metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5⋊6SD16, C5⋊3(C4×SD16), D4.D5⋊6C4, D4.2(C4×D5), C10.62(C4×D4), C2.1(D5×SD16), C4⋊C4.130D10, (C8×Dic5)⋊18C2, D4⋊C4.9D5, (C2×C8).198D10, C20.Q8⋊1C2, (D4×Dic5).2C2, C22.67(D4×D5), Dic10⋊11(C2×C4), Dic5⋊3Q8⋊2C2, (C2×D4).125D10, C10.20(C4○D8), C2.1(D8⋊3D5), C20.38(C22×C4), C10.18(C2×SD16), C20.144(C4○D4), C20.44D4⋊15C2, C4.45(D4⋊2D5), (C2×C40).177C22, (C2×C20).198C23, (C2×Dic5).268D4, (D4×C10).19C22, C4⋊Dic5.58C22, C2.16(Dic5⋊4D4), (C2×Dic10).54C22, (C4×Dic5).252C22, C4.3(C2×C4×D5), C5⋊2C8⋊18(C2×C4), (C5×D4).16(C2×C4), (C5×C4⋊C4).3C22, (C2×D4.D5).2C2, (C5×D4⋊C4).8C2, (C2×C10).211(C2×D4), (C2×C4).305(C22×D5), (C2×C5⋊2C8).216C22, SmallGroup(320,385)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5⋊6SD16
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c3 >
Subgroups: 422 in 122 conjugacy classes, 51 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C2×C10, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C5⋊2C8, C40, Dic10, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, C4×SD16, C2×C5⋊2C8, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, D4.D5, C23.D5, C5×C4⋊C4, C2×C40, C2×Dic10, C22×Dic5, D4×C10, C20.Q8, C8×Dic5, C20.44D4, C5×D4⋊C4, Dic5⋊3Q8, C2×D4.D5, D4×Dic5, Dic5⋊6SD16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, SD16, C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×SD16, C4○D8, C4×D5, C22×D5, C4×SD16, C2×C4×D5, D4×D5, D4⋊2D5, Dic5⋊4D4, D8⋊3D5, D5×SD16, Dic5⋊6SD16
►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)
(1 102 6 107)(2 101 7 106)(3 110 8 105)(4 109 9 104)(5 108 10 103)(11 71 16 76)(12 80 17 75)(13 79 18 74)(14 78 19 73)(15 77 20 72)(21 96 26 91)(22 95 27 100)(23 94 28 99)(24 93 29 98)(25 92 30 97)(31 127 36 122)(32 126 37 121)(33 125 38 130)(34 124 39 129)(35 123 40 128)(41 116 46 111)(42 115 47 120)(43 114 48 119)(44 113 49 118)(45 112 50 117)(51 147 56 142)(52 146 57 141)(53 145 58 150)(54 144 59 149)(55 143 60 148)(61 136 66 131)(62 135 67 140)(63 134 68 139)(64 133 69 138)(65 132 70 137)(81 153 86 158)(82 152 87 157)(83 151 88 156)(84 160 89 155)(85 159 90 154)
(1 154 36 134 28 16 48 147)(2 153 37 133 29 15 49 146)(3 152 38 132 30 14 50 145)(4 151 39 131 21 13 41 144)(5 160 40 140 22 12 42 143)(6 159 31 139 23 11 43 142)(7 158 32 138 24 20 44 141)(8 157 33 137 25 19 45 150)(9 156 34 136 26 18 46 149)(10 155 35 135 27 17 47 148)(51 107 90 127 63 94 71 114)(52 106 81 126 64 93 72 113)(53 105 82 125 65 92 73 112)(54 104 83 124 66 91 74 111)(55 103 84 123 67 100 75 120)(56 102 85 122 68 99 76 119)(57 101 86 121 69 98 77 118)(58 110 87 130 70 97 78 117)(59 109 88 129 61 96 79 116)(60 108 89 128 62 95 80 115)
(11 142)(12 143)(13 144)(14 145)(15 146)(16 147)(17 148)(18 149)(19 150)(20 141)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 41)(40 42)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 88)(62 89)(63 90)(64 81)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(111 124)(112 125)(113 126)(114 127)(115 128)(116 129)(117 130)(118 121)(119 122)(120 123)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)
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), (1,102,6,107)(2,101,7,106)(3,110,8,105)(4,109,9,104)(5,108,10,103)(11,71,16,76)(12,80,17,75)(13,79,18,74)(14,78,19,73)(15,77,20,72)(21,96,26,91)(22,95,27,100)(23,94,28,99)(24,93,29,98)(25,92,30,97)(31,127,36,122)(32,126,37,121)(33,125,38,130)(34,124,39,129)(35,123,40,128)(41,116,46,111)(42,115,47,120)(43,114,48,119)(44,113,49,118)(45,112,50,117)(51,147,56,142)(52,146,57,141)(53,145,58,150)(54,144,59,149)(55,143,60,148)(61,136,66,131)(62,135,67,140)(63,134,68,139)(64,133,69,138)(65,132,70,137)(81,153,86,158)(82,152,87,157)(83,151,88,156)(84,160,89,155)(85,159,90,154), (1,154,36,134,28,16,48,147)(2,153,37,133,29,15,49,146)(3,152,38,132,30,14,50,145)(4,151,39,131,21,13,41,144)(5,160,40,140,22,12,42,143)(6,159,31,139,23,11,43,142)(7,158,32,138,24,20,44,141)(8,157,33,137,25,19,45,150)(9,156,34,136,26,18,46,149)(10,155,35,135,27,17,47,148)(51,107,90,127,63,94,71,114)(52,106,81,126,64,93,72,113)(53,105,82,125,65,92,73,112)(54,104,83,124,66,91,74,111)(55,103,84,123,67,100,75,120)(56,102,85,122,68,99,76,119)(57,101,86,121,69,98,77,118)(58,110,87,130,70,97,78,117)(59,109,88,129,61,96,79,116)(60,108,89,128,62,95,80,115), (11,142)(12,143)(13,144)(14,145)(15,146)(16,147)(17,148)(18,149)(19,150)(20,141)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,41)(40,42)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,88)(62,89)(63,90)(64,81)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(111,124)(112,125)(113,126)(114,127)(115,128)(116,129)(117,130)(118,121)(119,122)(120,123)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160)>;
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), (1,102,6,107)(2,101,7,106)(3,110,8,105)(4,109,9,104)(5,108,10,103)(11,71,16,76)(12,80,17,75)(13,79,18,74)(14,78,19,73)(15,77,20,72)(21,96,26,91)(22,95,27,100)(23,94,28,99)(24,93,29,98)(25,92,30,97)(31,127,36,122)(32,126,37,121)(33,125,38,130)(34,124,39,129)(35,123,40,128)(41,116,46,111)(42,115,47,120)(43,114,48,119)(44,113,49,118)(45,112,50,117)(51,147,56,142)(52,146,57,141)(53,145,58,150)(54,144,59,149)(55,143,60,148)(61,136,66,131)(62,135,67,140)(63,134,68,139)(64,133,69,138)(65,132,70,137)(81,153,86,158)(82,152,87,157)(83,151,88,156)(84,160,89,155)(85,159,90,154), (1,154,36,134,28,16,48,147)(2,153,37,133,29,15,49,146)(3,152,38,132,30,14,50,145)(4,151,39,131,21,13,41,144)(5,160,40,140,22,12,42,143)(6,159,31,139,23,11,43,142)(7,158,32,138,24,20,44,141)(8,157,33,137,25,19,45,150)(9,156,34,136,26,18,46,149)(10,155,35,135,27,17,47,148)(51,107,90,127,63,94,71,114)(52,106,81,126,64,93,72,113)(53,105,82,125,65,92,73,112)(54,104,83,124,66,91,74,111)(55,103,84,123,67,100,75,120)(56,102,85,122,68,99,76,119)(57,101,86,121,69,98,77,118)(58,110,87,130,70,97,78,117)(59,109,88,129,61,96,79,116)(60,108,89,128,62,95,80,115), (11,142)(12,143)(13,144)(14,145)(15,146)(16,147)(17,148)(18,149)(19,150)(20,141)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,41)(40,42)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,88)(62,89)(63,90)(64,81)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(111,124)(112,125)(113,126)(114,127)(115,128)(116,129)(117,130)(118,121)(119,122)(120,123)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160) );
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)], [(1,102,6,107),(2,101,7,106),(3,110,8,105),(4,109,9,104),(5,108,10,103),(11,71,16,76),(12,80,17,75),(13,79,18,74),(14,78,19,73),(15,77,20,72),(21,96,26,91),(22,95,27,100),(23,94,28,99),(24,93,29,98),(25,92,30,97),(31,127,36,122),(32,126,37,121),(33,125,38,130),(34,124,39,129),(35,123,40,128),(41,116,46,111),(42,115,47,120),(43,114,48,119),(44,113,49,118),(45,112,50,117),(51,147,56,142),(52,146,57,141),(53,145,58,150),(54,144,59,149),(55,143,60,148),(61,136,66,131),(62,135,67,140),(63,134,68,139),(64,133,69,138),(65,132,70,137),(81,153,86,158),(82,152,87,157),(83,151,88,156),(84,160,89,155),(85,159,90,154)], [(1,154,36,134,28,16,48,147),(2,153,37,133,29,15,49,146),(3,152,38,132,30,14,50,145),(4,151,39,131,21,13,41,144),(5,160,40,140,22,12,42,143),(6,159,31,139,23,11,43,142),(7,158,32,138,24,20,44,141),(8,157,33,137,25,19,45,150),(9,156,34,136,26,18,46,149),(10,155,35,135,27,17,47,148),(51,107,90,127,63,94,71,114),(52,106,81,126,64,93,72,113),(53,105,82,125,65,92,73,112),(54,104,83,124,66,91,74,111),(55,103,84,123,67,100,75,120),(56,102,85,122,68,99,76,119),(57,101,86,121,69,98,77,118),(58,110,87,130,70,97,78,117),(59,109,88,129,61,96,79,116),(60,108,89,128,62,95,80,115)], [(11,142),(12,143),(13,144),(14,145),(15,146),(16,147),(17,148),(18,149),(19,150),(20,141),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(37,49),(38,50),(39,41),(40,42),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(61,88),(62,89),(63,90),(64,81),(65,82),(66,83),(67,84),(68,85),(69,86),(70,87),(111,124),(112,125),(113,126),(114,127),(115,128),(116,129),(117,130),(118,121),(119,122),(120,123),(131,151),(132,152),(133,153),(134,154),(135,155),(136,156),(137,157),(138,158),(139,159),(140,160)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | SD16 | C4○D4 | D10 | D10 | D10 | C4○D8 | C4×D5 | D4⋊2D5 | D4×D5 | D8⋊3D5 | D5×SD16 |
| kernel | Dic5⋊6SD16 | C20.Q8 | C8×Dic5 | C20.44D4 | C5×D4⋊C4 | Dic5⋊3Q8 | C2×D4.D5 | D4×Dic5 | D4.D5 | C2×Dic5 | D4⋊C4 | Dic5 | C20 | C4⋊C4 | C2×C8 | C2×D4 | C10 | D4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of Dic5⋊6SD16 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 0 | 0 | 0 | 0 | 5 | 35 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 26 |
| 0 | 0 | 0 | 0 | 15 | 29 |
| 15 | 15 | 0 | 0 | 0 | 0 |
| 26 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 19 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 26 |
| 0 | 0 | 0 | 0 | 15 | 29 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 21 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,5,0,0,0,0,1,35],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,15,0,0,0,0,26,29],[15,26,0,0,0,0,15,15,0,0,0,0,0,0,0,19,0,0,0,0,13,30,0,0,0,0,0,0,12,15,0,0,0,0,26,29],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,21,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
Dic5⋊6SD16 in GAP, Magma, Sage, TeX
{\rm Dic}_5\rtimes_6{\rm SD}_{16} % in TeX
G:=Group("Dic5:6SD16"); // GroupNames label
G:=SmallGroup(320,385);
// by ID
G=gap.SmallGroup(320,385);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,135,268,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^8=d^2=1,b^2=a^5,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations