metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊2Q16, C40.17D4, C8.19D20, C2.D8⋊6D5, C4⋊C4.52D10, C4.55(C2×D20), C2.15(D5×Q16), C20.135(C2×D4), (C2×C8).232D10, C5⋊2(C8.18D4), C10.25(C2×Q16), (C2×Dic20)⋊16C2, C20.42(C4○D4), C10.29(C4○D8), C10.Q16⋊21C2, (C2×C40).84C22, D10⋊2Q8.9C2, (C22×D5).87D4, C22.233(D4×D5), C2.14(D8⋊3D5), C2.21(C4⋊D20), C10.48(C4⋊D4), (C2×C20).303C23, C4.11(Q8⋊2D5), (C2×Dic5).149D4, (C2×Dic10).95C22, (D5×C2×C8).4C2, (C5×C2.D8)⋊6C2, (C2×C10).308(C2×D4), (C5×C4⋊C4).96C22, (C2×C4×D5).307C22, (C2×C4).406(C22×D5), (C2×C5⋊2C8).244C22, SmallGroup(320,514)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊2Q16
G = < a,b,c,d | a10=b2=c8=1, d2=c4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a3b, dcd-1=c-1 >
Subgroups: 454 in 114 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, Q8, C23, D5, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, Q8⋊C4, C2.D8, C22⋊Q8, C22×C8, C2×Q16, C5⋊2C8, C40, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C8.18D4, C8×D5, Dic20, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C10.Q16, C5×C2.D8, D10⋊2Q8, D5×C2×C8, C2×Dic20, D10⋊2Q16
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, C4○D4, D10, C4⋊D4, C2×Q16, C4○D8, D20, C22×D5, C8.18D4, C2×D20, D4×D5, Q8⋊2D5, C4⋊D20, D8⋊3D5, D5×Q16, D10⋊2Q16
►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 48)(12 47)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 50)(20 49)(21 143)(22 142)(23 141)(24 150)(25 149)(26 148)(27 147)(28 146)(29 145)(30 144)(51 72)(52 71)(53 80)(54 79)(55 78)(56 77)(57 76)(58 75)(59 74)(60 73)(61 87)(62 86)(63 85)(64 84)(65 83)(66 82)(67 81)(68 90)(69 89)(70 88)(91 117)(92 116)(93 115)(94 114)(95 113)(96 112)(97 111)(98 120)(99 119)(100 118)(101 122)(102 121)(103 130)(104 129)(105 128)(106 127)(107 126)(108 125)(109 124)(110 123)(131 152)(132 151)(133 160)(134 159)(135 158)(136 157)(137 156)(138 155)(139 154)(140 153)
(1 130 50 113 33 104 20 96)(2 121 41 114 34 105 11 97)(3 122 42 115 35 106 12 98)(4 123 43 116 36 107 13 99)(5 124 44 117 37 108 14 100)(6 125 45 118 38 109 15 91)(7 126 46 119 39 110 16 92)(8 127 47 120 40 101 17 93)(9 128 48 111 31 102 18 94)(10 129 49 112 32 103 19 95)(21 85 131 51 149 69 158 78)(22 86 132 52 150 70 159 79)(23 87 133 53 141 61 160 80)(24 88 134 54 142 62 151 71)(25 89 135 55 143 63 152 72)(26 90 136 56 144 64 153 73)(27 81 137 57 145 65 154 74)(28 82 138 58 146 66 155 75)(29 83 139 59 147 67 156 76)(30 84 140 60 148 68 157 77)
(1 51 33 78)(2 60 34 77)(3 59 35 76)(4 58 36 75)(5 57 37 74)(6 56 38 73)(7 55 39 72)(8 54 40 71)(9 53 31 80)(10 52 32 79)(11 68 41 84)(12 67 42 83)(13 66 43 82)(14 65 44 81)(15 64 45 90)(16 63 46 89)(17 62 47 88)(18 61 48 87)(19 70 49 86)(20 69 50 85)(21 96 149 113)(22 95 150 112)(23 94 141 111)(24 93 142 120)(25 92 143 119)(26 91 144 118)(27 100 145 117)(28 99 146 116)(29 98 147 115)(30 97 148 114)(101 151 127 134)(102 160 128 133)(103 159 129 132)(104 158 130 131)(105 157 121 140)(106 156 122 139)(107 155 123 138)(108 154 124 137)(109 153 125 136)(110 152 126 135)
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)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,50)(20,49)(21,143)(22,142)(23,141)(24,150)(25,149)(26,148)(27,147)(28,146)(29,145)(30,144)(51,72)(52,71)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,87)(62,86)(63,85)(64,84)(65,83)(66,82)(67,81)(68,90)(69,89)(70,88)(91,117)(92,116)(93,115)(94,114)(95,113)(96,112)(97,111)(98,120)(99,119)(100,118)(101,122)(102,121)(103,130)(104,129)(105,128)(106,127)(107,126)(108,125)(109,124)(110,123)(131,152)(132,151)(133,160)(134,159)(135,158)(136,157)(137,156)(138,155)(139,154)(140,153), (1,130,50,113,33,104,20,96)(2,121,41,114,34,105,11,97)(3,122,42,115,35,106,12,98)(4,123,43,116,36,107,13,99)(5,124,44,117,37,108,14,100)(6,125,45,118,38,109,15,91)(7,126,46,119,39,110,16,92)(8,127,47,120,40,101,17,93)(9,128,48,111,31,102,18,94)(10,129,49,112,32,103,19,95)(21,85,131,51,149,69,158,78)(22,86,132,52,150,70,159,79)(23,87,133,53,141,61,160,80)(24,88,134,54,142,62,151,71)(25,89,135,55,143,63,152,72)(26,90,136,56,144,64,153,73)(27,81,137,57,145,65,154,74)(28,82,138,58,146,66,155,75)(29,83,139,59,147,67,156,76)(30,84,140,60,148,68,157,77), (1,51,33,78)(2,60,34,77)(3,59,35,76)(4,58,36,75)(5,57,37,74)(6,56,38,73)(7,55,39,72)(8,54,40,71)(9,53,31,80)(10,52,32,79)(11,68,41,84)(12,67,42,83)(13,66,43,82)(14,65,44,81)(15,64,45,90)(16,63,46,89)(17,62,47,88)(18,61,48,87)(19,70,49,86)(20,69,50,85)(21,96,149,113)(22,95,150,112)(23,94,141,111)(24,93,142,120)(25,92,143,119)(26,91,144,118)(27,100,145,117)(28,99,146,116)(29,98,147,115)(30,97,148,114)(101,151,127,134)(102,160,128,133)(103,159,129,132)(104,158,130,131)(105,157,121,140)(106,156,122,139)(107,155,123,138)(108,154,124,137)(109,153,125,136)(110,152,126,135)>;
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)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,50)(20,49)(21,143)(22,142)(23,141)(24,150)(25,149)(26,148)(27,147)(28,146)(29,145)(30,144)(51,72)(52,71)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,87)(62,86)(63,85)(64,84)(65,83)(66,82)(67,81)(68,90)(69,89)(70,88)(91,117)(92,116)(93,115)(94,114)(95,113)(96,112)(97,111)(98,120)(99,119)(100,118)(101,122)(102,121)(103,130)(104,129)(105,128)(106,127)(107,126)(108,125)(109,124)(110,123)(131,152)(132,151)(133,160)(134,159)(135,158)(136,157)(137,156)(138,155)(139,154)(140,153), (1,130,50,113,33,104,20,96)(2,121,41,114,34,105,11,97)(3,122,42,115,35,106,12,98)(4,123,43,116,36,107,13,99)(5,124,44,117,37,108,14,100)(6,125,45,118,38,109,15,91)(7,126,46,119,39,110,16,92)(8,127,47,120,40,101,17,93)(9,128,48,111,31,102,18,94)(10,129,49,112,32,103,19,95)(21,85,131,51,149,69,158,78)(22,86,132,52,150,70,159,79)(23,87,133,53,141,61,160,80)(24,88,134,54,142,62,151,71)(25,89,135,55,143,63,152,72)(26,90,136,56,144,64,153,73)(27,81,137,57,145,65,154,74)(28,82,138,58,146,66,155,75)(29,83,139,59,147,67,156,76)(30,84,140,60,148,68,157,77), (1,51,33,78)(2,60,34,77)(3,59,35,76)(4,58,36,75)(5,57,37,74)(6,56,38,73)(7,55,39,72)(8,54,40,71)(9,53,31,80)(10,52,32,79)(11,68,41,84)(12,67,42,83)(13,66,43,82)(14,65,44,81)(15,64,45,90)(16,63,46,89)(17,62,47,88)(18,61,48,87)(19,70,49,86)(20,69,50,85)(21,96,149,113)(22,95,150,112)(23,94,141,111)(24,93,142,120)(25,92,143,119)(26,91,144,118)(27,100,145,117)(28,99,146,116)(29,98,147,115)(30,97,148,114)(101,151,127,134)(102,160,128,133)(103,159,129,132)(104,158,130,131)(105,157,121,140)(106,156,122,139)(107,155,123,138)(108,154,124,137)(109,153,125,136)(110,152,126,135) );
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),(11,48),(12,47),(13,46),(14,45),(15,44),(16,43),(17,42),(18,41),(19,50),(20,49),(21,143),(22,142),(23,141),(24,150),(25,149),(26,148),(27,147),(28,146),(29,145),(30,144),(51,72),(52,71),(53,80),(54,79),(55,78),(56,77),(57,76),(58,75),(59,74),(60,73),(61,87),(62,86),(63,85),(64,84),(65,83),(66,82),(67,81),(68,90),(69,89),(70,88),(91,117),(92,116),(93,115),(94,114),(95,113),(96,112),(97,111),(98,120),(99,119),(100,118),(101,122),(102,121),(103,130),(104,129),(105,128),(106,127),(107,126),(108,125),(109,124),(110,123),(131,152),(132,151),(133,160),(134,159),(135,158),(136,157),(137,156),(138,155),(139,154),(140,153)], [(1,130,50,113,33,104,20,96),(2,121,41,114,34,105,11,97),(3,122,42,115,35,106,12,98),(4,123,43,116,36,107,13,99),(5,124,44,117,37,108,14,100),(6,125,45,118,38,109,15,91),(7,126,46,119,39,110,16,92),(8,127,47,120,40,101,17,93),(9,128,48,111,31,102,18,94),(10,129,49,112,32,103,19,95),(21,85,131,51,149,69,158,78),(22,86,132,52,150,70,159,79),(23,87,133,53,141,61,160,80),(24,88,134,54,142,62,151,71),(25,89,135,55,143,63,152,72),(26,90,136,56,144,64,153,73),(27,81,137,57,145,65,154,74),(28,82,138,58,146,66,155,75),(29,83,139,59,147,67,156,76),(30,84,140,60,148,68,157,77)], [(1,51,33,78),(2,60,34,77),(3,59,35,76),(4,58,36,75),(5,57,37,74),(6,56,38,73),(7,55,39,72),(8,54,40,71),(9,53,31,80),(10,52,32,79),(11,68,41,84),(12,67,42,83),(13,66,43,82),(14,65,44,81),(15,64,45,90),(16,63,46,89),(17,62,47,88),(18,61,48,87),(19,70,49,86),(20,69,50,85),(21,96,149,113),(22,95,150,112),(23,94,141,111),(24,93,142,120),(25,92,143,119),(26,91,144,118),(27,100,145,117),(28,99,146,116),(29,98,147,115),(30,97,148,114),(101,151,127,134),(102,160,128,133),(103,159,129,132),(104,158,130,131),(105,157,121,140),(106,156,122,139),(107,155,123,138),(108,154,124,137),(109,153,125,136),(110,152,126,135)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | Q16 | D10 | D10 | C4○D8 | D20 | Q8⋊2D5 | D4×D5 | D8⋊3D5 | D5×Q16 |
| kernel | D10⋊2Q16 | C10.Q16 | C5×C2.D8 | D10⋊2Q8 | D5×C2×C8 | C2×Dic20 | C40 | C2×Dic5 | C22×D5 | C2.D8 | C20 | D10 | C4⋊C4 | C2×C8 | C10 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 4 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of D10⋊2Q16 ►in GL4(𝔽41) generated by
| 1 | 6 | 0 | 0 |
| 35 | 6 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 6 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 30 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 14 | 0 |
| 0 | 0 | 1 | 3 |
| 25 | 25 | 0 | 0 |
| 39 | 16 | 0 | 0 |
| 0 | 0 | 19 | 37 |
| 0 | 0 | 29 | 22 |
G:=sub<GL(4,GF(41))| [1,35,0,0,6,6,0,0,0,0,40,0,0,0,0,40],[40,6,0,0,0,1,0,0,0,0,1,30,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,14,1,0,0,0,3],[25,39,0,0,25,16,0,0,0,0,19,29,0,0,37,22] >;
D10⋊2Q16 in GAP, Magma, Sage, TeX
D_{10}\rtimes_2Q_{16} % in TeX
G:=Group("D10:2Q16"); // GroupNames label
G:=SmallGroup(320,514);
// by ID
G=gap.SmallGroup(320,514);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,226,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=c^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations