metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊2SD16, Dic10⋊8D4, C42.39D10, C4⋊C8⋊10D5, C4⋊2(C40⋊C2), C5⋊2(C4⋊SD16), C4.133(D4×D5), C20⋊4D4.6C2, (C2×C8).133D10, (C2×C20).124D4, (C2×C4).135D20, C20.342(C2×D4), D20⋊5C4⋊13C2, (C4×Dic10)⋊18C2, (C4×C20).74C22, C10.12(C2×SD16), C20.331(C4○D4), C10.41(C4⋊D4), C2.20(C8⋊D10), C2.14(C4⋊D20), C10.17(C8⋊C22), (C2×C40).140C22, (C2×C20).758C23, C4.47(Q8⋊2D5), (C2×D20).19C22, C22.121(C2×D20), C4⋊Dic5.276C22, (C2×Dic10).221C22, (C5×C4⋊C8)⋊12C2, (C2×C40⋊C2)⋊20C2, C2.15(C2×C40⋊C2), (C2×C10).141(C2×D4), (C2×C4).703(C22×D5), SmallGroup(320,475)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊8D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a5b, dcd=c-1 >
Subgroups: 710 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, D5, C10, C42, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×Q8, Dic5, C20, C20, C20, D10, C2×C10, D4⋊C4, C4⋊C8, C4×Q8, C4⋊1D4, C2×SD16, C40, Dic10, Dic10, D20, C2×Dic5, C2×C20, C22×D5, C4⋊SD16, C40⋊C2, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C2×C40, C2×Dic10, C2×D20, C2×D20, D20⋊5C4, C5×C4⋊C8, C4×Dic10, C20⋊4D4, C2×C40⋊C2, Dic10⋊8D4
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C4⋊D4, C2×SD16, C8⋊C22, D20, C22×D5, C4⋊SD16, C40⋊C2, C2×D20, D4×D5, Q8⋊2D5, C4⋊D20, C2×C40⋊C2, C8⋊D10, Dic10⋊8D4
►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 50 11 60)(2 49 12 59)(3 48 13 58)(4 47 14 57)(5 46 15 56)(6 45 16 55)(7 44 17 54)(8 43 18 53)(9 42 19 52)(10 41 20 51)(21 74 31 64)(22 73 32 63)(23 72 33 62)(24 71 34 61)(25 70 35 80)(26 69 36 79)(27 68 37 78)(28 67 38 77)(29 66 39 76)(30 65 40 75)(81 132 91 122)(82 131 92 121)(83 130 93 140)(84 129 94 139)(85 128 95 138)(86 127 96 137)(87 126 97 136)(88 125 98 135)(89 124 99 134)(90 123 100 133)(101 151 111 141)(102 150 112 160)(103 149 113 159)(104 148 114 158)(105 147 115 157)(106 146 116 156)(107 145 117 155)(108 144 118 154)(109 143 119 153)(110 142 120 152)
(1 111 74 99)(2 112 75 100)(3 113 76 81)(4 114 77 82)(5 115 78 83)(6 116 79 84)(7 117 80 85)(8 118 61 86)(9 119 62 87)(10 120 63 88)(11 101 64 89)(12 102 65 90)(13 103 66 91)(14 104 67 92)(15 105 68 93)(16 106 69 94)(17 107 70 95)(18 108 71 96)(19 109 72 97)(20 110 73 98)(21 124 60 151)(22 125 41 152)(23 126 42 153)(24 127 43 154)(25 128 44 155)(26 129 45 156)(27 130 46 157)(28 131 47 158)(29 132 48 159)(30 133 49 160)(31 134 50 141)(32 135 51 142)(33 136 52 143)(34 137 53 144)(35 138 54 145)(36 139 55 146)(37 140 56 147)(38 121 57 148)(39 122 58 149)(40 123 59 150)
(1 111)(2 110)(3 109)(4 108)(5 107)(6 106)(7 105)(8 104)(9 103)(10 102)(11 101)(12 120)(13 119)(14 118)(15 117)(16 116)(17 115)(18 114)(19 113)(20 112)(21 139)(22 138)(23 137)(24 136)(25 135)(26 134)(27 133)(28 132)(29 131)(30 130)(31 129)(32 128)(33 127)(34 126)(35 125)(36 124)(37 123)(38 122)(39 121)(40 140)(41 145)(42 144)(43 143)(44 142)(45 141)(46 160)(47 159)(48 158)(49 157)(50 156)(51 155)(52 154)(53 153)(54 152)(55 151)(56 150)(57 149)(58 148)(59 147)(60 146)(61 92)(62 91)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 100)(74 99)(75 98)(76 97)(77 96)(78 95)(79 94)(80 93)
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), (1,50,11,60)(2,49,12,59)(3,48,13,58)(4,47,14,57)(5,46,15,56)(6,45,16,55)(7,44,17,54)(8,43,18,53)(9,42,19,52)(10,41,20,51)(21,74,31,64)(22,73,32,63)(23,72,33,62)(24,71,34,61)(25,70,35,80)(26,69,36,79)(27,68,37,78)(28,67,38,77)(29,66,39,76)(30,65,40,75)(81,132,91,122)(82,131,92,121)(83,130,93,140)(84,129,94,139)(85,128,95,138)(86,127,96,137)(87,126,97,136)(88,125,98,135)(89,124,99,134)(90,123,100,133)(101,151,111,141)(102,150,112,160)(103,149,113,159)(104,148,114,158)(105,147,115,157)(106,146,116,156)(107,145,117,155)(108,144,118,154)(109,143,119,153)(110,142,120,152), (1,111,74,99)(2,112,75,100)(3,113,76,81)(4,114,77,82)(5,115,78,83)(6,116,79,84)(7,117,80,85)(8,118,61,86)(9,119,62,87)(10,120,63,88)(11,101,64,89)(12,102,65,90)(13,103,66,91)(14,104,67,92)(15,105,68,93)(16,106,69,94)(17,107,70,95)(18,108,71,96)(19,109,72,97)(20,110,73,98)(21,124,60,151)(22,125,41,152)(23,126,42,153)(24,127,43,154)(25,128,44,155)(26,129,45,156)(27,130,46,157)(28,131,47,158)(29,132,48,159)(30,133,49,160)(31,134,50,141)(32,135,51,142)(33,136,52,143)(34,137,53,144)(35,138,54,145)(36,139,55,146)(37,140,56,147)(38,121,57,148)(39,122,58,149)(40,123,59,150), (1,111)(2,110)(3,109)(4,108)(5,107)(6,106)(7,105)(8,104)(9,103)(10,102)(11,101)(12,120)(13,119)(14,118)(15,117)(16,116)(17,115)(18,114)(19,113)(20,112)(21,139)(22,138)(23,137)(24,136)(25,135)(26,134)(27,133)(28,132)(29,131)(30,130)(31,129)(32,128)(33,127)(34,126)(35,125)(36,124)(37,123)(38,122)(39,121)(40,140)(41,145)(42,144)(43,143)(44,142)(45,141)(46,160)(47,159)(48,158)(49,157)(50,156)(51,155)(52,154)(53,153)(54,152)(55,151)(56,150)(57,149)(58,148)(59,147)(60,146)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)>;
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), (1,50,11,60)(2,49,12,59)(3,48,13,58)(4,47,14,57)(5,46,15,56)(6,45,16,55)(7,44,17,54)(8,43,18,53)(9,42,19,52)(10,41,20,51)(21,74,31,64)(22,73,32,63)(23,72,33,62)(24,71,34,61)(25,70,35,80)(26,69,36,79)(27,68,37,78)(28,67,38,77)(29,66,39,76)(30,65,40,75)(81,132,91,122)(82,131,92,121)(83,130,93,140)(84,129,94,139)(85,128,95,138)(86,127,96,137)(87,126,97,136)(88,125,98,135)(89,124,99,134)(90,123,100,133)(101,151,111,141)(102,150,112,160)(103,149,113,159)(104,148,114,158)(105,147,115,157)(106,146,116,156)(107,145,117,155)(108,144,118,154)(109,143,119,153)(110,142,120,152), (1,111,74,99)(2,112,75,100)(3,113,76,81)(4,114,77,82)(5,115,78,83)(6,116,79,84)(7,117,80,85)(8,118,61,86)(9,119,62,87)(10,120,63,88)(11,101,64,89)(12,102,65,90)(13,103,66,91)(14,104,67,92)(15,105,68,93)(16,106,69,94)(17,107,70,95)(18,108,71,96)(19,109,72,97)(20,110,73,98)(21,124,60,151)(22,125,41,152)(23,126,42,153)(24,127,43,154)(25,128,44,155)(26,129,45,156)(27,130,46,157)(28,131,47,158)(29,132,48,159)(30,133,49,160)(31,134,50,141)(32,135,51,142)(33,136,52,143)(34,137,53,144)(35,138,54,145)(36,139,55,146)(37,140,56,147)(38,121,57,148)(39,122,58,149)(40,123,59,150), (1,111)(2,110)(3,109)(4,108)(5,107)(6,106)(7,105)(8,104)(9,103)(10,102)(11,101)(12,120)(13,119)(14,118)(15,117)(16,116)(17,115)(18,114)(19,113)(20,112)(21,139)(22,138)(23,137)(24,136)(25,135)(26,134)(27,133)(28,132)(29,131)(30,130)(31,129)(32,128)(33,127)(34,126)(35,125)(36,124)(37,123)(38,122)(39,121)(40,140)(41,145)(42,144)(43,143)(44,142)(45,141)(46,160)(47,159)(48,158)(49,157)(50,156)(51,155)(52,154)(53,153)(54,152)(55,151)(56,150)(57,149)(58,148)(59,147)(60,146)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93) );
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)], [(1,50,11,60),(2,49,12,59),(3,48,13,58),(4,47,14,57),(5,46,15,56),(6,45,16,55),(7,44,17,54),(8,43,18,53),(9,42,19,52),(10,41,20,51),(21,74,31,64),(22,73,32,63),(23,72,33,62),(24,71,34,61),(25,70,35,80),(26,69,36,79),(27,68,37,78),(28,67,38,77),(29,66,39,76),(30,65,40,75),(81,132,91,122),(82,131,92,121),(83,130,93,140),(84,129,94,139),(85,128,95,138),(86,127,96,137),(87,126,97,136),(88,125,98,135),(89,124,99,134),(90,123,100,133),(101,151,111,141),(102,150,112,160),(103,149,113,159),(104,148,114,158),(105,147,115,157),(106,146,116,156),(107,145,117,155),(108,144,118,154),(109,143,119,153),(110,142,120,152)], [(1,111,74,99),(2,112,75,100),(3,113,76,81),(4,114,77,82),(5,115,78,83),(6,116,79,84),(7,117,80,85),(8,118,61,86),(9,119,62,87),(10,120,63,88),(11,101,64,89),(12,102,65,90),(13,103,66,91),(14,104,67,92),(15,105,68,93),(16,106,69,94),(17,107,70,95),(18,108,71,96),(19,109,72,97),(20,110,73,98),(21,124,60,151),(22,125,41,152),(23,126,42,153),(24,127,43,154),(25,128,44,155),(26,129,45,156),(27,130,46,157),(28,131,47,158),(29,132,48,159),(30,133,49,160),(31,134,50,141),(32,135,51,142),(33,136,52,143),(34,137,53,144),(35,138,54,145),(36,139,55,146),(37,140,56,147),(38,121,57,148),(39,122,58,149),(40,123,59,150)], [(1,111),(2,110),(3,109),(4,108),(5,107),(6,106),(7,105),(8,104),(9,103),(10,102),(11,101),(12,120),(13,119),(14,118),(15,117),(16,116),(17,115),(18,114),(19,113),(20,112),(21,139),(22,138),(23,137),(24,136),(25,135),(26,134),(27,133),(28,132),(29,131),(30,130),(31,129),(32,128),(33,127),(34,126),(35,125),(36,124),(37,123),(38,122),(39,121),(40,140),(41,145),(42,144),(43,143),(44,142),(45,141),(46,160),(47,159),(48,158),(49,157),(50,156),(51,155),(52,154),(53,153),(54,152),(55,151),(56,150),(57,149),(58,148),(59,147),(60,146),(61,92),(62,91),(63,90),(64,89),(65,88),(66,87),(67,86),(68,85),(69,84),(70,83),(71,82),(72,81),(73,100),(74,99),(75,98),(76,97),(77,96),(78,95),(79,94),(80,93)]])
59 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 | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 40 | 40 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 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 | D4 | D4 | D5 | SD16 | C4○D4 | D10 | D10 | D20 | C40⋊C2 | C8⋊C22 | D4×D5 | Q8⋊2D5 | C8⋊D10 |
| kernel | Dic10⋊8D4 | D20⋊5C4 | C5×C4⋊C8 | C4×Dic10 | C20⋊4D4 | C2×C40⋊C2 | Dic10 | C2×C20 | C4⋊C8 | C20 | C20 | C42 | C2×C8 | C2×C4 | C4 | C10 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 16 | 1 | 2 | 2 | 4 |
Matrix representation of Dic10⋊8D4 ►in GL6(𝔽41)
| 1 | 9 | 0 | 0 | 0 | 0 |
| 18 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 13 | 0 | 0 |
| 0 | 0 | 25 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 4 |
| 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 | 32 | 5 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 23 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 36 |
| 0 | 0 | 0 | 0 | 16 | 32 |
G:=sub<GL(6,GF(41))| [1,18,0,0,0,0,9,40,0,0,0,0,0,0,0,1,0,0,0,0,40,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,17,0,0,0,0,12,0,0,0,0,0,0,0,2,25,0,0,0,0,13,39,0,0,0,0,0,0,1,0,0,0,0,0,4,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,5,9],[40,23,0,0,0,0,0,1,0,0,0,0,0,0,40,35,0,0,0,0,0,1,0,0,0,0,0,0,9,16,0,0,0,0,36,32] >;
Dic10⋊8D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_8D_4 % in TeX
G:=Group("Dic10:8D4"); // GroupNames label
G:=SmallGroup(320,475);
// by ID
G=gap.SmallGroup(320,475);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,254,219,58,1123,136,12550]);
// 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,a*c=c*a,b*c=c*b,d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations