metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.1Dic10, C4⋊C4.4D10, C5⋊1(D4.Q8), (C5×D4).1Q8, C20.3(C2×Q8), C40⋊6C4⋊11C2, D4⋊C4.6D5, (C2×C8).114D10, C10.D8⋊4C2, C4.Dic10⋊1C2, (D4×Dic5).5C2, C4.3(C2×Dic10), (C2×D4).128D10, C10.37(C4○D8), C20.8Q8⋊11C2, D4⋊Dic5.3C2, C22.165(D4×D5), C10.9(C22⋊Q8), C20.147(C4○D4), C4.76(D4⋊2D5), C2.10(D8⋊D5), C10.27(C8⋊C22), (C2×C40).125C22, (C2×C20).203C23, (C2×Dic5).193D4, (D4×C10).24C22, C4⋊Dic5.62C22, C2.8(SD16⋊3D5), (C4×Dic5).13C22, C2.14(Dic5.14D4), (C5×C4⋊C4).8C22, (C5×D4⋊C4).6C2, (C2×C10).216(C2×D4), (C2×C5⋊2C8).9C22, (C2×C4).310(C22×D5), SmallGroup(320,390)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.Dic10
G = < a,b,c,d | a4=b2=c20=1, d2=c10, bab=cac-1=a-1, ad=da, cbc-1=a-1b, bd=db, dcd-1=a2c-1 >
Subgroups: 374 in 102 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, C2×C10, C2×C10, D4⋊C4, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C5⋊2C8, C40, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, D4.Q8, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C23.D5, C5×C4⋊C4, C2×C40, C22×Dic5, D4×C10, C10.D8, C20.8Q8, C40⋊6C4, D4⋊Dic5, C5×D4⋊C4, C4.Dic10, D4×Dic5, D4.Dic10
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C4○D8, C8⋊C22, Dic10, C22×D5, D4.Q8, C2×Dic10, D4×D5, D4⋊2D5, Dic5.14D4, D8⋊D5, SD16⋊3D5, D4.Dic10
►On 160 points
Generators in S160
(1 104 159 56)(2 57 160 105)(3 106 141 58)(4 59 142 107)(5 108 143 60)(6 41 144 109)(7 110 145 42)(8 43 146 111)(9 112 147 44)(10 45 148 113)(11 114 149 46)(12 47 150 115)(13 116 151 48)(14 49 152 117)(15 118 153 50)(16 51 154 119)(17 120 155 52)(18 53 156 101)(19 102 157 54)(20 55 158 103)(21 72 89 140)(22 121 90 73)(23 74 91 122)(24 123 92 75)(25 76 93 124)(26 125 94 77)(27 78 95 126)(28 127 96 79)(29 80 97 128)(30 129 98 61)(31 62 99 130)(32 131 100 63)(33 64 81 132)(34 133 82 65)(35 66 83 134)(36 135 84 67)(37 68 85 136)(38 137 86 69)(39 70 87 138)(40 139 88 71)
(1 46)(2 12)(3 48)(4 14)(5 50)(6 16)(7 52)(8 18)(9 54)(10 20)(11 56)(13 58)(15 60)(17 42)(19 44)(21 130)(22 32)(23 132)(24 34)(25 134)(26 36)(27 136)(28 38)(29 138)(30 40)(31 140)(33 122)(35 124)(37 126)(39 128)(41 119)(43 101)(45 103)(47 105)(49 107)(51 109)(53 111)(55 113)(57 115)(59 117)(61 139)(62 89)(63 121)(64 91)(65 123)(66 93)(67 125)(68 95)(69 127)(70 97)(71 129)(72 99)(73 131)(74 81)(75 133)(76 83)(77 135)(78 85)(79 137)(80 87)(82 92)(84 94)(86 96)(88 98)(90 100)(102 147)(104 149)(106 151)(108 153)(110 155)(112 157)(114 159)(116 141)(118 143)(120 145)(142 152)(144 154)(146 156)(148 158)(150 160)
(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 29 11 39)(2 96 12 86)(3 27 13 37)(4 94 14 84)(5 25 15 35)(6 92 16 82)(7 23 17 33)(8 90 18 100)(9 21 19 31)(10 88 20 98)(22 156 32 146)(24 154 34 144)(26 152 36 142)(28 150 38 160)(30 148 40 158)(41 75 51 65)(42 122 52 132)(43 73 53 63)(44 140 54 130)(45 71 55 61)(46 138 56 128)(47 69 57 79)(48 136 58 126)(49 67 59 77)(50 134 60 124)(62 112 72 102)(64 110 74 120)(66 108 76 118)(68 106 78 116)(70 104 80 114)(81 145 91 155)(83 143 93 153)(85 141 95 151)(87 159 97 149)(89 157 99 147)(101 131 111 121)(103 129 113 139)(105 127 115 137)(107 125 117 135)(109 123 119 133)
G:=sub<Sym(160)| (1,104,159,56)(2,57,160,105)(3,106,141,58)(4,59,142,107)(5,108,143,60)(6,41,144,109)(7,110,145,42)(8,43,146,111)(9,112,147,44)(10,45,148,113)(11,114,149,46)(12,47,150,115)(13,116,151,48)(14,49,152,117)(15,118,153,50)(16,51,154,119)(17,120,155,52)(18,53,156,101)(19,102,157,54)(20,55,158,103)(21,72,89,140)(22,121,90,73)(23,74,91,122)(24,123,92,75)(25,76,93,124)(26,125,94,77)(27,78,95,126)(28,127,96,79)(29,80,97,128)(30,129,98,61)(31,62,99,130)(32,131,100,63)(33,64,81,132)(34,133,82,65)(35,66,83,134)(36,135,84,67)(37,68,85,136)(38,137,86,69)(39,70,87,138)(40,139,88,71), (1,46)(2,12)(3,48)(4,14)(5,50)(6,16)(7,52)(8,18)(9,54)(10,20)(11,56)(13,58)(15,60)(17,42)(19,44)(21,130)(22,32)(23,132)(24,34)(25,134)(26,36)(27,136)(28,38)(29,138)(30,40)(31,140)(33,122)(35,124)(37,126)(39,128)(41,119)(43,101)(45,103)(47,105)(49,107)(51,109)(53,111)(55,113)(57,115)(59,117)(61,139)(62,89)(63,121)(64,91)(65,123)(66,93)(67,125)(68,95)(69,127)(70,97)(71,129)(72,99)(73,131)(74,81)(75,133)(76,83)(77,135)(78,85)(79,137)(80,87)(82,92)(84,94)(86,96)(88,98)(90,100)(102,147)(104,149)(106,151)(108,153)(110,155)(112,157)(114,159)(116,141)(118,143)(120,145)(142,152)(144,154)(146,156)(148,158)(150,160), (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), (1,29,11,39)(2,96,12,86)(3,27,13,37)(4,94,14,84)(5,25,15,35)(6,92,16,82)(7,23,17,33)(8,90,18,100)(9,21,19,31)(10,88,20,98)(22,156,32,146)(24,154,34,144)(26,152,36,142)(28,150,38,160)(30,148,40,158)(41,75,51,65)(42,122,52,132)(43,73,53,63)(44,140,54,130)(45,71,55,61)(46,138,56,128)(47,69,57,79)(48,136,58,126)(49,67,59,77)(50,134,60,124)(62,112,72,102)(64,110,74,120)(66,108,76,118)(68,106,78,116)(70,104,80,114)(81,145,91,155)(83,143,93,153)(85,141,95,151)(87,159,97,149)(89,157,99,147)(101,131,111,121)(103,129,113,139)(105,127,115,137)(107,125,117,135)(109,123,119,133)>;
G:=Group( (1,104,159,56)(2,57,160,105)(3,106,141,58)(4,59,142,107)(5,108,143,60)(6,41,144,109)(7,110,145,42)(8,43,146,111)(9,112,147,44)(10,45,148,113)(11,114,149,46)(12,47,150,115)(13,116,151,48)(14,49,152,117)(15,118,153,50)(16,51,154,119)(17,120,155,52)(18,53,156,101)(19,102,157,54)(20,55,158,103)(21,72,89,140)(22,121,90,73)(23,74,91,122)(24,123,92,75)(25,76,93,124)(26,125,94,77)(27,78,95,126)(28,127,96,79)(29,80,97,128)(30,129,98,61)(31,62,99,130)(32,131,100,63)(33,64,81,132)(34,133,82,65)(35,66,83,134)(36,135,84,67)(37,68,85,136)(38,137,86,69)(39,70,87,138)(40,139,88,71), (1,46)(2,12)(3,48)(4,14)(5,50)(6,16)(7,52)(8,18)(9,54)(10,20)(11,56)(13,58)(15,60)(17,42)(19,44)(21,130)(22,32)(23,132)(24,34)(25,134)(26,36)(27,136)(28,38)(29,138)(30,40)(31,140)(33,122)(35,124)(37,126)(39,128)(41,119)(43,101)(45,103)(47,105)(49,107)(51,109)(53,111)(55,113)(57,115)(59,117)(61,139)(62,89)(63,121)(64,91)(65,123)(66,93)(67,125)(68,95)(69,127)(70,97)(71,129)(72,99)(73,131)(74,81)(75,133)(76,83)(77,135)(78,85)(79,137)(80,87)(82,92)(84,94)(86,96)(88,98)(90,100)(102,147)(104,149)(106,151)(108,153)(110,155)(112,157)(114,159)(116,141)(118,143)(120,145)(142,152)(144,154)(146,156)(148,158)(150,160), (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), (1,29,11,39)(2,96,12,86)(3,27,13,37)(4,94,14,84)(5,25,15,35)(6,92,16,82)(7,23,17,33)(8,90,18,100)(9,21,19,31)(10,88,20,98)(22,156,32,146)(24,154,34,144)(26,152,36,142)(28,150,38,160)(30,148,40,158)(41,75,51,65)(42,122,52,132)(43,73,53,63)(44,140,54,130)(45,71,55,61)(46,138,56,128)(47,69,57,79)(48,136,58,126)(49,67,59,77)(50,134,60,124)(62,112,72,102)(64,110,74,120)(66,108,76,118)(68,106,78,116)(70,104,80,114)(81,145,91,155)(83,143,93,153)(85,141,95,151)(87,159,97,149)(89,157,99,147)(101,131,111,121)(103,129,113,139)(105,127,115,137)(107,125,117,135)(109,123,119,133) );
G=PermutationGroup([[(1,104,159,56),(2,57,160,105),(3,106,141,58),(4,59,142,107),(5,108,143,60),(6,41,144,109),(7,110,145,42),(8,43,146,111),(9,112,147,44),(10,45,148,113),(11,114,149,46),(12,47,150,115),(13,116,151,48),(14,49,152,117),(15,118,153,50),(16,51,154,119),(17,120,155,52),(18,53,156,101),(19,102,157,54),(20,55,158,103),(21,72,89,140),(22,121,90,73),(23,74,91,122),(24,123,92,75),(25,76,93,124),(26,125,94,77),(27,78,95,126),(28,127,96,79),(29,80,97,128),(30,129,98,61),(31,62,99,130),(32,131,100,63),(33,64,81,132),(34,133,82,65),(35,66,83,134),(36,135,84,67),(37,68,85,136),(38,137,86,69),(39,70,87,138),(40,139,88,71)], [(1,46),(2,12),(3,48),(4,14),(5,50),(6,16),(7,52),(8,18),(9,54),(10,20),(11,56),(13,58),(15,60),(17,42),(19,44),(21,130),(22,32),(23,132),(24,34),(25,134),(26,36),(27,136),(28,38),(29,138),(30,40),(31,140),(33,122),(35,124),(37,126),(39,128),(41,119),(43,101),(45,103),(47,105),(49,107),(51,109),(53,111),(55,113),(57,115),(59,117),(61,139),(62,89),(63,121),(64,91),(65,123),(66,93),(67,125),(68,95),(69,127),(70,97),(71,129),(72,99),(73,131),(74,81),(75,133),(76,83),(77,135),(78,85),(79,137),(80,87),(82,92),(84,94),(86,96),(88,98),(90,100),(102,147),(104,149),(106,151),(108,153),(110,155),(112,157),(114,159),(116,141),(118,143),(120,145),(142,152),(144,154),(146,156),(148,158),(150,160)], [(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)], [(1,29,11,39),(2,96,12,86),(3,27,13,37),(4,94,14,84),(5,25,15,35),(6,92,16,82),(7,23,17,33),(8,90,18,100),(9,21,19,31),(10,88,20,98),(22,156,32,146),(24,154,34,144),(26,152,36,142),(28,150,38,160),(30,148,40,158),(41,75,51,65),(42,122,52,132),(43,73,53,63),(44,140,54,130),(45,71,55,61),(46,138,56,128),(47,69,57,79),(48,136,58,126),(49,67,59,77),(50,134,60,124),(62,112,72,102),(64,110,74,120),(66,108,76,118),(68,106,78,116),(70,104,80,114),(81,145,91,155),(83,143,93,153),(85,141,95,151),(87,159,97,149),(89,157,99,147),(101,131,111,121),(103,129,113,139),(105,127,115,137),(107,125,117,135),(109,123,119,133)]])
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 | 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 | 5 | 5 | 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 | 8 | 10 | 10 | 20 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D5 | C4○D4 | D10 | D10 | D10 | C4○D8 | Dic10 | C8⋊C22 | D4⋊2D5 | D4×D5 | D8⋊D5 | SD16⋊3D5 |
| kernel | D4.Dic10 | C10.D8 | C20.8Q8 | C40⋊6C4 | D4⋊Dic5 | C5×D4⋊C4 | C4.Dic10 | D4×Dic5 | C2×Dic5 | C5×D4 | D4⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×D4 | C10 | D4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D4.Dic10 ►in GL4(𝔽41) generated by
| 1 | 18 | 0 | 0 |
| 9 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 23 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 11 | 17 | 0 | 0 |
| 29 | 30 | 0 | 0 |
| 0 | 0 | 30 | 39 |
| 0 | 0 | 16 | 14 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 18 | 37 |
| 0 | 0 | 30 | 23 |
G:=sub<GL(4,GF(41))| [1,9,0,0,18,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,23,1,0,0,0,0,40,0,0,0,0,40],[11,29,0,0,17,30,0,0,0,0,30,16,0,0,39,14],[32,0,0,0,0,32,0,0,0,0,18,30,0,0,37,23] >;
D4.Dic10 in GAP, Magma, Sage, TeX
D_4.{\rm Dic}_{10} % in TeX
G:=Group("D4.Dic10"); // GroupNames label
G:=SmallGroup(320,390);
// by ID
G=gap.SmallGroup(320,390);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,926,219,58,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^20=1,d^2=c^10,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^-1*b,b*d=d*b,d*c*d^-1=a^2*c^-1>;
// generators/relations