metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊5Q16, Dic10.17D4, (C2×Q16)⋊3D5, C4.67(D4×D5), (C5×Q8).9D4, C20.52(C2×D4), (C2×C8).38D10, C2.18(D5×Q16), (C10×Q16)⋊13C2, (C2×Q8).62D10, C10.29(C2×Q16), C5⋊4(C22⋊Q16), Q8.8(C5⋊D4), C10.61C22≀C2, Q8⋊Dic5⋊33C2, D10⋊3Q8.8C2, (C2×Dic5).84D4, C22.277(D4×D5), D10⋊1C8.12C2, C20.44D4⋊30C2, (C2×C40).252C22, (C2×C20).460C23, (C22×D5).130D4, (Q8×C10).89C22, C2.29(C23⋊D10), C2.28(Q16⋊D5), C10.78(C8.C22), C4⋊Dic5.183C22, (C2×Dic10).136C22, (C2×Q8×D5).5C2, C4.48(C2×C5⋊D4), (C2×C5⋊Q16)⋊20C2, (C2×C4×D5).56C22, (C2×C10).371(C2×D4), (C2×C4).548(C22×D5), (C2×C5⋊2C8).165C22, SmallGroup(320,813)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊5Q16
G = < a,b,c,d | a10=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a5b, dcd-1=c-1 >
Subgroups: 574 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, C2×C8, C2×C8, Q16, C22×C4, C2×Q8, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, C22⋊C8, Q8⋊C4, C22⋊Q8, C2×Q16, 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, C2×C5⋊2C8, C10.D4, C4⋊Dic5, D10⋊C4, C5⋊Q16, C2×C40, C5×Q16, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, Q8×D5, Q8×C10, C20.44D4, D10⋊1C8, Q8⋊Dic5, C2×C5⋊Q16, D10⋊3Q8, C10×Q16, C2×Q8×D5, D10⋊5Q16
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, D10, C22≀C2, C2×Q16, C8.C22, C5⋊D4, C22×D5, C22⋊Q16, D4×D5, C2×C5⋊D4, D5×Q16, Q16⋊D5, C23⋊D10, D10⋊5Q16
►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 44)(12 43)(13 42)(14 41)(15 50)(16 49)(17 48)(18 47)(19 46)(20 45)(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)
(1 142 16 153 33 26 50 134)(2 143 17 154 34 27 41 135)(3 144 18 155 35 28 42 136)(4 145 19 156 36 29 43 137)(5 146 20 157 37 30 44 138)(6 147 11 158 38 21 45 139)(7 148 12 159 39 22 46 140)(8 149 13 160 40 23 47 131)(9 150 14 151 31 24 48 132)(10 141 15 152 32 25 49 133)(51 109 87 91 80 127 69 120)(52 110 88 92 71 128 70 111)(53 101 89 93 72 129 61 112)(54 102 90 94 73 130 62 113)(55 103 81 95 74 121 63 114)(56 104 82 96 75 122 64 115)(57 105 83 97 76 123 65 116)(58 106 84 98 77 124 66 117)(59 107 85 99 78 125 67 118)(60 108 86 100 79 126 68 119)
(1 73 33 54)(2 74 34 55)(3 75 35 56)(4 76 36 57)(5 77 37 58)(6 78 38 59)(7 79 39 60)(8 80 40 51)(9 71 31 52)(10 72 32 53)(11 85 45 67)(12 86 46 68)(13 87 47 69)(14 88 48 70)(15 89 49 61)(16 90 50 62)(17 81 41 63)(18 82 42 64)(19 83 43 65)(20 84 44 66)(21 118 147 99)(22 119 148 100)(23 120 149 91)(24 111 150 92)(25 112 141 93)(26 113 142 94)(27 114 143 95)(28 115 144 96)(29 116 145 97)(30 117 146 98)(101 133 129 152)(102 134 130 153)(103 135 121 154)(104 136 122 155)(105 137 123 156)(106 138 124 157)(107 139 125 158)(108 140 126 159)(109 131 127 160)(110 132 128 151)
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)(11,44)(12,43)(13,42)(14,41)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(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), (1,142,16,153,33,26,50,134)(2,143,17,154,34,27,41,135)(3,144,18,155,35,28,42,136)(4,145,19,156,36,29,43,137)(5,146,20,157,37,30,44,138)(6,147,11,158,38,21,45,139)(7,148,12,159,39,22,46,140)(8,149,13,160,40,23,47,131)(9,150,14,151,31,24,48,132)(10,141,15,152,32,25,49,133)(51,109,87,91,80,127,69,120)(52,110,88,92,71,128,70,111)(53,101,89,93,72,129,61,112)(54,102,90,94,73,130,62,113)(55,103,81,95,74,121,63,114)(56,104,82,96,75,122,64,115)(57,105,83,97,76,123,65,116)(58,106,84,98,77,124,66,117)(59,107,85,99,78,125,67,118)(60,108,86,100,79,126,68,119), (1,73,33,54)(2,74,34,55)(3,75,35,56)(4,76,36,57)(5,77,37,58)(6,78,38,59)(7,79,39,60)(8,80,40,51)(9,71,31,52)(10,72,32,53)(11,85,45,67)(12,86,46,68)(13,87,47,69)(14,88,48,70)(15,89,49,61)(16,90,50,62)(17,81,41,63)(18,82,42,64)(19,83,43,65)(20,84,44,66)(21,118,147,99)(22,119,148,100)(23,120,149,91)(24,111,150,92)(25,112,141,93)(26,113,142,94)(27,114,143,95)(28,115,144,96)(29,116,145,97)(30,117,146,98)(101,133,129,152)(102,134,130,153)(103,135,121,154)(104,136,122,155)(105,137,123,156)(106,138,124,157)(107,139,125,158)(108,140,126,159)(109,131,127,160)(110,132,128,151)>;
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)(11,44)(12,43)(13,42)(14,41)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(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), (1,142,16,153,33,26,50,134)(2,143,17,154,34,27,41,135)(3,144,18,155,35,28,42,136)(4,145,19,156,36,29,43,137)(5,146,20,157,37,30,44,138)(6,147,11,158,38,21,45,139)(7,148,12,159,39,22,46,140)(8,149,13,160,40,23,47,131)(9,150,14,151,31,24,48,132)(10,141,15,152,32,25,49,133)(51,109,87,91,80,127,69,120)(52,110,88,92,71,128,70,111)(53,101,89,93,72,129,61,112)(54,102,90,94,73,130,62,113)(55,103,81,95,74,121,63,114)(56,104,82,96,75,122,64,115)(57,105,83,97,76,123,65,116)(58,106,84,98,77,124,66,117)(59,107,85,99,78,125,67,118)(60,108,86,100,79,126,68,119), (1,73,33,54)(2,74,34,55)(3,75,35,56)(4,76,36,57)(5,77,37,58)(6,78,38,59)(7,79,39,60)(8,80,40,51)(9,71,31,52)(10,72,32,53)(11,85,45,67)(12,86,46,68)(13,87,47,69)(14,88,48,70)(15,89,49,61)(16,90,50,62)(17,81,41,63)(18,82,42,64)(19,83,43,65)(20,84,44,66)(21,118,147,99)(22,119,148,100)(23,120,149,91)(24,111,150,92)(25,112,141,93)(26,113,142,94)(27,114,143,95)(28,115,144,96)(29,116,145,97)(30,117,146,98)(101,133,129,152)(102,134,130,153)(103,135,121,154)(104,136,122,155)(105,137,123,156)(106,138,124,157)(107,139,125,158)(108,140,126,159)(109,131,127,160)(110,132,128,151) );
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),(11,44),(12,43),(13,42),(14,41),(15,50),(16,49),(17,48),(18,47),(19,46),(20,45),(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)], [(1,142,16,153,33,26,50,134),(2,143,17,154,34,27,41,135),(3,144,18,155,35,28,42,136),(4,145,19,156,36,29,43,137),(5,146,20,157,37,30,44,138),(6,147,11,158,38,21,45,139),(7,148,12,159,39,22,46,140),(8,149,13,160,40,23,47,131),(9,150,14,151,31,24,48,132),(10,141,15,152,32,25,49,133),(51,109,87,91,80,127,69,120),(52,110,88,92,71,128,70,111),(53,101,89,93,72,129,61,112),(54,102,90,94,73,130,62,113),(55,103,81,95,74,121,63,114),(56,104,82,96,75,122,64,115),(57,105,83,97,76,123,65,116),(58,106,84,98,77,124,66,117),(59,107,85,99,78,125,67,118),(60,108,86,100,79,126,68,119)], [(1,73,33,54),(2,74,34,55),(3,75,35,56),(4,76,36,57),(5,77,37,58),(6,78,38,59),(7,79,39,60),(8,80,40,51),(9,71,31,52),(10,72,32,53),(11,85,45,67),(12,86,46,68),(13,87,47,69),(14,88,48,70),(15,89,49,61),(16,90,50,62),(17,81,41,63),(18,82,42,64),(19,83,43,65),(20,84,44,66),(21,118,147,99),(22,119,148,100),(23,120,149,91),(24,111,150,92),(25,112,141,93),(26,113,142,94),(27,114,143,95),(28,115,144,96),(29,116,145,97),(30,117,146,98),(101,133,129,152),(102,134,130,153),(103,135,121,154),(104,136,122,155),(105,137,123,156),(106,138,124,157),(107,139,125,158),(108,140,126,159),(109,131,127,160),(110,132,128,151)]])
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 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | Q16 | D10 | D10 | C5⋊D4 | C8.C22 | D4×D5 | D4×D5 | D5×Q16 | Q16⋊D5 |
| kernel | D10⋊5Q16 | C20.44D4 | D10⋊1C8 | Q8⋊Dic5 | C2×C5⋊Q16 | D10⋊3Q8 | C10×Q16 | C2×Q8×D5 | Dic10 | C2×Dic5 | C5×Q8 | C22×D5 | C2×Q16 | D10 | 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 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D10⋊5Q16 ►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 | 40 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 9 | 0 | 0 |
| 0 | 0 | 40 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 |
| 0 | 0 | 0 | 0 | 34 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 9 | 0 | 0 |
| 0 | 0 | 40 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 17 |
| 0 | 0 | 0 | 0 | 15 | 21 |
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],[35,40,0,0,0,0,35,6,0,0,0,0,0,0,40,40,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,16,40,0,0,0,0,9,25,0,0,0,0,0,0,0,34,0,0,0,0,6,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,40,0,0,0,0,9,25,0,0,0,0,0,0,20,15,0,0,0,0,17,21] >;
D10⋊5Q16 in GAP, Magma, Sage, TeX
D_{10}\rtimes_5Q_{16} % in TeX
G:=Group("D10:5Q16"); // GroupNames label
G:=SmallGroup(320,813);
// by ID
G=gap.SmallGroup(320,813);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,184,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations