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