 127 138)(106 160 128 137)(107 159 129 136)(108 158 130 135)(109 157 121 134)(110 156 122 133)
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)(11,42)(12,41)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,147)(22,146)(23,145)(24,144)(25,143)(26,142)(27,141)(28,150)(29,149)(30,148)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,80)(58,79)(59,78)(60,77)(61,81)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,84)(69,83)(70,82)(91,111)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,114)(99,113)(100,112)(101,126)(102,125)(103,124)(104,123)(105,122)(106,121)(107,130)(108,129)(109,128)(110,127)(131,156)(132,155)(133,154)(134,153)(135,152)(136,151)(137,160)(138,159)(139,158)(140,157), (1,21,14,135,33,143,50,158)(2,30,15,134,34,142,41,157)(3,29,16,133,35,141,42,156)(4,28,17,132,36,150,43,155)(5,27,18,131,37,149,44,154)(6,26,19,140,38,148,45,153)(7,25,20,139,39,147,46,152)(8,24,11,138,40,146,47,151)(9,23,12,137,31,145,48,160)(10,22,13,136,32,144,49,159)(51,124,81,117,74,102,69,94)(52,123,82,116,75,101,70,93)(53,122,83,115,76,110,61,92)(54,121,84,114,77,109,62,91)(55,130,85,113,78,108,63,100)(56,129,86,112,79,107,64,99)(57,128,87,111,80,106,65,98)(58,127,88,120,71,105,66,97)(59,126,89,119,72,104,67,96)(60,125,90,118,73,103,68,95), (1,55,33,78)(2,54,34,77)(3,53,35,76)(4,52,36,75)(5,51,37,74)(6,60,38,73)(7,59,39,72)(8,58,40,71)(9,57,31,80)(10,56,32,79)(11,66,47,88)(12,65,48,87)(13,64,49,86)(14,63,50,85)(15,62,41,84)(16,61,42,83)(17,70,43,82)(18,69,44,81)(19,68,45,90)(20,67,46,89)(21,100,143,113)(22,99,144,112)(23,98,145,111)(24,97,146,120)(25,96,147,119)(26,95,148,118)(27,94,149,117)(28,93,150,116)(29,92,141,115)(30,91,142,114)(101,155,123,132)(102,154,124,131)(103,153,125,140)(104,152,126,139)(105,151,127,138)(106,160,128,137)(107,159,129,136)(108,158,130,135)(109,157,121,134)(110,156,122,133)>;
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)(11,42)(12,41)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,147)(22,146)(23,145)(24,144)(25,143)(26,142)(27,141)(28,150)(29,149)(30,148)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,80)(58,79)(59,78)(60,77)(61,81)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,84)(69,83)(70,82)(91,111)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,114)(99,113)(100,112)(101,126)(102,125)(103,124)(104,123)(105,122)(106,121)(107,130)(108,129)(109,128)(110,127)(131,156)(132,155)(133,154)(134,153)(135,152)(136,151)(137,160)(138,159)(139,158)(140,157), (1,21,14,135,33,143,50,158)(2,30,15,134,34,142,41,157)(3,29,16,133,35,141,42,156)(4,28,17,132,36,150,43,155)(5,27,18,131,37,149,44,154)(6,26,19,140,38,148,45,153)(7,25,20,139,39,147,46,152)(8,24,11,138,40,146,47,151)(9,23,12,137,31,145,48,160)(10,22,13,136,32,144,49,159)(51,124,81,117,74,102,69,94)(52,123,82,116,75,101,70,93)(53,122,83,115,76,110,61,92)(54,121,84,114,77,109,62,91)(55,130,85,113,78,108,63,100)(56,129,86,112,79,107,64,99)(57,128,87,111,80,106,65,98)(58,127,88,120,71,105,66,97)(59,126,89,119,72,104,67,96)(60,125,90,118,73,103,68,95), (1,55,33,78)(2,54,34,77)(3,53,35,76)(4,52,36,75)(5,51,37,74)(6,60,38,73)(7,59,39,72)(8,58,40,71)(9,57,31,80)(10,56,32,79)(11,66,47,88)(12,65,48,87)(13,64,49,86)(14,63,50,85)(15,62,41,84)(16,61,42,83)(17,70,43,82)(18,69,44,81)(19,68,45,90)(20,67,46,89)(21,100,143,113)(22,99,144,112)(23,98,145,111)(24,97,146,120)(25,96,147,119)(26,95,148,118)(27,94,149,117)(28,93,150,116)(29,92,141,115)(30,91,142,114)(101,155,123,132)(102,154,124,131)(103,153,125,140)(104,152,126,139)(105,151,127,138)(106,160,128,137)(107,159,129,136)(108,158,130,135)(109,157,121,134)(110,156,122,133) );
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),(11,42),(12,41),(13,50),(14,49),(15,48),(16,47),(17,46),(18,45),(19,44),(20,43),(21,147),(22,146),(23,145),(24,144),(25,143),(26,142),(27,141),(28,150),(29,149),(30,148),(51,76),(52,75),(53,74),(54,73),(55,72),(56,71),(57,80),(58,79),(59,78),(60,77),(61,81),(62,90),(63,89),(64,88),(65,87),(66,86),(67,85),(68,84),(69,83),(70,82),(91,111),(92,120),(93,119),(94,118),(95,117),(96,116),(97,115),(98,114),(99,113),(100,112),(101,126),(102,125),(103,124),(104,123),(105,122),(106,121),(107,130),(108,129),(109,128),(110,127),(131,156),(132,155),(133,154),(134,153),(135,152),(136,151),(137,160),(138,159),(139,158),(140,157)], [(1,21,14,135,33,143,50,158),(2,30,15,134,34,142,41,157),(3,29,16,133,35,141,42,156),(4,28,17,132,36,150,43,155),(5,27,18,131,37,149,44,154),(6,26,19,140,38,148,45,153),(7,25,20,139,39,147,46,152),(8,24,11,138,40,146,47,151),(9,23,12,137,31,145,48,160),(10,22,13,136,32,144,49,159),(51,124,81,117,74,102,69,94),(52,123,82,116,75,101,70,93),(53,122,83,115,76,110,61,92),(54,121,84,114,77,109,62,91),(55,130,85,113,78,108,63,100),(56,129,86,112,79,107,64,99),(57,128,87,111,80,106,65,98),(58,127,88,120,71,105,66,97),(59,126,89,119,72,104,67,96),(60,125,90,118,73,103,68,95)], [(1,55,33,78),(2,54,34,77),(3,53,35,76),(4,52,36,75),(5,51,37,74),(6,60,38,73),(7,59,39,72),(8,58,40,71),(9,57,31,80),(10,56,32,79),(11,66,47,88),(12,65,48,87),(13,64,49,86),(14,63,50,85),(15,62,41,84),(16,61,42,83),(17,70,43,82),(18,69,44,81),(19,68,45,90),(20,67,46,89),(21,100,143,113),(22,99,144,112),(23,98,145,111),(24,97,146,120),(25,96,147,119),(26,95,148,118),(27,94,149,117),(28,93,150,116),(29,92,141,115),(30,91,142,114),(101,155,123,132),(102,154,124,131),(103,153,125,140),(104,152,126,139),(105,151,127,138),(106,160,128,137),(107,159,129,136),(108,158,130,135),(109,157,121,134),(110,156,122,133)])
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