metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊1Q16, C4⋊C4.29D10, Q8⋊C4⋊3D5, C4.168(D4×D5), C5⋊2C8.44D4, C2.11(D5×Q16), C20.126(C2×D4), (C2×C8).211D10, C5⋊1(C8.18D4), (C2×Q8).19D10, C10.19(C2×Q16), C4.34(C4○D20), C20.21(C4○D4), C10.49(C4○D8), C10.D8⋊11C2, D10⋊2Q8.3C2, D10⋊3Q8.4C2, (C22×D5).82D4, C22.203(D4×D5), C20.44D4⋊24C2, C10.25(C4⋊D4), (C2×C20).253C23, (C2×C40).202C22, (C2×Dic5).140D4, C4⋊Dic5.97C22, (Q8×C10).36C22, C2.28(D10⋊D4), C2.18(SD16⋊3D5), (C2×Dic10).77C22, (D5×C2×C8).16C2, (C2×C5⋊Q16)⋊5C2, (C5×Q8⋊C4)⋊24C2, (C2×C10).266(C2×D4), (C5×C4⋊C4).54C22, (C2×C4×D5).299C22, (C2×C4).360(C22×D5), (C2×C5⋊2C8).229C22, SmallGroup(320,440)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊Q16
G = < a,b,c,d | a10=b2=c8=1, d2=c4, bab=cac-1=a-1, ad=da, cbc-1=a8b, dbd-1=a5b, dcd-1=c-1 >
Subgroups: 438 in 114 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, 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, Q8⋊C4, Q8⋊C4, C2.D8, C22⋊Q8, C22×C8, C2×Q16, C5⋊2C8, C40, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C8.18D4, C8×D5, C2×C5⋊2C8, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C5⋊Q16, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, Q8×C10, C10.D8, C20.44D4, C5×Q8⋊C4, D10⋊2Q8, D5×C2×C8, C2×C5⋊Q16, D10⋊3Q8, D10⋊Q16
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, C4○D4, D10, C4⋊D4, C2×Q16, C4○D8, C22×D5, C8.18D4, C4○D20, D4×D5, D10⋊D4, SD16⋊3D5, D5×Q16, D10⋊Q16
►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 10)(2 9)(3 8)(4 7)(5 6)(11 18)(12 17)(13 16)(14 15)(19 20)(21 24)(22 23)(25 30)(26 29)(27 28)(31 35)(32 34)(36 40)(37 39)(41 43)(44 50)(45 49)(46 48)(51 58)(52 57)(53 56)(54 55)(59 60)(61 69)(62 68)(63 67)(64 66)(71 79)(72 78)(73 77)(74 76)(81 88)(82 87)(83 86)(84 85)(89 90)(91 93)(94 100)(95 99)(96 98)(101 104)(102 103)(105 110)(106 109)(107 108)(111 114)(112 113)(115 120)(116 119)(117 118)(121 123)(124 130)(125 129)(126 128)(131 138)(132 137)(133 136)(134 135)(139 140)(141 149)(142 148)(143 147)(144 146)(151 159)(152 158)(153 157)(154 156)
(1 118 28 135 55 90 15 108)(2 117 29 134 56 89 16 107)(3 116 30 133 57 88 17 106)(4 115 21 132 58 87 18 105)(5 114 22 131 59 86 19 104)(6 113 23 140 60 85 20 103)(7 112 24 139 51 84 11 102)(8 111 25 138 52 83 12 101)(9 120 26 137 53 82 13 110)(10 119 27 136 54 81 14 109)(31 95 153 78 143 68 125 50)(32 94 154 77 144 67 126 49)(33 93 155 76 145 66 127 48)(34 92 156 75 146 65 128 47)(35 91 157 74 147 64 129 46)(36 100 158 73 148 63 130 45)(37 99 159 72 149 62 121 44)(38 98 160 71 150 61 122 43)(39 97 151 80 141 70 123 42)(40 96 152 79 142 69 124 41)
(1 95 55 68)(2 96 56 69)(3 97 57 70)(4 98 58 61)(5 99 59 62)(6 100 60 63)(7 91 51 64)(8 92 52 65)(9 93 53 66)(10 94 54 67)(11 74 24 46)(12 75 25 47)(13 76 26 48)(14 77 27 49)(15 78 28 50)(16 79 29 41)(17 80 30 42)(18 71 21 43)(19 72 22 44)(20 73 23 45)(31 90 143 118)(32 81 144 119)(33 82 145 120)(34 83 146 111)(35 84 147 112)(36 85 148 113)(37 86 149 114)(38 87 150 115)(39 88 141 116)(40 89 142 117)(101 156 138 128)(102 157 139 129)(103 158 140 130)(104 159 131 121)(105 160 132 122)(106 151 133 123)(107 152 134 124)(108 153 135 125)(109 154 136 126)(110 155 137 127)
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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,18)(12,17)(13,16)(14,15)(19,20)(21,24)(22,23)(25,30)(26,29)(27,28)(31,35)(32,34)(36,40)(37,39)(41,43)(44,50)(45,49)(46,48)(51,58)(52,57)(53,56)(54,55)(59,60)(61,69)(62,68)(63,67)(64,66)(71,79)(72,78)(73,77)(74,76)(81,88)(82,87)(83,86)(84,85)(89,90)(91,93)(94,100)(95,99)(96,98)(101,104)(102,103)(105,110)(106,109)(107,108)(111,114)(112,113)(115,120)(116,119)(117,118)(121,123)(124,130)(125,129)(126,128)(131,138)(132,137)(133,136)(134,135)(139,140)(141,149)(142,148)(143,147)(144,146)(151,159)(152,158)(153,157)(154,156), (1,118,28,135,55,90,15,108)(2,117,29,134,56,89,16,107)(3,116,30,133,57,88,17,106)(4,115,21,132,58,87,18,105)(5,114,22,131,59,86,19,104)(6,113,23,140,60,85,20,103)(7,112,24,139,51,84,11,102)(8,111,25,138,52,83,12,101)(9,120,26,137,53,82,13,110)(10,119,27,136,54,81,14,109)(31,95,153,78,143,68,125,50)(32,94,154,77,144,67,126,49)(33,93,155,76,145,66,127,48)(34,92,156,75,146,65,128,47)(35,91,157,74,147,64,129,46)(36,100,158,73,148,63,130,45)(37,99,159,72,149,62,121,44)(38,98,160,71,150,61,122,43)(39,97,151,80,141,70,123,42)(40,96,152,79,142,69,124,41), (1,95,55,68)(2,96,56,69)(3,97,57,70)(4,98,58,61)(5,99,59,62)(6,100,60,63)(7,91,51,64)(8,92,52,65)(9,93,53,66)(10,94,54,67)(11,74,24,46)(12,75,25,47)(13,76,26,48)(14,77,27,49)(15,78,28,50)(16,79,29,41)(17,80,30,42)(18,71,21,43)(19,72,22,44)(20,73,23,45)(31,90,143,118)(32,81,144,119)(33,82,145,120)(34,83,146,111)(35,84,147,112)(36,85,148,113)(37,86,149,114)(38,87,150,115)(39,88,141,116)(40,89,142,117)(101,156,138,128)(102,157,139,129)(103,158,140,130)(104,159,131,121)(105,160,132,122)(106,151,133,123)(107,152,134,124)(108,153,135,125)(109,154,136,126)(110,155,137,127)>;
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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,18)(12,17)(13,16)(14,15)(19,20)(21,24)(22,23)(25,30)(26,29)(27,28)(31,35)(32,34)(36,40)(37,39)(41,43)(44,50)(45,49)(46,48)(51,58)(52,57)(53,56)(54,55)(59,60)(61,69)(62,68)(63,67)(64,66)(71,79)(72,78)(73,77)(74,76)(81,88)(82,87)(83,86)(84,85)(89,90)(91,93)(94,100)(95,99)(96,98)(101,104)(102,103)(105,110)(106,109)(107,108)(111,114)(112,113)(115,120)(116,119)(117,118)(121,123)(124,130)(125,129)(126,128)(131,138)(132,137)(133,136)(134,135)(139,140)(141,149)(142,148)(143,147)(144,146)(151,159)(152,158)(153,157)(154,156), (1,118,28,135,55,90,15,108)(2,117,29,134,56,89,16,107)(3,116,30,133,57,88,17,106)(4,115,21,132,58,87,18,105)(5,114,22,131,59,86,19,104)(6,113,23,140,60,85,20,103)(7,112,24,139,51,84,11,102)(8,111,25,138,52,83,12,101)(9,120,26,137,53,82,13,110)(10,119,27,136,54,81,14,109)(31,95,153,78,143,68,125,50)(32,94,154,77,144,67,126,49)(33,93,155,76,145,66,127,48)(34,92,156,75,146,65,128,47)(35,91,157,74,147,64,129,46)(36,100,158,73,148,63,130,45)(37,99,159,72,149,62,121,44)(38,98,160,71,150,61,122,43)(39,97,151,80,141,70,123,42)(40,96,152,79,142,69,124,41), (1,95,55,68)(2,96,56,69)(3,97,57,70)(4,98,58,61)(5,99,59,62)(6,100,60,63)(7,91,51,64)(8,92,52,65)(9,93,53,66)(10,94,54,67)(11,74,24,46)(12,75,25,47)(13,76,26,48)(14,77,27,49)(15,78,28,50)(16,79,29,41)(17,80,30,42)(18,71,21,43)(19,72,22,44)(20,73,23,45)(31,90,143,118)(32,81,144,119)(33,82,145,120)(34,83,146,111)(35,84,147,112)(36,85,148,113)(37,86,149,114)(38,87,150,115)(39,88,141,116)(40,89,142,117)(101,156,138,128)(102,157,139,129)(103,158,140,130)(104,159,131,121)(105,160,132,122)(106,151,133,123)(107,152,134,124)(108,153,135,125)(109,154,136,126)(110,155,137,127) );
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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,18),(12,17),(13,16),(14,15),(19,20),(21,24),(22,23),(25,30),(26,29),(27,28),(31,35),(32,34),(36,40),(37,39),(41,43),(44,50),(45,49),(46,48),(51,58),(52,57),(53,56),(54,55),(59,60),(61,69),(62,68),(63,67),(64,66),(71,79),(72,78),(73,77),(74,76),(81,88),(82,87),(83,86),(84,85),(89,90),(91,93),(94,100),(95,99),(96,98),(101,104),(102,103),(105,110),(106,109),(107,108),(111,114),(112,113),(115,120),(116,119),(117,118),(121,123),(124,130),(125,129),(126,128),(131,138),(132,137),(133,136),(134,135),(139,140),(141,149),(142,148),(143,147),(144,146),(151,159),(152,158),(153,157),(154,156)], [(1,118,28,135,55,90,15,108),(2,117,29,134,56,89,16,107),(3,116,30,133,57,88,17,106),(4,115,21,132,58,87,18,105),(5,114,22,131,59,86,19,104),(6,113,23,140,60,85,20,103),(7,112,24,139,51,84,11,102),(8,111,25,138,52,83,12,101),(9,120,26,137,53,82,13,110),(10,119,27,136,54,81,14,109),(31,95,153,78,143,68,125,50),(32,94,154,77,144,67,126,49),(33,93,155,76,145,66,127,48),(34,92,156,75,146,65,128,47),(35,91,157,74,147,64,129,46),(36,100,158,73,148,63,130,45),(37,99,159,72,149,62,121,44),(38,98,160,71,150,61,122,43),(39,97,151,80,141,70,123,42),(40,96,152,79,142,69,124,41)], [(1,95,55,68),(2,96,56,69),(3,97,57,70),(4,98,58,61),(5,99,59,62),(6,100,60,63),(7,91,51,64),(8,92,52,65),(9,93,53,66),(10,94,54,67),(11,74,24,46),(12,75,25,47),(13,76,26,48),(14,77,27,49),(15,78,28,50),(16,79,29,41),(17,80,30,42),(18,71,21,43),(19,72,22,44),(20,73,23,45),(31,90,143,118),(32,81,144,119),(33,82,145,120),(34,83,146,111),(35,84,147,112),(36,85,148,113),(37,86,149,114),(38,87,150,115),(39,88,141,116),(40,89,142,117),(101,156,138,128),(102,157,139,129),(103,158,140,130),(104,159,131,121),(105,160,132,122),(106,151,133,123),(107,152,134,124),(108,153,135,125),(109,154,136,126),(110,155,137,127)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 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 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 8 | 8 | 10 | 10 | 40 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | Q16 | D10 | D10 | D10 | C4○D8 | C4○D20 | D4×D5 | D4×D5 | SD16⋊3D5 | D5×Q16 |
| kernel | D10⋊Q16 | C10.D8 | C20.44D4 | C5×Q8⋊C4 | D10⋊2Q8 | D5×C2×C8 | C2×C5⋊Q16 | D10⋊3Q8 | C5⋊2C8 | C2×Dic5 | C22×D5 | Q8⋊C4 | C20 | D10 | C4⋊C4 | C2×C8 | C2×Q8 | C10 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of D10⋊Q16 ►in GL6(𝔽41)
| 6 | 7 | 0 | 0 | 0 | 0 |
| 35 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 6 | 7 | 0 | 0 | 0 | 0 |
| 36 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 13 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 38 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 5 | 0 | 0 |
| 0 | 0 | 29 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
G:=sub<GL(6,GF(41))| [6,35,0,0,0,0,7,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[35,6,0,0,0,0,1,6,0,0,0,0,0,0,40,35,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[6,36,0,0,0,0,7,35,0,0,0,0,0,0,9,13,0,0,0,0,0,32,0,0,0,0,0,0,27,0,0,0,0,0,0,38],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,26,29,0,0,0,0,5,15,0,0,0,0,0,0,0,40,0,0,0,0,1,0] >;
D10⋊Q16 in GAP, Magma, Sage, TeX
D_{10}\rtimes Q_{16} % in TeX
G:=Group("D10:Q16"); // GroupNames label
G:=SmallGroup(320,440);
// by ID
G=gap.SmallGroup(320,440);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,555,184,297,136,438,102,12550]);
// 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=a^-1,a*d=d*a,c*b*c^-1=a^8*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations