metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊19D4, C10.692- 1+4, C4⋊D4⋊6D5, C5⋊3(Q8⋊5D4), C4.108(D4×D5), C4⋊C4.176D10, (D4×Dic5)⋊15C2, C20.224(C2×D4), C22⋊C4.5D10, D10⋊2Q8⋊19C2, Dic5⋊D4⋊9C2, (C2×D4).151D10, Dic5.44(C2×D4), C10.61(C22×D4), Dic5⋊3Q8⋊19C2, C20.17D4⋊14C2, C22⋊1(D4⋊2D5), C23.9(C22×D5), (C2×C20).500C23, (C2×C10).142C24, (C22×C4).218D10, Dic5.5D4⋊17C2, (C22×Dic10)⋊16C2, (D4×C10).116C22, C4⋊Dic5.204C22, (C22×C10).13C23, (C4×Dic5).97C22, (C2×Dic5).65C23, (C22×D5).61C23, C22.163(C23×D5), Dic5.14D4⋊16C2, C23.D5.20C22, D10⋊C4.11C22, (C22×C20).236C22, C10.D4.13C22, C2.27(D4.10D10), (C2×Dic10).253C22, (C22×Dic5).103C22, C2.34(C2×D4×D5), (C5×C4⋊D4)⋊7C2, (C4×C5⋊D4)⋊14C2, (C2×C10)⋊4(C4○D4), C10.80(C2×C4○D4), (C2×D4⋊2D5)⋊10C2, (C2×C4×D5).90C22, C2.31(C2×D4⋊2D5), (C5×C4⋊C4).138C22, (C2×C4).173(C22×D5), (C5×C22⋊C4).7C22, (C2×C5⋊D4).126C22, SmallGroup(320,1270)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊19D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, cac-1=a11, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 982 in 290 conjugacy classes, 107 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, 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, C2×C10, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C4.4D4, C22×Q8, C2×C4○D4, Dic10, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, Q8⋊5D4, C4×Dic5, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C2×Dic10, C2×C4×D5, D4⋊2D5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, Dic5.14D4, Dic5.5D4, Dic5⋊3Q8, D10⋊2Q8, C4×C5⋊D4, D4×Dic5, C20.17D4, Dic5⋊D4, C5×C4⋊D4, C22×Dic10, C2×D4⋊2D5, Dic10⋊19D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, 2- 1+4, C22×D5, Q8⋊5D4, D4×D5, D4⋊2D5, C23×D5, C2×D4×D5, C2×D4⋊2D5, D4.10D10, Dic10⋊19D4
►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)
(1 129 11 139)(2 128 12 138)(3 127 13 137)(4 126 14 136)(5 125 15 135)(6 124 16 134)(7 123 17 133)(8 122 18 132)(9 121 19 131)(10 140 20 130)(21 78 31 68)(22 77 32 67)(23 76 33 66)(24 75 34 65)(25 74 35 64)(26 73 36 63)(27 72 37 62)(28 71 38 61)(29 70 39 80)(30 69 40 79)(41 90 51 100)(42 89 52 99)(43 88 53 98)(44 87 54 97)(45 86 55 96)(46 85 56 95)(47 84 57 94)(48 83 58 93)(49 82 59 92)(50 81 60 91)(101 148 111 158)(102 147 112 157)(103 146 113 156)(104 145 114 155)(105 144 115 154)(106 143 116 153)(107 142 117 152)(108 141 118 151)(109 160 119 150)(110 159 120 149)
(1 156 90 28)(2 147 91 39)(3 158 92 30)(4 149 93 21)(5 160 94 32)(6 151 95 23)(7 142 96 34)(8 153 97 25)(9 144 98 36)(10 155 99 27)(11 146 100 38)(12 157 81 29)(13 148 82 40)(14 159 83 31)(15 150 84 22)(16 141 85 33)(17 152 86 24)(18 143 87 35)(19 154 88 26)(20 145 89 37)(41 61 139 113)(42 72 140 104)(43 63 121 115)(44 74 122 106)(45 65 123 117)(46 76 124 108)(47 67 125 119)(48 78 126 110)(49 69 127 101)(50 80 128 112)(51 71 129 103)(52 62 130 114)(53 73 131 105)(54 64 132 116)(55 75 133 107)(56 66 134 118)(57 77 135 109)(58 68 136 120)(59 79 137 111)(60 70 138 102)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 73)(9 74)(10 75)(11 76)(12 77)(13 78)(14 79)(15 80)(16 61)(17 62)(18 63)(19 64)(20 65)(21 127)(22 128)(23 129)(24 130)(25 131)(26 132)(27 133)(28 134)(29 135)(30 136)(31 137)(32 138)(33 139)(34 140)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(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)(81 109)(82 110)(83 111)(84 112)(85 113)(86 114)(87 115)(88 116)(89 117)(90 118)(91 119)(92 120)(93 101)(94 102)(95 103)(96 104)(97 105)(98 106)(99 107)(100 108)
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), (1,129,11,139)(2,128,12,138)(3,127,13,137)(4,126,14,136)(5,125,15,135)(6,124,16,134)(7,123,17,133)(8,122,18,132)(9,121,19,131)(10,140,20,130)(21,78,31,68)(22,77,32,67)(23,76,33,66)(24,75,34,65)(25,74,35,64)(26,73,36,63)(27,72,37,62)(28,71,38,61)(29,70,39,80)(30,69,40,79)(41,90,51,100)(42,89,52,99)(43,88,53,98)(44,87,54,97)(45,86,55,96)(46,85,56,95)(47,84,57,94)(48,83,58,93)(49,82,59,92)(50,81,60,91)(101,148,111,158)(102,147,112,157)(103,146,113,156)(104,145,114,155)(105,144,115,154)(106,143,116,153)(107,142,117,152)(108,141,118,151)(109,160,119,150)(110,159,120,149), (1,156,90,28)(2,147,91,39)(3,158,92,30)(4,149,93,21)(5,160,94,32)(6,151,95,23)(7,142,96,34)(8,153,97,25)(9,144,98,36)(10,155,99,27)(11,146,100,38)(12,157,81,29)(13,148,82,40)(14,159,83,31)(15,150,84,22)(16,141,85,33)(17,152,86,24)(18,143,87,35)(19,154,88,26)(20,145,89,37)(41,61,139,113)(42,72,140,104)(43,63,121,115)(44,74,122,106)(45,65,123,117)(46,76,124,108)(47,67,125,119)(48,78,126,110)(49,69,127,101)(50,80,128,112)(51,71,129,103)(52,62,130,114)(53,73,131,105)(54,64,132,116)(55,75,133,107)(56,66,134,118)(57,77,135,109)(58,68,136,120)(59,79,137,111)(60,70,138,102), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,61)(17,62)(18,63)(19,64)(20,65)(21,127)(22,128)(23,129)(24,130)(25,131)(26,132)(27,133)(28,134)(29,135)(30,136)(31,137)(32,138)(33,139)(34,140)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(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)(81,109)(82,110)(83,111)(84,112)(85,113)(86,114)(87,115)(88,116)(89,117)(90,118)(91,119)(92,120)(93,101)(94,102)(95,103)(96,104)(97,105)(98,106)(99,107)(100,108)>;
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), (1,129,11,139)(2,128,12,138)(3,127,13,137)(4,126,14,136)(5,125,15,135)(6,124,16,134)(7,123,17,133)(8,122,18,132)(9,121,19,131)(10,140,20,130)(21,78,31,68)(22,77,32,67)(23,76,33,66)(24,75,34,65)(25,74,35,64)(26,73,36,63)(27,72,37,62)(28,71,38,61)(29,70,39,80)(30,69,40,79)(41,90,51,100)(42,89,52,99)(43,88,53,98)(44,87,54,97)(45,86,55,96)(46,85,56,95)(47,84,57,94)(48,83,58,93)(49,82,59,92)(50,81,60,91)(101,148,111,158)(102,147,112,157)(103,146,113,156)(104,145,114,155)(105,144,115,154)(106,143,116,153)(107,142,117,152)(108,141,118,151)(109,160,119,150)(110,159,120,149), (1,156,90,28)(2,147,91,39)(3,158,92,30)(4,149,93,21)(5,160,94,32)(6,151,95,23)(7,142,96,34)(8,153,97,25)(9,144,98,36)(10,155,99,27)(11,146,100,38)(12,157,81,29)(13,148,82,40)(14,159,83,31)(15,150,84,22)(16,141,85,33)(17,152,86,24)(18,143,87,35)(19,154,88,26)(20,145,89,37)(41,61,139,113)(42,72,140,104)(43,63,121,115)(44,74,122,106)(45,65,123,117)(46,76,124,108)(47,67,125,119)(48,78,126,110)(49,69,127,101)(50,80,128,112)(51,71,129,103)(52,62,130,114)(53,73,131,105)(54,64,132,116)(55,75,133,107)(56,66,134,118)(57,77,135,109)(58,68,136,120)(59,79,137,111)(60,70,138,102), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,61)(17,62)(18,63)(19,64)(20,65)(21,127)(22,128)(23,129)(24,130)(25,131)(26,132)(27,133)(28,134)(29,135)(30,136)(31,137)(32,138)(33,139)(34,140)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(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)(81,109)(82,110)(83,111)(84,112)(85,113)(86,114)(87,115)(88,116)(89,117)(90,118)(91,119)(92,120)(93,101)(94,102)(95,103)(96,104)(97,105)(98,106)(99,107)(100,108) );
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)], [(1,129,11,139),(2,128,12,138),(3,127,13,137),(4,126,14,136),(5,125,15,135),(6,124,16,134),(7,123,17,133),(8,122,18,132),(9,121,19,131),(10,140,20,130),(21,78,31,68),(22,77,32,67),(23,76,33,66),(24,75,34,65),(25,74,35,64),(26,73,36,63),(27,72,37,62),(28,71,38,61),(29,70,39,80),(30,69,40,79),(41,90,51,100),(42,89,52,99),(43,88,53,98),(44,87,54,97),(45,86,55,96),(46,85,56,95),(47,84,57,94),(48,83,58,93),(49,82,59,92),(50,81,60,91),(101,148,111,158),(102,147,112,157),(103,146,113,156),(104,145,114,155),(105,144,115,154),(106,143,116,153),(107,142,117,152),(108,141,118,151),(109,160,119,150),(110,159,120,149)], [(1,156,90,28),(2,147,91,39),(3,158,92,30),(4,149,93,21),(5,160,94,32),(6,151,95,23),(7,142,96,34),(8,153,97,25),(9,144,98,36),(10,155,99,27),(11,146,100,38),(12,157,81,29),(13,148,82,40),(14,159,83,31),(15,150,84,22),(16,141,85,33),(17,152,86,24),(18,143,87,35),(19,154,88,26),(20,145,89,37),(41,61,139,113),(42,72,140,104),(43,63,121,115),(44,74,122,106),(45,65,123,117),(46,76,124,108),(47,67,125,119),(48,78,126,110),(49,69,127,101),(50,80,128,112),(51,71,129,103),(52,62,130,114),(53,73,131,105),(54,64,132,116),(55,75,133,107),(56,66,134,118),(57,77,135,109),(58,68,136,120),(59,79,137,111),(60,70,138,102)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,73),(9,74),(10,75),(11,76),(12,77),(13,78),(14,79),(15,80),(16,61),(17,62),(18,63),(19,64),(20,65),(21,127),(22,128),(23,129),(24,130),(25,131),(26,132),(27,133),(28,134),(29,135),(30,136),(31,137),(32,138),(33,139),(34,140),(35,121),(36,122),(37,123),(38,124),(39,125),(40,126),(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),(81,109),(82,110),(83,111),(84,112),(85,113),(86,114),(87,115),(88,116),(89,117),(90,118),(91,119),(92,120),(93,101),(94,102),(95,103),(96,104),(97,105),(98,106),(99,107),(100,108)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 4N | 4O | 4P | 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 | 4 | 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 | 2 | 2 | 4 | 4 | 20 | 2 | 2 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 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 | 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 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2- 1+4 | D4×D5 | D4⋊2D5 | D4.10D10 |
| kernel | Dic10⋊19D4 | Dic5.14D4 | Dic5.5D4 | Dic5⋊3Q8 | D10⋊2Q8 | C4×C5⋊D4 | D4×Dic5 | C20.17D4 | Dic5⋊D4 | C5×C4⋊D4 | C22×Dic10 | C2×D4⋊2D5 | Dic10 | C4⋊D4 | C2×C10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 2 | 2 | 6 | 1 | 4 | 4 | 4 |
Matrix representation of Dic10⋊19D4 ►in GL6(𝔽41)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 31 |
| 0 | 0 | 0 | 0 | 12 | 18 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 10 |
| 0 | 0 | 0 | 0 | 21 | 23 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,23,12,0,0,0,0,31,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,21,0,0,0,0,10,23] >;
Dic10⋊19D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_{19}D_4 % in TeX
G:=Group("Dic10:19D4"); // GroupNames label
G:=SmallGroup(320,1270);
// by ID
G=gap.SmallGroup(320,1270);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,477,232,570,185,12550]);
// 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,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations