metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊3Q16, C40.27D4, (C2×Q16)⋊5D5, (C10×Q16)⋊5C2, C40⋊5C4⋊24C2, C2.19(D5×Q16), (C2×C8).243D10, C20.186(C2×D4), C8.28(C5⋊D4), C5⋊5(C8.18D4), (C2×Q8).64D10, C10.30(C2×Q16), C10.81(C4○D8), Q8⋊Dic5⋊35C2, (C2×C40).95C22, D10⋊3Q8.9C2, (C22×D5).91D4, C22.279(D4×D5), C20.107(C4○D4), C4.36(D4⋊2D5), C2.22(C20⋊2D4), (C2×C20).462C23, (C2×Dic5).162D4, (Q8×C10).91C22, C2.18(Q8.D10), C10.121(C4⋊D4), C4⋊Dic5.185C22, (D5×C2×C8).5C2, C4.85(C2×C5⋊D4), (C2×C10).373(C2×D4), (C2×C4×D5).311C22, (C2×C4).550(C22×D5), (C2×C5⋊2C8).286C22, SmallGroup(320,815)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊3Q16
G = < a,b,c,d | a10=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a5b, dcd-1=c-1 >
Subgroups: 406 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, 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, 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, D10⋊C4, C2×C40, C5×Q16, C2×C4×D5, Q8×C10, C40⋊5C4, Q8⋊Dic5, D5×C2×C8, D10⋊3Q8, C10×Q16, D10⋊3Q16
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, C4○D4, D10, C4⋊D4, C2×Q16, C4○D8, C5⋊D4, C22×D5, C8.18D4, D4×D5, D4⋊2D5, C2×C5⋊D4, D5×Q16, Q8.D10, C20⋊2D4, D10⋊3Q16
►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)(21 147)(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)
(1 130 50 113 33 108 16 100)(2 121 41 114 34 109 17 91)(3 122 42 115 35 110 18 92)(4 123 43 116 36 101 19 93)(5 124 44 117 37 102 20 94)(6 125 45 118 38 103 11 95)(7 126 46 119 39 104 12 96)(8 127 47 120 40 105 13 97)(9 128 48 111 31 106 14 98)(10 129 49 112 32 107 15 99)(21 85 135 55 143 63 158 78)(22 86 136 56 144 64 159 79)(23 87 137 57 145 65 160 80)(24 88 138 58 146 66 151 71)(25 89 139 59 147 67 152 72)(26 90 140 60 148 68 153 73)(27 81 131 51 149 69 154 74)(28 82 132 52 150 70 155 75)(29 83 133 53 141 61 156 76)(30 84 134 54 142 62 157 77)
(1 73 33 60)(2 74 34 51)(3 75 35 52)(4 76 36 53)(5 77 37 54)(6 78 38 55)(7 79 39 56)(8 80 40 57)(9 71 31 58)(10 72 32 59)(11 85 45 63)(12 86 46 64)(13 87 47 65)(14 88 48 66)(15 89 49 67)(16 90 50 68)(17 81 41 69)(18 82 42 70)(19 83 43 61)(20 84 44 62)(21 118 143 95)(22 119 144 96)(23 120 145 97)(24 111 146 98)(25 112 147 99)(26 113 148 100)(27 114 149 91)(28 115 150 92)(29 116 141 93)(30 117 142 94)(101 133 123 156)(102 134 124 157)(103 135 125 158)(104 136 126 159)(105 137 127 160)(106 138 128 151)(107 139 129 152)(108 140 130 153)(109 131 121 154)(110 132 122 155)
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)(21,147)(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), (1,130,50,113,33,108,16,100)(2,121,41,114,34,109,17,91)(3,122,42,115,35,110,18,92)(4,123,43,116,36,101,19,93)(5,124,44,117,37,102,20,94)(6,125,45,118,38,103,11,95)(7,126,46,119,39,104,12,96)(8,127,47,120,40,105,13,97)(9,128,48,111,31,106,14,98)(10,129,49,112,32,107,15,99)(21,85,135,55,143,63,158,78)(22,86,136,56,144,64,159,79)(23,87,137,57,145,65,160,80)(24,88,138,58,146,66,151,71)(25,89,139,59,147,67,152,72)(26,90,140,60,148,68,153,73)(27,81,131,51,149,69,154,74)(28,82,132,52,150,70,155,75)(29,83,133,53,141,61,156,76)(30,84,134,54,142,62,157,77), (1,73,33,60)(2,74,34,51)(3,75,35,52)(4,76,36,53)(5,77,37,54)(6,78,38,55)(7,79,39,56)(8,80,40,57)(9,71,31,58)(10,72,32,59)(11,85,45,63)(12,86,46,64)(13,87,47,65)(14,88,48,66)(15,89,49,67)(16,90,50,68)(17,81,41,69)(18,82,42,70)(19,83,43,61)(20,84,44,62)(21,118,143,95)(22,119,144,96)(23,120,145,97)(24,111,146,98)(25,112,147,99)(26,113,148,100)(27,114,149,91)(28,115,150,92)(29,116,141,93)(30,117,142,94)(101,133,123,156)(102,134,124,157)(103,135,125,158)(104,136,126,159)(105,137,127,160)(106,138,128,151)(107,139,129,152)(108,140,130,153)(109,131,121,154)(110,132,122,155)>;
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)(21,147)(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), (1,130,50,113,33,108,16,100)(2,121,41,114,34,109,17,91)(3,122,42,115,35,110,18,92)(4,123,43,116,36,101,19,93)(5,124,44,117,37,102,20,94)(6,125,45,118,38,103,11,95)(7,126,46,119,39,104,12,96)(8,127,47,120,40,105,13,97)(9,128,48,111,31,106,14,98)(10,129,49,112,32,107,15,99)(21,85,135,55,143,63,158,78)(22,86,136,56,144,64,159,79)(23,87,137,57,145,65,160,80)(24,88,138,58,146,66,151,71)(25,89,139,59,147,67,152,72)(26,90,140,60,148,68,153,73)(27,81,131,51,149,69,154,74)(28,82,132,52,150,70,155,75)(29,83,133,53,141,61,156,76)(30,84,134,54,142,62,157,77), (1,73,33,60)(2,74,34,51)(3,75,35,52)(4,76,36,53)(5,77,37,54)(6,78,38,55)(7,79,39,56)(8,80,40,57)(9,71,31,58)(10,72,32,59)(11,85,45,63)(12,86,46,64)(13,87,47,65)(14,88,48,66)(15,89,49,67)(16,90,50,68)(17,81,41,69)(18,82,42,70)(19,83,43,61)(20,84,44,62)(21,118,143,95)(22,119,144,96)(23,120,145,97)(24,111,146,98)(25,112,147,99)(26,113,148,100)(27,114,149,91)(28,115,150,92)(29,116,141,93)(30,117,142,94)(101,133,123,156)(102,134,124,157)(103,135,125,158)(104,136,126,159)(105,137,127,160)(106,138,128,151)(107,139,129,152)(108,140,130,153)(109,131,121,154)(110,132,122,155) );
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),(21,147),(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)], [(1,130,50,113,33,108,16,100),(2,121,41,114,34,109,17,91),(3,122,42,115,35,110,18,92),(4,123,43,116,36,101,19,93),(5,124,44,117,37,102,20,94),(6,125,45,118,38,103,11,95),(7,126,46,119,39,104,12,96),(8,127,47,120,40,105,13,97),(9,128,48,111,31,106,14,98),(10,129,49,112,32,107,15,99),(21,85,135,55,143,63,158,78),(22,86,136,56,144,64,159,79),(23,87,137,57,145,65,160,80),(24,88,138,58,146,66,151,71),(25,89,139,59,147,67,152,72),(26,90,140,60,148,68,153,73),(27,81,131,51,149,69,154,74),(28,82,132,52,150,70,155,75),(29,83,133,53,141,61,156,76),(30,84,134,54,142,62,157,77)], [(1,73,33,60),(2,74,34,51),(3,75,35,52),(4,76,36,53),(5,77,37,54),(6,78,38,55),(7,79,39,56),(8,80,40,57),(9,71,31,58),(10,72,32,59),(11,85,45,63),(12,86,46,64),(13,87,47,65),(14,88,48,66),(15,89,49,67),(16,90,50,68),(17,81,41,69),(18,82,42,70),(19,83,43,61),(20,84,44,62),(21,118,143,95),(22,119,144,96),(23,120,145,97),(24,111,146,98),(25,112,147,99),(26,113,148,100),(27,114,149,91),(28,115,150,92),(29,116,141,93),(30,117,142,94),(101,133,123,156),(102,134,124,157),(103,135,125,158),(104,136,126,159),(105,137,127,160),(106,138,128,151),(107,139,129,152),(108,140,130,153),(109,131,121,154),(110,132,122,155)]])
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 | C5⋊D4 | D4⋊2D5 | D4×D5 | D5×Q16 | Q8.D10 |
| kernel | D10⋊3Q16 | C40⋊5C4 | Q8⋊Dic5 | D5×C2×C8 | D10⋊3Q8 | C10×Q16 | C40 | C2×Dic5 | C22×D5 | C2×Q16 | C20 | D10 | C2×C8 | C2×Q8 | C10 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of D10⋊3Q16 ►in GL4(𝔽41) generated by
| 6 | 6 | 0 | 0 |
| 35 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 6 | 6 | 0 | 0 |
| 1 | 35 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 26 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 21 | 38 |
| 18 | 35 | 0 | 0 |
| 6 | 23 | 0 | 0 |
| 0 | 0 | 12 | 18 |
| 0 | 0 | 17 | 29 |
G:=sub<GL(4,GF(41))| [6,35,0,0,6,1,0,0,0,0,40,0,0,0,0,40],[6,1,0,0,6,35,0,0,0,0,1,26,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,27,21,0,0,0,38],[18,6,0,0,35,23,0,0,0,0,12,17,0,0,18,29] >;
D10⋊3Q16 in GAP, Magma, Sage, TeX
D_{10}\rtimes_3Q_{16} % in TeX
G:=Group("D10:3Q16"); // GroupNames label
G:=SmallGroup(320,815);
// by ID
G=gap.SmallGroup(320,815);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,184,438,102,12550]);
// 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,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations