metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊2D4, Dic5⋊4SD16, C4.82(D4×D5), D4⋊C4⋊9D5, C5⋊1(C4⋊SD16), D20⋊6C4⋊2C2, C4⋊C4.131D10, (C2×D4).18D10, C20.2(C4○D4), C4.1(C4○D20), C20⋊D4.5C2, (C2×C8).113D10, C20.101(C2×D4), C2.10(D5×SD16), Dic5⋊3Q8⋊3C2, C2.9(D8⋊D5), C20.8Q8⋊10C2, C10.21(C2×SD16), C22.164(D4×D5), C10.13(C4⋊D4), C10.26(C8⋊C22), (C2×C40).124C22, (C2×C20).202C23, (C2×Dic5).192D4, (C2×D20).49C22, (D4×C10).23C22, C2.16(D10⋊D4), (C4×Dic5).12C22, (C2×Dic10).55C22, (C2×D4.D5)⋊2C2, (C5×D4⋊C4)⋊9C2, (C2×C40⋊C2)⋊13C2, (C5×C4⋊C4).7C22, (C2×C10).215(C2×D4), (C2×C5⋊2C8).8C22, (C2×C4).309(C22×D5), SmallGroup(320,389)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊2D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=dad=a-1, cac-1=a9, bc=cb, dbd=a15b, dcd=c-1 >
Subgroups: 614 in 128 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C2×D4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, D4⋊C4, C4⋊C8, C4×Q8, C4⋊1D4, C2×SD16, C5⋊2C8, C40, Dic10, Dic10, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C4⋊SD16, C40⋊C2, C2×C5⋊2C8, C4×Dic5, C4×Dic5, C10.D4, D4.D5, C5×C4⋊C4, C2×C40, C2×Dic10, C2×D20, C2×C5⋊D4, D4×C10, D20⋊6C4, C20.8Q8, C5×D4⋊C4, Dic5⋊3Q8, C2×C40⋊C2, C2×D4.D5, C20⋊D4, Dic10⋊2D4
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C4⋊D4, C2×SD16, C8⋊C22, C22×D5, C4⋊SD16, C4○D20, D4×D5, D10⋊D4, D8⋊D5, D5×SD16, Dic10⋊2D4
►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 42 11 52)(2 41 12 51)(3 60 13 50)(4 59 14 49)(5 58 15 48)(6 57 16 47)(7 56 17 46)(8 55 18 45)(9 54 19 44)(10 53 20 43)(21 76 31 66)(22 75 32 65)(23 74 33 64)(24 73 34 63)(25 72 35 62)(26 71 36 61)(27 70 37 80)(28 69 38 79)(29 68 39 78)(30 67 40 77)(81 159 91 149)(82 158 92 148)(83 157 93 147)(84 156 94 146)(85 155 95 145)(86 154 96 144)(87 153 97 143)(88 152 98 142)(89 151 99 141)(90 150 100 160)(101 137 111 127)(102 136 112 126)(103 135 113 125)(104 134 114 124)(105 133 115 123)(106 132 116 122)(107 131 117 121)(108 130 118 140)(109 129 119 139)(110 128 120 138)
(1 23 114 152)(2 32 115 141)(3 21 116 150)(4 30 117 159)(5 39 118 148)(6 28 119 157)(7 37 120 146)(8 26 101 155)(9 35 102 144)(10 24 103 153)(11 33 104 142)(12 22 105 151)(13 31 106 160)(14 40 107 149)(15 29 108 158)(16 38 109 147)(17 27 110 156)(18 36 111 145)(19 25 112 154)(20 34 113 143)(41 65 123 89)(42 74 124 98)(43 63 125 87)(44 72 126 96)(45 61 127 85)(46 70 128 94)(47 79 129 83)(48 68 130 92)(49 77 131 81)(50 66 132 90)(51 75 133 99)(52 64 134 88)(53 73 135 97)(54 62 136 86)(55 71 137 95)(56 80 138 84)(57 69 139 93)(58 78 140 82)(59 67 121 91)(60 76 122 100)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 154)(22 153)(23 152)(24 151)(25 150)(26 149)(27 148)(28 147)(29 146)(30 145)(31 144)(32 143)(33 142)(34 141)(35 160)(36 159)(37 158)(38 157)(39 156)(40 155)(41 48)(42 47)(43 46)(44 45)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 96)(62 95)(63 94)(64 93)(65 92)(66 91)(67 90)(68 89)(69 88)(70 87)(71 86)(72 85)(73 84)(74 83)(75 82)(76 81)(77 100)(78 99)(79 98)(80 97)(101 107)(102 106)(103 105)(108 120)(109 119)(110 118)(111 117)(112 116)(113 115)(121 132)(122 131)(123 130)(124 129)(125 128)(126 127)(133 140)(134 139)(135 138)(136 137)
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), (1,42,11,52)(2,41,12,51)(3,60,13,50)(4,59,14,49)(5,58,15,48)(6,57,16,47)(7,56,17,46)(8,55,18,45)(9,54,19,44)(10,53,20,43)(21,76,31,66)(22,75,32,65)(23,74,33,64)(24,73,34,63)(25,72,35,62)(26,71,36,61)(27,70,37,80)(28,69,38,79)(29,68,39,78)(30,67,40,77)(81,159,91,149)(82,158,92,148)(83,157,93,147)(84,156,94,146)(85,155,95,145)(86,154,96,144)(87,153,97,143)(88,152,98,142)(89,151,99,141)(90,150,100,160)(101,137,111,127)(102,136,112,126)(103,135,113,125)(104,134,114,124)(105,133,115,123)(106,132,116,122)(107,131,117,121)(108,130,118,140)(109,129,119,139)(110,128,120,138), (1,23,114,152)(2,32,115,141)(3,21,116,150)(4,30,117,159)(5,39,118,148)(6,28,119,157)(7,37,120,146)(8,26,101,155)(9,35,102,144)(10,24,103,153)(11,33,104,142)(12,22,105,151)(13,31,106,160)(14,40,107,149)(15,29,108,158)(16,38,109,147)(17,27,110,156)(18,36,111,145)(19,25,112,154)(20,34,113,143)(41,65,123,89)(42,74,124,98)(43,63,125,87)(44,72,126,96)(45,61,127,85)(46,70,128,94)(47,79,129,83)(48,68,130,92)(49,77,131,81)(50,66,132,90)(51,75,133,99)(52,64,134,88)(53,73,135,97)(54,62,136,86)(55,71,137,95)(56,80,138,84)(57,69,139,93)(58,78,140,82)(59,67,121,91)(60,76,122,100), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,154)(22,153)(23,152)(24,151)(25,150)(26,149)(27,148)(28,147)(29,146)(30,145)(31,144)(32,143)(33,142)(34,141)(35,160)(36,159)(37,158)(38,157)(39,156)(40,155)(41,48)(42,47)(43,46)(44,45)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,100)(78,99)(79,98)(80,97)(101,107)(102,106)(103,105)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(133,140)(134,139)(135,138)(136,137)>;
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), (1,42,11,52)(2,41,12,51)(3,60,13,50)(4,59,14,49)(5,58,15,48)(6,57,16,47)(7,56,17,46)(8,55,18,45)(9,54,19,44)(10,53,20,43)(21,76,31,66)(22,75,32,65)(23,74,33,64)(24,73,34,63)(25,72,35,62)(26,71,36,61)(27,70,37,80)(28,69,38,79)(29,68,39,78)(30,67,40,77)(81,159,91,149)(82,158,92,148)(83,157,93,147)(84,156,94,146)(85,155,95,145)(86,154,96,144)(87,153,97,143)(88,152,98,142)(89,151,99,141)(90,150,100,160)(101,137,111,127)(102,136,112,126)(103,135,113,125)(104,134,114,124)(105,133,115,123)(106,132,116,122)(107,131,117,121)(108,130,118,140)(109,129,119,139)(110,128,120,138), (1,23,114,152)(2,32,115,141)(3,21,116,150)(4,30,117,159)(5,39,118,148)(6,28,119,157)(7,37,120,146)(8,26,101,155)(9,35,102,144)(10,24,103,153)(11,33,104,142)(12,22,105,151)(13,31,106,160)(14,40,107,149)(15,29,108,158)(16,38,109,147)(17,27,110,156)(18,36,111,145)(19,25,112,154)(20,34,113,143)(41,65,123,89)(42,74,124,98)(43,63,125,87)(44,72,126,96)(45,61,127,85)(46,70,128,94)(47,79,129,83)(48,68,130,92)(49,77,131,81)(50,66,132,90)(51,75,133,99)(52,64,134,88)(53,73,135,97)(54,62,136,86)(55,71,137,95)(56,80,138,84)(57,69,139,93)(58,78,140,82)(59,67,121,91)(60,76,122,100), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,154)(22,153)(23,152)(24,151)(25,150)(26,149)(27,148)(28,147)(29,146)(30,145)(31,144)(32,143)(33,142)(34,141)(35,160)(36,159)(37,158)(38,157)(39,156)(40,155)(41,48)(42,47)(43,46)(44,45)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,100)(78,99)(79,98)(80,97)(101,107)(102,106)(103,105)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(133,140)(134,139)(135,138)(136,137) );
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)], [(1,42,11,52),(2,41,12,51),(3,60,13,50),(4,59,14,49),(5,58,15,48),(6,57,16,47),(7,56,17,46),(8,55,18,45),(9,54,19,44),(10,53,20,43),(21,76,31,66),(22,75,32,65),(23,74,33,64),(24,73,34,63),(25,72,35,62),(26,71,36,61),(27,70,37,80),(28,69,38,79),(29,68,39,78),(30,67,40,77),(81,159,91,149),(82,158,92,148),(83,157,93,147),(84,156,94,146),(85,155,95,145),(86,154,96,144),(87,153,97,143),(88,152,98,142),(89,151,99,141),(90,150,100,160),(101,137,111,127),(102,136,112,126),(103,135,113,125),(104,134,114,124),(105,133,115,123),(106,132,116,122),(107,131,117,121),(108,130,118,140),(109,129,119,139),(110,128,120,138)], [(1,23,114,152),(2,32,115,141),(3,21,116,150),(4,30,117,159),(5,39,118,148),(6,28,119,157),(7,37,120,146),(8,26,101,155),(9,35,102,144),(10,24,103,153),(11,33,104,142),(12,22,105,151),(13,31,106,160),(14,40,107,149),(15,29,108,158),(16,38,109,147),(17,27,110,156),(18,36,111,145),(19,25,112,154),(20,34,113,143),(41,65,123,89),(42,74,124,98),(43,63,125,87),(44,72,126,96),(45,61,127,85),(46,70,128,94),(47,79,129,83),(48,68,130,92),(49,77,131,81),(50,66,132,90),(51,75,133,99),(52,64,134,88),(53,73,135,97),(54,62,136,86),(55,71,137,95),(56,80,138,84),(57,69,139,93),(58,78,140,82),(59,67,121,91),(60,76,122,100)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,154),(22,153),(23,152),(24,151),(25,150),(26,149),(27,148),(28,147),(29,146),(30,145),(31,144),(32,143),(33,142),(34,141),(35,160),(36,159),(37,158),(38,157),(39,156),(40,155),(41,48),(42,47),(43,46),(44,45),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(61,96),(62,95),(63,94),(64,93),(65,92),(66,91),(67,90),(68,89),(69,88),(70,87),(71,86),(72,85),(73,84),(74,83),(75,82),(76,81),(77,100),(78,99),(79,98),(80,97),(101,107),(102,106),(103,105),(108,120),(109,119),(110,118),(111,117),(112,116),(113,115),(121,132),(122,131),(123,130),(124,129),(125,128),(126,127),(133,140),(134,139),(135,138),(136,137)]])
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 | 8 | 40 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 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 | D4 | D5 | SD16 | C4○D4 | D10 | D10 | D10 | C4○D20 | C8⋊C22 | D4×D5 | D4×D5 | D8⋊D5 | D5×SD16 |
| kernel | Dic10⋊2D4 | D20⋊6C4 | C20.8Q8 | C5×D4⋊C4 | Dic5⋊3Q8 | C2×C40⋊C2 | C2×D4.D5 | C20⋊D4 | Dic10 | C2×Dic5 | D4⋊C4 | Dic5 | C20 | C4⋊C4 | C2×C8 | C2×D4 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of Dic10⋊2D4 ►in GL4(𝔽41) generated by
| 35 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 39 |
| 0 | 0 | 1 | 1 |
| 25 | 25 | 0 | 0 |
| 39 | 16 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 26 | 0 |
| 21 | 21 | 0 | 0 |
| 18 | 20 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 6 | 35 | 0 | 0 |
| 40 | 35 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 40 | 40 |
G:=sub<GL(4,GF(41))| [35,1,0,0,40,0,0,0,0,0,40,1,0,0,39,1],[25,39,0,0,25,16,0,0,0,0,0,26,0,0,11,0],[21,18,0,0,21,20,0,0,0,0,40,0,0,0,0,40],[6,40,0,0,35,35,0,0,0,0,1,40,0,0,0,40] >;
Dic10⋊2D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_2D_4 % in TeX
G:=Group("Dic10:2D4"); // GroupNames label
G:=SmallGroup(320,389);
// by ID
G=gap.SmallGroup(320,389);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,135,268,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a^9,b*c=c*b,d*b*d=a^15*b,d*c*d=c^-1>;
// generators/relations