metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊3SD16, Dic10⋊9D4, C42.75D10, C4.54(D4×D5), C4⋊2(D4.D5), C20⋊3C8⋊32C2, C5⋊4(C4⋊SD16), C4⋊1D4.5D5, C20.33(C2×D4), (C2×D4).58D10, (C2×C20).149D4, (C4×Dic10)⋊22C2, C20.77(C4○D4), D4⋊Dic5⋊23C2, C4.4(D4⋊2D5), C10.58(C2×SD16), C2.13(C20⋊2D4), C10.96(C8⋊C22), (C4×C20).123C22, (C2×C20).393C23, (D4×C10).74C22, C10.104(C4⋊D4), C4⋊Dic5.345C22, C2.17(D4.D10), (C2×Dic10).282C22, (C2×D4.D5)⋊14C2, (C5×C4⋊1D4).4C2, C2.12(C2×D4.D5), (C2×C10).524(C2×D4), (C2×C4).186(C5⋊D4), (C2×C4).491(C22×D5), C22.197(C2×C5⋊D4), (C2×C5⋊2C8).132C22, SmallGroup(320,702)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊9D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, ac=ca, dad=a11, bc=cb, dbd=a5b, dcd=c-1 >
Subgroups: 438 in 128 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×D4, C2×Q8, Dic5, C20, C20, C20, C2×C10, C2×C10, D4⋊C4, C4⋊C8, C4×Q8, C4⋊1D4, C2×SD16, C5⋊2C8, Dic10, Dic10, C2×Dic5, C2×C20, C5×D4, C22×C10, C4⋊SD16, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, D4.D5, C4×C20, C2×Dic10, D4×C10, D4×C10, C20⋊3C8, D4⋊Dic5, C4×Dic10, C2×D4.D5, C5×C4⋊1D4, Dic10⋊9D4
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C4⋊D4, C2×SD16, C8⋊C22, C5⋊D4, C22×D5, C4⋊SD16, D4.D5, D4×D5, D4⋊2D5, C2×C5⋊D4, D4.D10, C2×D4.D5, C20⋊2D4, Dic10⋊9D4
►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 61 11 71)(2 80 12 70)(3 79 13 69)(4 78 14 68)(5 77 15 67)(6 76 16 66)(7 75 17 65)(8 74 18 64)(9 73 19 63)(10 72 20 62)(21 58 31 48)(22 57 32 47)(23 56 33 46)(24 55 34 45)(25 54 35 44)(26 53 36 43)(27 52 37 42)(28 51 38 41)(29 50 39 60)(30 49 40 59)(81 145 91 155)(82 144 92 154)(83 143 93 153)(84 142 94 152)(85 141 95 151)(86 160 96 150)(87 159 97 149)(88 158 98 148)(89 157 99 147)(90 156 100 146)(101 121 111 131)(102 140 112 130)(103 139 113 129)(104 138 114 128)(105 137 115 127)(106 136 116 126)(107 135 117 125)(108 134 118 124)(109 133 119 123)(110 132 120 122)
(1 32 103 151)(2 33 104 152)(3 34 105 153)(4 35 106 154)(5 36 107 155)(6 37 108 156)(7 38 109 157)(8 39 110 158)(9 40 111 159)(10 21 112 160)(11 22 113 141)(12 23 114 142)(13 24 115 143)(14 25 116 144)(15 26 117 145)(16 27 118 146)(17 28 119 147)(18 29 120 148)(19 30 101 149)(20 31 102 150)(41 133 99 75)(42 134 100 76)(43 135 81 77)(44 136 82 78)(45 137 83 79)(46 138 84 80)(47 139 85 61)(48 140 86 62)(49 121 87 63)(50 122 88 64)(51 123 89 65)(52 124 90 66)(53 125 91 67)(54 126 92 68)(55 127 93 69)(56 128 94 70)(57 129 95 71)(58 130 96 72)(59 131 97 73)(60 132 98 74)
(2 12)(4 14)(6 16)(8 18)(10 20)(21 150)(22 141)(23 152)(24 143)(25 154)(26 145)(27 156)(28 147)(29 158)(30 149)(31 160)(32 151)(33 142)(34 153)(35 144)(36 155)(37 146)(38 157)(39 148)(40 159)(41 94)(42 85)(43 96)(44 87)(45 98)(46 89)(47 100)(48 91)(49 82)(50 93)(51 84)(52 95)(53 86)(54 97)(55 88)(56 99)(57 90)(58 81)(59 92)(60 83)(61 76)(62 67)(63 78)(64 69)(65 80)(66 71)(68 73)(70 75)(72 77)(74 79)(102 112)(104 114)(106 116)(108 118)(110 120)(121 136)(122 127)(123 138)(124 129)(125 140)(126 131)(128 133)(130 135)(132 137)(134 139)
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), (1,61,11,71)(2,80,12,70)(3,79,13,69)(4,78,14,68)(5,77,15,67)(6,76,16,66)(7,75,17,65)(8,74,18,64)(9,73,19,63)(10,72,20,62)(21,58,31,48)(22,57,32,47)(23,56,33,46)(24,55,34,45)(25,54,35,44)(26,53,36,43)(27,52,37,42)(28,51,38,41)(29,50,39,60)(30,49,40,59)(81,145,91,155)(82,144,92,154)(83,143,93,153)(84,142,94,152)(85,141,95,151)(86,160,96,150)(87,159,97,149)(88,158,98,148)(89,157,99,147)(90,156,100,146)(101,121,111,131)(102,140,112,130)(103,139,113,129)(104,138,114,128)(105,137,115,127)(106,136,116,126)(107,135,117,125)(108,134,118,124)(109,133,119,123)(110,132,120,122), (1,32,103,151)(2,33,104,152)(3,34,105,153)(4,35,106,154)(5,36,107,155)(6,37,108,156)(7,38,109,157)(8,39,110,158)(9,40,111,159)(10,21,112,160)(11,22,113,141)(12,23,114,142)(13,24,115,143)(14,25,116,144)(15,26,117,145)(16,27,118,146)(17,28,119,147)(18,29,120,148)(19,30,101,149)(20,31,102,150)(41,133,99,75)(42,134,100,76)(43,135,81,77)(44,136,82,78)(45,137,83,79)(46,138,84,80)(47,139,85,61)(48,140,86,62)(49,121,87,63)(50,122,88,64)(51,123,89,65)(52,124,90,66)(53,125,91,67)(54,126,92,68)(55,127,93,69)(56,128,94,70)(57,129,95,71)(58,130,96,72)(59,131,97,73)(60,132,98,74), (2,12)(4,14)(6,16)(8,18)(10,20)(21,150)(22,141)(23,152)(24,143)(25,154)(26,145)(27,156)(28,147)(29,158)(30,149)(31,160)(32,151)(33,142)(34,153)(35,144)(36,155)(37,146)(38,157)(39,148)(40,159)(41,94)(42,85)(43,96)(44,87)(45,98)(46,89)(47,100)(48,91)(49,82)(50,93)(51,84)(52,95)(53,86)(54,97)(55,88)(56,99)(57,90)(58,81)(59,92)(60,83)(61,76)(62,67)(63,78)(64,69)(65,80)(66,71)(68,73)(70,75)(72,77)(74,79)(102,112)(104,114)(106,116)(108,118)(110,120)(121,136)(122,127)(123,138)(124,129)(125,140)(126,131)(128,133)(130,135)(132,137)(134,139)>;
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), (1,61,11,71)(2,80,12,70)(3,79,13,69)(4,78,14,68)(5,77,15,67)(6,76,16,66)(7,75,17,65)(8,74,18,64)(9,73,19,63)(10,72,20,62)(21,58,31,48)(22,57,32,47)(23,56,33,46)(24,55,34,45)(25,54,35,44)(26,53,36,43)(27,52,37,42)(28,51,38,41)(29,50,39,60)(30,49,40,59)(81,145,91,155)(82,144,92,154)(83,143,93,153)(84,142,94,152)(85,141,95,151)(86,160,96,150)(87,159,97,149)(88,158,98,148)(89,157,99,147)(90,156,100,146)(101,121,111,131)(102,140,112,130)(103,139,113,129)(104,138,114,128)(105,137,115,127)(106,136,116,126)(107,135,117,125)(108,134,118,124)(109,133,119,123)(110,132,120,122), (1,32,103,151)(2,33,104,152)(3,34,105,153)(4,35,106,154)(5,36,107,155)(6,37,108,156)(7,38,109,157)(8,39,110,158)(9,40,111,159)(10,21,112,160)(11,22,113,141)(12,23,114,142)(13,24,115,143)(14,25,116,144)(15,26,117,145)(16,27,118,146)(17,28,119,147)(18,29,120,148)(19,30,101,149)(20,31,102,150)(41,133,99,75)(42,134,100,76)(43,135,81,77)(44,136,82,78)(45,137,83,79)(46,138,84,80)(47,139,85,61)(48,140,86,62)(49,121,87,63)(50,122,88,64)(51,123,89,65)(52,124,90,66)(53,125,91,67)(54,126,92,68)(55,127,93,69)(56,128,94,70)(57,129,95,71)(58,130,96,72)(59,131,97,73)(60,132,98,74), (2,12)(4,14)(6,16)(8,18)(10,20)(21,150)(22,141)(23,152)(24,143)(25,154)(26,145)(27,156)(28,147)(29,158)(30,149)(31,160)(32,151)(33,142)(34,153)(35,144)(36,155)(37,146)(38,157)(39,148)(40,159)(41,94)(42,85)(43,96)(44,87)(45,98)(46,89)(47,100)(48,91)(49,82)(50,93)(51,84)(52,95)(53,86)(54,97)(55,88)(56,99)(57,90)(58,81)(59,92)(60,83)(61,76)(62,67)(63,78)(64,69)(65,80)(66,71)(68,73)(70,75)(72,77)(74,79)(102,112)(104,114)(106,116)(108,118)(110,120)(121,136)(122,127)(123,138)(124,129)(125,140)(126,131)(128,133)(130,135)(132,137)(134,139) );
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)], [(1,61,11,71),(2,80,12,70),(3,79,13,69),(4,78,14,68),(5,77,15,67),(6,76,16,66),(7,75,17,65),(8,74,18,64),(9,73,19,63),(10,72,20,62),(21,58,31,48),(22,57,32,47),(23,56,33,46),(24,55,34,45),(25,54,35,44),(26,53,36,43),(27,52,37,42),(28,51,38,41),(29,50,39,60),(30,49,40,59),(81,145,91,155),(82,144,92,154),(83,143,93,153),(84,142,94,152),(85,141,95,151),(86,160,96,150),(87,159,97,149),(88,158,98,148),(89,157,99,147),(90,156,100,146),(101,121,111,131),(102,140,112,130),(103,139,113,129),(104,138,114,128),(105,137,115,127),(106,136,116,126),(107,135,117,125),(108,134,118,124),(109,133,119,123),(110,132,120,122)], [(1,32,103,151),(2,33,104,152),(3,34,105,153),(4,35,106,154),(5,36,107,155),(6,37,108,156),(7,38,109,157),(8,39,110,158),(9,40,111,159),(10,21,112,160),(11,22,113,141),(12,23,114,142),(13,24,115,143),(14,25,116,144),(15,26,117,145),(16,27,118,146),(17,28,119,147),(18,29,120,148),(19,30,101,149),(20,31,102,150),(41,133,99,75),(42,134,100,76),(43,135,81,77),(44,136,82,78),(45,137,83,79),(46,138,84,80),(47,139,85,61),(48,140,86,62),(49,121,87,63),(50,122,88,64),(51,123,89,65),(52,124,90,66),(53,125,91,67),(54,126,92,68),(55,127,93,69),(56,128,94,70),(57,129,95,71),(58,130,96,72),(59,131,97,73),(60,132,98,74)], [(2,12),(4,14),(6,16),(8,18),(10,20),(21,150),(22,141),(23,152),(24,143),(25,154),(26,145),(27,156),(28,147),(29,158),(30,149),(31,160),(32,151),(33,142),(34,153),(35,144),(36,155),(37,146),(38,157),(39,148),(40,159),(41,94),(42,85),(43,96),(44,87),(45,98),(46,89),(47,100),(48,91),(49,82),(50,93),(51,84),(52,95),(53,86),(54,97),(55,88),(56,99),(57,90),(58,81),(59,92),(60,83),(61,76),(62,67),(63,78),(64,69),(65,80),(66,71),(68,73),(70,75),(72,77),(74,79),(102,112),(104,114),(106,116),(108,118),(110,120),(121,136),(122,127),(123,138),(124,129),(125,140),(126,131),(128,133),(130,135),(132,137),(134,139)]])
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 | 10G | ··· | 10N | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | SD16 | C4○D4 | D10 | D10 | C5⋊D4 | C8⋊C22 | D4.D5 | D4×D5 | D4⋊2D5 | D4.D10 |
| kernel | Dic10⋊9D4 | C20⋊3C8 | D4⋊Dic5 | C4×Dic10 | C2×D4.D5 | C5×C4⋊1D4 | Dic10 | C2×C20 | C4⋊1D4 | C20 | C20 | C42 | C2×D4 | C2×C4 | C10 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 1 | 4 | 2 | 2 | 4 |
Matrix representation of Dic10⋊9D4 ►in GL6(𝔽41)
| 1 | 37 | 0 | 0 | 0 | 0 |
| 21 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 0 | 0 |
| 0 | 0 | 9 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 22 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 25 | 0 | 0 |
| 0 | 0 | 2 | 22 | 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 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 18 |
| 0 | 0 | 0 | 0 | 29 | 25 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 21 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 28 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 21 | 40 |
G:=sub<GL(6,GF(41))| [1,21,0,0,0,0,37,40,0,0,0,0,0,0,31,9,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,13,0,0,0,0,22,0,0,0,0,0,0,0,19,2,0,0,0,0,25,22,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,16,29,0,0,0,0,18,25],[1,21,0,0,0,0,0,40,0,0,0,0,0,0,1,28,0,0,0,0,0,40,0,0,0,0,0,0,1,21,0,0,0,0,0,40] >;
Dic10⋊9D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_9D_4 % in TeX
G:=Group("Dic10:9D4"); // GroupNames label
G:=SmallGroup(320,702);
// by ID
G=gap.SmallGroup(320,702);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,254,219,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^11,b*c=c*b,d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations