metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊20D4, C10.332+ 1+4, C4⋊D4⋊7D5, C5⋊3(Q8⋊6D4), C4.109(D4×D5), C20⋊D4⋊15C2, C4⋊D20⋊19C2, C4⋊C4.177D10, (C2×D4).90D10, C20.225(C2×D4), D10⋊D4⋊17C2, Dic5⋊8(C4○D4), Dic5⋊4D4⋊6C2, (C2×C20).35C23, C22⋊C4.46D10, Dic5.45(C2×D4), C10.62(C22×D4), Dic5⋊3Q8⋊20C2, Dic5⋊D4⋊10C2, (C2×C10).143C24, (C22×C4).219D10, C2.35(D4⋊6D10), C23.10(C22×D5), (C2×D20).148C22, (D4×C10).117C22, (C22×C10).14C23, (C4×Dic5).98C22, (C22×D5).62C23, C22.164(C23×D5), D10⋊C4.12C22, (C22×C20).237C22, (C2×Dic5).235C23, C10.D4.14C22, C23.D5.110C22, (C2×Dic10).301C22, (C22×Dic5).104C22, C2.35(C2×D4×D5), (C5×C4⋊D4)⋊8C2, (C4×C5⋊D4)⋊15C2, C2.34(D5×C4○D4), (C2×C4○D20)⋊19C2, (C2×D4⋊2D5)⋊11C2, (C2×C4×D5).91C22, C10.148(C2×C4○D4), (C5×C4⋊C4).139C22, (C2×C4).585(C22×D5), (C2×C5⋊D4).25C22, (C5×C22⋊C4).8C22, SmallGroup(320,1271)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊20D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, cac-1=a11, ad=da, cbc-1=dbd=a10b, dcd=c-1 >
Subgroups: 1222 in 312 conjugacy classes, 105 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C4⋊1D4, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22×C10, Q8⋊6D4, C4×Dic5, C4×Dic5, C10.D4, C10.D4, D10⋊C4, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C4○D20, D4⋊2D5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, Dic5⋊4D4, D10⋊D4, Dic5⋊3Q8, C4⋊D20, C4×C5⋊D4, Dic5⋊D4, C20⋊D4, C20⋊D4, C5×C4⋊D4, C2×C4○D20, C2×D4⋊2D5, Dic10⋊20D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, 2+ 1+4, C22×D5, Q8⋊6D4, D4×D5, C23×D5, C2×D4×D5, D4⋊6D10, D5×C4○D4, Dic10⋊20D4
►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 80 11 70)(2 79 12 69)(3 78 13 68)(4 77 14 67)(5 76 15 66)(6 75 16 65)(7 74 17 64)(8 73 18 63)(9 72 19 62)(10 71 20 61)(21 137 31 127)(22 136 32 126)(23 135 33 125)(24 134 34 124)(25 133 35 123)(26 132 36 122)(27 131 37 121)(28 130 38 140)(29 129 39 139)(30 128 40 138)(41 152 51 142)(42 151 52 141)(43 150 53 160)(44 149 54 159)(45 148 55 158)(46 147 56 157)(47 146 57 156)(48 145 58 155)(49 144 59 154)(50 143 60 153)(81 109 91 119)(82 108 92 118)(83 107 93 117)(84 106 94 116)(85 105 95 115)(86 104 96 114)(87 103 97 113)(88 102 98 112)(89 101 99 111)(90 120 100 110)
(1 30 159 114)(2 21 160 105)(3 32 141 116)(4 23 142 107)(5 34 143 118)(6 25 144 109)(7 36 145 120)(8 27 146 111)(9 38 147 102)(10 29 148 113)(11 40 149 104)(12 31 150 115)(13 22 151 106)(14 33 152 117)(15 24 153 108)(16 35 154 119)(17 26 155 110)(18 37 156 101)(19 28 157 112)(20 39 158 103)(41 83 77 125)(42 94 78 136)(43 85 79 127)(44 96 80 138)(45 87 61 129)(46 98 62 140)(47 89 63 131)(48 100 64 122)(49 91 65 133)(50 82 66 124)(51 93 67 135)(52 84 68 126)(53 95 69 137)(54 86 70 128)(55 97 71 139)(56 88 72 130)(57 99 73 121)(58 90 74 132)(59 81 75 123)(60 92 76 134)
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 73)(20 74)(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(27 98)(28 99)(29 100)(30 81)(31 82)(32 83)(33 84)(34 85)(35 86)(36 87)(37 88)(38 89)(39 90)(40 91)(41 141)(42 142)(43 143)(44 144)(45 145)(46 146)(47 147)(48 148)(49 149)(50 150)(51 151)(52 152)(53 153)(54 154)(55 155)(56 156)(57 157)(58 158)(59 159)(60 160)(101 130)(102 131)(103 132)(104 133)(105 134)(106 135)(107 136)(108 137)(109 138)(110 139)(111 140)(112 121)(113 122)(114 123)(115 124)(116 125)(117 126)(118 127)(119 128)(120 129)
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), (1,80,11,70)(2,79,12,69)(3,78,13,68)(4,77,14,67)(5,76,15,66)(6,75,16,65)(7,74,17,64)(8,73,18,63)(9,72,19,62)(10,71,20,61)(21,137,31,127)(22,136,32,126)(23,135,33,125)(24,134,34,124)(25,133,35,123)(26,132,36,122)(27,131,37,121)(28,130,38,140)(29,129,39,139)(30,128,40,138)(41,152,51,142)(42,151,52,141)(43,150,53,160)(44,149,54,159)(45,148,55,158)(46,147,56,157)(47,146,57,156)(48,145,58,155)(49,144,59,154)(50,143,60,153)(81,109,91,119)(82,108,92,118)(83,107,93,117)(84,106,94,116)(85,105,95,115)(86,104,96,114)(87,103,97,113)(88,102,98,112)(89,101,99,111)(90,120,100,110), (1,30,159,114)(2,21,160,105)(3,32,141,116)(4,23,142,107)(5,34,143,118)(6,25,144,109)(7,36,145,120)(8,27,146,111)(9,38,147,102)(10,29,148,113)(11,40,149,104)(12,31,150,115)(13,22,151,106)(14,33,152,117)(15,24,153,108)(16,35,154,119)(17,26,155,110)(18,37,156,101)(19,28,157,112)(20,39,158,103)(41,83,77,125)(42,94,78,136)(43,85,79,127)(44,96,80,138)(45,87,61,129)(46,98,62,140)(47,89,63,131)(48,100,64,122)(49,91,65,133)(50,82,66,124)(51,93,67,135)(52,84,68,126)(53,95,69,137)(54,86,70,128)(55,97,71,139)(56,88,72,130)(57,99,73,121)(58,90,74,132)(59,81,75,123)(60,92,76,134), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,81)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,141)(42,142)(43,143)(44,144)(45,145)(46,146)(47,147)(48,148)(49,149)(50,150)(51,151)(52,152)(53,153)(54,154)(55,155)(56,156)(57,157)(58,158)(59,159)(60,160)(101,130)(102,131)(103,132)(104,133)(105,134)(106,135)(107,136)(108,137)(109,138)(110,139)(111,140)(112,121)(113,122)(114,123)(115,124)(116,125)(117,126)(118,127)(119,128)(120,129)>;
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), (1,80,11,70)(2,79,12,69)(3,78,13,68)(4,77,14,67)(5,76,15,66)(6,75,16,65)(7,74,17,64)(8,73,18,63)(9,72,19,62)(10,71,20,61)(21,137,31,127)(22,136,32,126)(23,135,33,125)(24,134,34,124)(25,133,35,123)(26,132,36,122)(27,131,37,121)(28,130,38,140)(29,129,39,139)(30,128,40,138)(41,152,51,142)(42,151,52,141)(43,150,53,160)(44,149,54,159)(45,148,55,158)(46,147,56,157)(47,146,57,156)(48,145,58,155)(49,144,59,154)(50,143,60,153)(81,109,91,119)(82,108,92,118)(83,107,93,117)(84,106,94,116)(85,105,95,115)(86,104,96,114)(87,103,97,113)(88,102,98,112)(89,101,99,111)(90,120,100,110), (1,30,159,114)(2,21,160,105)(3,32,141,116)(4,23,142,107)(5,34,143,118)(6,25,144,109)(7,36,145,120)(8,27,146,111)(9,38,147,102)(10,29,148,113)(11,40,149,104)(12,31,150,115)(13,22,151,106)(14,33,152,117)(15,24,153,108)(16,35,154,119)(17,26,155,110)(18,37,156,101)(19,28,157,112)(20,39,158,103)(41,83,77,125)(42,94,78,136)(43,85,79,127)(44,96,80,138)(45,87,61,129)(46,98,62,140)(47,89,63,131)(48,100,64,122)(49,91,65,133)(50,82,66,124)(51,93,67,135)(52,84,68,126)(53,95,69,137)(54,86,70,128)(55,97,71,139)(56,88,72,130)(57,99,73,121)(58,90,74,132)(59,81,75,123)(60,92,76,134), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,81)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,141)(42,142)(43,143)(44,144)(45,145)(46,146)(47,147)(48,148)(49,149)(50,150)(51,151)(52,152)(53,153)(54,154)(55,155)(56,156)(57,157)(58,158)(59,159)(60,160)(101,130)(102,131)(103,132)(104,133)(105,134)(106,135)(107,136)(108,137)(109,138)(110,139)(111,140)(112,121)(113,122)(114,123)(115,124)(116,125)(117,126)(118,127)(119,128)(120,129) );
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)], [(1,80,11,70),(2,79,12,69),(3,78,13,68),(4,77,14,67),(5,76,15,66),(6,75,16,65),(7,74,17,64),(8,73,18,63),(9,72,19,62),(10,71,20,61),(21,137,31,127),(22,136,32,126),(23,135,33,125),(24,134,34,124),(25,133,35,123),(26,132,36,122),(27,131,37,121),(28,130,38,140),(29,129,39,139),(30,128,40,138),(41,152,51,142),(42,151,52,141),(43,150,53,160),(44,149,54,159),(45,148,55,158),(46,147,56,157),(47,146,57,156),(48,145,58,155),(49,144,59,154),(50,143,60,153),(81,109,91,119),(82,108,92,118),(83,107,93,117),(84,106,94,116),(85,105,95,115),(86,104,96,114),(87,103,97,113),(88,102,98,112),(89,101,99,111),(90,120,100,110)], [(1,30,159,114),(2,21,160,105),(3,32,141,116),(4,23,142,107),(5,34,143,118),(6,25,144,109),(7,36,145,120),(8,27,146,111),(9,38,147,102),(10,29,148,113),(11,40,149,104),(12,31,150,115),(13,22,151,106),(14,33,152,117),(15,24,153,108),(16,35,154,119),(17,26,155,110),(18,37,156,101),(19,28,157,112),(20,39,158,103),(41,83,77,125),(42,94,78,136),(43,85,79,127),(44,96,80,138),(45,87,61,129),(46,98,62,140),(47,89,63,131),(48,100,64,122),(49,91,65,133),(50,82,66,124),(51,93,67,135),(52,84,68,126),(53,95,69,137),(54,86,70,128),(55,97,71,139),(56,88,72,130),(57,99,73,121),(58,90,74,132),(59,81,75,123),(60,92,76,134)], [(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,61),(8,62),(9,63),(10,64),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,73),(20,74),(21,92),(22,93),(23,94),(24,95),(25,96),(26,97),(27,98),(28,99),(29,100),(30,81),(31,82),(32,83),(33,84),(34,85),(35,86),(36,87),(37,88),(38,89),(39,90),(40,91),(41,141),(42,142),(43,143),(44,144),(45,145),(46,146),(47,147),(48,148),(49,149),(50,150),(51,151),(52,152),(53,153),(54,154),(55,155),(56,156),(57,157),(58,158),(59,159),(60,160),(101,130),(102,131),(103,132),(104,133),(105,134),(106,135),(107,136),(108,137),(109,138),(110,139),(111,140),(112,121),(113,122),(114,123),(115,124),(116,125),(117,126),(118,127),(119,128),(120,129)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 20A | ··· | 20H | 20I | 20J | 20K | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | ··· | 10 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ 1+4 | D4×D5 | D4⋊6D10 | D5×C4○D4 |
| kernel | Dic10⋊20D4 | Dic5⋊4D4 | D10⋊D4 | Dic5⋊3Q8 | C4⋊D20 | C4×C5⋊D4 | Dic5⋊D4 | C20⋊D4 | C5×C4⋊D4 | C2×C4○D20 | C2×D4⋊2D5 | Dic10 | C4⋊D4 | Dic5 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 3 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 2 | 2 | 6 | 1 | 4 | 4 | 4 |
Matrix representation of Dic10⋊20D4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 2 | 0 | 0 |
| 0 | 0 | 24 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 7 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 34 |
| 0 | 0 | 0 | 0 | 1 | 34 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 2 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 23 | 0 | 0 |
| 0 | 0 | 21 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 2 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 23 | 0 | 0 |
| 0 | 0 | 30 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,22,24,0,0,0,0,2,19,0,0,0,0,0,0,7,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,7,0,0,0,0,0,32,0,0,0,0,0,0,7,1,0,0,0,0,34,34],[1,2,0,0,0,0,40,40,0,0,0,0,0,0,7,21,0,0,0,0,23,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,2,0,0,0,0,0,40,0,0,0,0,0,0,7,30,0,0,0,0,23,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
Dic10⋊20D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_{20}D_4 % in TeX
G:=Group("Dic10:20D4"); // GroupNames label
G:=SmallGroup(320,1271);
// by ID
G=gap.SmallGroup(320,1271);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,477,232,184,570,185,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations