metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10.11D4, C4.91(D4×D5), Q8⋊C4⋊9D5, C4⋊C4.146D10, C4.6(C4○D20), C20.115(C2×D4), (C2×C8).120D10, (C2×Q8).13D10, C5⋊2(Q8.D4), Dic5⋊3Q8⋊5C2, D20⋊6C4.2C2, C10.45(C4○D8), C20.17(C4○D4), C20.8Q8⋊13C2, (C2×Dic5).36D4, C22.192(D4×D5), C10.22(C4⋊D4), (C2×C20).238C23, (C2×C40).131C22, C20.23D4.5C2, (C2×D20).62C22, (Q8×C10).21C22, C2.25(D10⋊D4), C2.11(Q16⋊D5), C10.56(C8.C22), (C4×Dic5).28C22, C2.14(SD16⋊3D5), (C2×Dic10).71C22, (C2×C5⋊Q16)⋊2C2, (C5×Q8⋊C4)⋊9C2, (C2×C40⋊C2).3C2, (C2×C10).251(C2×D4), (C5×C4⋊C4).39C22, (C2×C5⋊2C8).33C22, (C2×C4).345(C22×D5), SmallGroup(320,425)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10.11D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=dad=a-1, cac-1=a9, bc=cb, dbd=a15b, dcd=a10c-1 >
Subgroups: 486 in 112 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, D10, C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C5⋊2C8, C40, Dic10, Dic10, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, Q8.D4, C40⋊C2, C2×C5⋊2C8, C4×Dic5, C4×Dic5, C10.D4, D10⋊C4, C5⋊Q16, C5×C4⋊C4, C2×C40, C2×Dic10, C2×D20, Q8×C10, D20⋊6C4, C20.8Q8, C5×Q8⋊C4, Dic5⋊3Q8, C2×C40⋊C2, C2×C5⋊Q16, C20.23D4, Dic10.11D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C4○D8, C8.C22, C22×D5, Q8.D4, C4○D20, D4×D5, D10⋊D4, SD16⋊3D5, Q16⋊D5, Dic10.11D4
►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 28 11 38)(2 27 12 37)(3 26 13 36)(4 25 14 35)(5 24 15 34)(6 23 16 33)(7 22 17 32)(8 21 18 31)(9 40 19 30)(10 39 20 29)(41 67 51 77)(42 66 52 76)(43 65 53 75)(44 64 54 74)(45 63 55 73)(46 62 56 72)(47 61 57 71)(48 80 58 70)(49 79 59 69)(50 78 60 68)(81 138 91 128)(82 137 92 127)(83 136 93 126)(84 135 94 125)(85 134 95 124)(86 133 96 123)(87 132 97 122)(88 131 98 121)(89 130 99 140)(90 129 100 139)(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 96 61 147)(2 85 62 156)(3 94 63 145)(4 83 64 154)(5 92 65 143)(6 81 66 152)(7 90 67 141)(8 99 68 150)(9 88 69 159)(10 97 70 148)(11 86 71 157)(12 95 72 146)(13 84 73 155)(14 93 74 144)(15 82 75 153)(16 91 76 142)(17 100 77 151)(18 89 78 160)(19 98 79 149)(20 87 80 158)(21 140 50 112)(22 129 51 101)(23 138 52 110)(24 127 53 119)(25 136 54 108)(26 125 55 117)(27 134 56 106)(28 123 57 115)(29 132 58 104)(30 121 59 113)(31 130 60 102)(32 139 41 111)(33 128 42 120)(34 137 43 109)(35 126 44 118)(36 135 45 107)(37 124 46 116)(38 133 47 105)(39 122 48 114)(40 131 49 103)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(59 60)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(81 152)(82 151)(83 150)(84 149)(85 148)(86 147)(87 146)(88 145)(89 144)(90 143)(91 142)(92 141)(93 160)(94 159)(95 158)(96 157)(97 156)(98 155)(99 154)(100 153)(101 132)(102 131)(103 130)(104 129)(105 128)(106 127)(107 126)(108 125)(109 124)(110 123)(111 122)(112 121)(113 140)(114 139)(115 138)(116 137)(117 136)(118 135)(119 134)(120 133)
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), (1,28,11,38)(2,27,12,37)(3,26,13,36)(4,25,14,35)(5,24,15,34)(6,23,16,33)(7,22,17,32)(8,21,18,31)(9,40,19,30)(10,39,20,29)(41,67,51,77)(42,66,52,76)(43,65,53,75)(44,64,54,74)(45,63,55,73)(46,62,56,72)(47,61,57,71)(48,80,58,70)(49,79,59,69)(50,78,60,68)(81,138,91,128)(82,137,92,127)(83,136,93,126)(84,135,94,125)(85,134,95,124)(86,133,96,123)(87,132,97,122)(88,131,98,121)(89,130,99,140)(90,129,100,139)(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), (1,96,61,147)(2,85,62,156)(3,94,63,145)(4,83,64,154)(5,92,65,143)(6,81,66,152)(7,90,67,141)(8,99,68,150)(9,88,69,159)(10,97,70,148)(11,86,71,157)(12,95,72,146)(13,84,73,155)(14,93,74,144)(15,82,75,153)(16,91,76,142)(17,100,77,151)(18,89,78,160)(19,98,79,149)(20,87,80,158)(21,140,50,112)(22,129,51,101)(23,138,52,110)(24,127,53,119)(25,136,54,108)(26,125,55,117)(27,134,56,106)(28,123,57,115)(29,132,58,104)(30,121,59,113)(31,130,60,102)(32,139,41,111)(33,128,42,120)(34,137,43,109)(35,126,44,118)(36,135,45,107)(37,124,46,116)(38,133,47,105)(39,122,48,114)(40,131,49,103), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(81,152)(82,151)(83,150)(84,149)(85,148)(86,147)(87,146)(88,145)(89,144)(90,143)(91,142)(92,141)(93,160)(94,159)(95,158)(96,157)(97,156)(98,155)(99,154)(100,153)(101,132)(102,131)(103,130)(104,129)(105,128)(106,127)(107,126)(108,125)(109,124)(110,123)(111,122)(112,121)(113,140)(114,139)(115,138)(116,137)(117,136)(118,135)(119,134)(120,133)>;
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), (1,28,11,38)(2,27,12,37)(3,26,13,36)(4,25,14,35)(5,24,15,34)(6,23,16,33)(7,22,17,32)(8,21,18,31)(9,40,19,30)(10,39,20,29)(41,67,51,77)(42,66,52,76)(43,65,53,75)(44,64,54,74)(45,63,55,73)(46,62,56,72)(47,61,57,71)(48,80,58,70)(49,79,59,69)(50,78,60,68)(81,138,91,128)(82,137,92,127)(83,136,93,126)(84,135,94,125)(85,134,95,124)(86,133,96,123)(87,132,97,122)(88,131,98,121)(89,130,99,140)(90,129,100,139)(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), (1,96,61,147)(2,85,62,156)(3,94,63,145)(4,83,64,154)(5,92,65,143)(6,81,66,152)(7,90,67,141)(8,99,68,150)(9,88,69,159)(10,97,70,148)(11,86,71,157)(12,95,72,146)(13,84,73,155)(14,93,74,144)(15,82,75,153)(16,91,76,142)(17,100,77,151)(18,89,78,160)(19,98,79,149)(20,87,80,158)(21,140,50,112)(22,129,51,101)(23,138,52,110)(24,127,53,119)(25,136,54,108)(26,125,55,117)(27,134,56,106)(28,123,57,115)(29,132,58,104)(30,121,59,113)(31,130,60,102)(32,139,41,111)(33,128,42,120)(34,137,43,109)(35,126,44,118)(36,135,45,107)(37,124,46,116)(38,133,47,105)(39,122,48,114)(40,131,49,103), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(81,152)(82,151)(83,150)(84,149)(85,148)(86,147)(87,146)(88,145)(89,144)(90,143)(91,142)(92,141)(93,160)(94,159)(95,158)(96,157)(97,156)(98,155)(99,154)(100,153)(101,132)(102,131)(103,130)(104,129)(105,128)(106,127)(107,126)(108,125)(109,124)(110,123)(111,122)(112,121)(113,140)(114,139)(115,138)(116,137)(117,136)(118,135)(119,134)(120,133) );
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)], [(1,28,11,38),(2,27,12,37),(3,26,13,36),(4,25,14,35),(5,24,15,34),(6,23,16,33),(7,22,17,32),(8,21,18,31),(9,40,19,30),(10,39,20,29),(41,67,51,77),(42,66,52,76),(43,65,53,75),(44,64,54,74),(45,63,55,73),(46,62,56,72),(47,61,57,71),(48,80,58,70),(49,79,59,69),(50,78,60,68),(81,138,91,128),(82,137,92,127),(83,136,93,126),(84,135,94,125),(85,134,95,124),(86,133,96,123),(87,132,97,122),(88,131,98,121),(89,130,99,140),(90,129,100,139),(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)], [(1,96,61,147),(2,85,62,156),(3,94,63,145),(4,83,64,154),(5,92,65,143),(6,81,66,152),(7,90,67,141),(8,99,68,150),(9,88,69,159),(10,97,70,148),(11,86,71,157),(12,95,72,146),(13,84,73,155),(14,93,74,144),(15,82,75,153),(16,91,76,142),(17,100,77,151),(18,89,78,160),(19,98,79,149),(20,87,80,158),(21,140,50,112),(22,129,51,101),(23,138,52,110),(24,127,53,119),(25,136,54,108),(26,125,55,117),(27,134,56,106),(28,123,57,115),(29,132,58,104),(30,121,59,113),(31,130,60,102),(32,139,41,111),(33,128,42,120),(34,137,43,109),(35,126,44,118),(36,135,45,107),(37,124,46,116),(38,133,47,105),(39,122,48,114),(40,131,49,103)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(41,58),(42,57),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50),(59,60),(62,80),(63,79),(64,78),(65,77),(66,76),(67,75),(68,74),(69,73),(70,72),(81,152),(82,151),(83,150),(84,149),(85,148),(86,147),(87,146),(88,145),(89,144),(90,143),(91,142),(92,141),(93,160),(94,159),(95,158),(96,157),(97,156),(98,155),(99,154),(100,153),(101,132),(102,131),(103,130),(104,129),(105,128),(106,127),(107,126),(108,125),(109,124),(110,123),(111,122),(112,121),(113,140),(114,139),(115,138),(116,137),(117,136),(118,135),(119,134),(120,133)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 40 | 2 | 2 | 4 | 4 | 8 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 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 | C4○D4 | D10 | D10 | D10 | C4○D8 | C4○D20 | C8.C22 | D4×D5 | D4×D5 | SD16⋊3D5 | Q16⋊D5 |
| kernel | Dic10.11D4 | D20⋊6C4 | C20.8Q8 | C5×Q8⋊C4 | Dic5⋊3Q8 | C2×C40⋊C2 | C2×C5⋊Q16 | C20.23D4 | Dic10 | C2×Dic5 | Q8⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×Q8 | C10 | C4 | 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 Dic10.11D4 ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 2 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 23 | 0 | 0 |
| 0 | 0 | 9 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 0 | 0 | 0 | 0 | 26 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 2 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 40 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,40,0,0,0,0,2,1],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,9,9,0,0,0,0,23,32,0,0,0,0,0,0,0,26,0,0,0,0,11,0],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,40,40,0,0,0,0,2,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,0,40] >;
Dic10.11D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}._{11}D_4 % in TeX
G:=Group("Dic10.11D4"); // GroupNames label
G:=SmallGroup(320,425);
// by ID
G=gap.SmallGroup(320,425);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,344,1094,135,184,570,297,136,12550]);
// 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=a^10*c^-1>;
// generators/relations