metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊10D4, C42.142D10, C10.912- 1+4, C4.71(D4×D5), (C4×D20)⋊44C2, C5⋊5(Q8⋊5D4), C20.64(C2×D4), C20⋊2D4⋊34C2, C4.4D4⋊11D5, D10⋊15(C4○D4), D10⋊D4⋊41C2, D10⋊3Q8⋊29C2, (C4×Dic10)⋊45C2, (C2×D4).174D10, (C2×C20).81C23, (C2×Q8).137D10, C22⋊C4.73D10, Dic5.52(C2×D4), C10.91(C22×D4), Dic5⋊4D4⋊30C2, (C2×C10).221C24, (C4×C20).186C22, C23.43(C22×D5), Dic5.5D4⋊40C2, (D4×C10).156C22, (C2×D20).231C22, C4⋊Dic5.377C22, (C22×C10).51C23, (Q8×C10).127C22, C22.242(C23×D5), Dic5.14D4⋊41C2, C23.D5.55C22, (C4×Dic5).234C22, (C2×Dic5).263C23, (C22×D5).226C23, C2.52(D4.10D10), D10⋊C4.135C22, (C2×Dic10).305C22, C10.D4.121C22, (C22×Dic5).143C22, (C2×Q8×D5)⋊11C2, C2.64(C2×D4×D5), C2.77(D5×C4○D4), (C2×D4⋊2D5)⋊19C2, C10.188(C2×C4○D4), (C5×C4.4D4)⋊13C2, (C2×C4×D5).130C22, (C2×C4).196(C22×D5), (C2×C5⋊D4).60C22, (C5×C22⋊C4).65C22, SmallGroup(320,1349)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊10D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, ac=ca, dad=a9, cbc-1=dbd=a10b, dcd=c-1 >
Subgroups: 1046 in 290 conjugacy classes, 105 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C4.4D4, C4.4D4, C22×Q8, C2×C4○D4, Dic10, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×C10, Q8⋊5D4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, D4⋊2D5, Q8×D5, C22×Dic5, C2×C5⋊D4, D4×C10, Q8×C10, C4×Dic10, C4×D20, Dic5.14D4, Dic5⋊4D4, D10⋊D4, Dic5.5D4, C20⋊2D4, D10⋊3Q8, C5×C4.4D4, C2×D4⋊2D5, C2×Q8×D5, Dic10⋊10D4
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, C23×D5, C2×D4×D5, D5×C4○D4, D4.10D10, Dic10⋊10D4
►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)
(1 156 11 146)(2 155 12 145)(3 154 13 144)(4 153 14 143)(5 152 15 142)(6 151 16 141)(7 150 17 160)(8 149 18 159)(9 148 19 158)(10 147 20 157)(21 70 31 80)(22 69 32 79)(23 68 33 78)(24 67 34 77)(25 66 35 76)(26 65 36 75)(27 64 37 74)(28 63 38 73)(29 62 39 72)(30 61 40 71)(41 139 51 129)(42 138 52 128)(43 137 53 127)(44 136 54 126)(45 135 55 125)(46 134 56 124)(47 133 57 123)(48 132 58 122)(49 131 59 121)(50 130 60 140)(81 103 91 113)(82 102 92 112)(83 101 93 111)(84 120 94 110)(85 119 95 109)(86 118 96 108)(87 117 97 107)(88 116 98 106)(89 115 99 105)(90 114 100 104)
(1 85 24 55)(2 86 25 56)(3 87 26 57)(4 88 27 58)(5 89 28 59)(6 90 29 60)(7 91 30 41)(8 92 31 42)(9 93 32 43)(10 94 33 44)(11 95 34 45)(12 96 35 46)(13 97 36 47)(14 98 37 48)(15 99 38 49)(16 100 39 50)(17 81 40 51)(18 82 21 52)(19 83 22 53)(20 84 23 54)(61 129 150 103)(62 130 151 104)(63 131 152 105)(64 132 153 106)(65 133 154 107)(66 134 155 108)(67 135 156 109)(68 136 157 110)(69 137 158 111)(70 138 159 112)(71 139 160 113)(72 140 141 114)(73 121 142 115)(74 122 143 116)(75 123 144 117)(76 124 145 118)(77 125 146 119)(78 126 147 120)(79 127 148 101)(80 128 149 102)
(1 24)(2 33)(3 22)(4 31)(5 40)(6 29)(7 38)(8 27)(9 36)(10 25)(11 34)(12 23)(13 32)(14 21)(15 30)(16 39)(17 28)(18 37)(19 26)(20 35)(41 49)(42 58)(43 47)(44 56)(46 54)(48 52)(51 59)(53 57)(61 152)(62 141)(63 150)(64 159)(65 148)(66 157)(67 146)(68 155)(69 144)(70 153)(71 142)(72 151)(73 160)(74 149)(75 158)(76 147)(77 156)(78 145)(79 154)(80 143)(81 89)(82 98)(83 87)(84 96)(86 94)(88 92)(91 99)(93 97)(101 107)(102 116)(103 105)(104 114)(106 112)(108 110)(109 119)(111 117)(113 115)(118 120)(121 139)(122 128)(123 137)(124 126)(125 135)(127 133)(129 131)(130 140)(132 138)(134 136)
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), (1,156,11,146)(2,155,12,145)(3,154,13,144)(4,153,14,143)(5,152,15,142)(6,151,16,141)(7,150,17,160)(8,149,18,159)(9,148,19,158)(10,147,20,157)(21,70,31,80)(22,69,32,79)(23,68,33,78)(24,67,34,77)(25,66,35,76)(26,65,36,75)(27,64,37,74)(28,63,38,73)(29,62,39,72)(30,61,40,71)(41,139,51,129)(42,138,52,128)(43,137,53,127)(44,136,54,126)(45,135,55,125)(46,134,56,124)(47,133,57,123)(48,132,58,122)(49,131,59,121)(50,130,60,140)(81,103,91,113)(82,102,92,112)(83,101,93,111)(84,120,94,110)(85,119,95,109)(86,118,96,108)(87,117,97,107)(88,116,98,106)(89,115,99,105)(90,114,100,104), (1,85,24,55)(2,86,25,56)(3,87,26,57)(4,88,27,58)(5,89,28,59)(6,90,29,60)(7,91,30,41)(8,92,31,42)(9,93,32,43)(10,94,33,44)(11,95,34,45)(12,96,35,46)(13,97,36,47)(14,98,37,48)(15,99,38,49)(16,100,39,50)(17,81,40,51)(18,82,21,52)(19,83,22,53)(20,84,23,54)(61,129,150,103)(62,130,151,104)(63,131,152,105)(64,132,153,106)(65,133,154,107)(66,134,155,108)(67,135,156,109)(68,136,157,110)(69,137,158,111)(70,138,159,112)(71,139,160,113)(72,140,141,114)(73,121,142,115)(74,122,143,116)(75,123,144,117)(76,124,145,118)(77,125,146,119)(78,126,147,120)(79,127,148,101)(80,128,149,102), (1,24)(2,33)(3,22)(4,31)(5,40)(6,29)(7,38)(8,27)(9,36)(10,25)(11,34)(12,23)(13,32)(14,21)(15,30)(16,39)(17,28)(18,37)(19,26)(20,35)(41,49)(42,58)(43,47)(44,56)(46,54)(48,52)(51,59)(53,57)(61,152)(62,141)(63,150)(64,159)(65,148)(66,157)(67,146)(68,155)(69,144)(70,153)(71,142)(72,151)(73,160)(74,149)(75,158)(76,147)(77,156)(78,145)(79,154)(80,143)(81,89)(82,98)(83,87)(84,96)(86,94)(88,92)(91,99)(93,97)(101,107)(102,116)(103,105)(104,114)(106,112)(108,110)(109,119)(111,117)(113,115)(118,120)(121,139)(122,128)(123,137)(124,126)(125,135)(127,133)(129,131)(130,140)(132,138)(134,136)>;
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), (1,156,11,146)(2,155,12,145)(3,154,13,144)(4,153,14,143)(5,152,15,142)(6,151,16,141)(7,150,17,160)(8,149,18,159)(9,148,19,158)(10,147,20,157)(21,70,31,80)(22,69,32,79)(23,68,33,78)(24,67,34,77)(25,66,35,76)(26,65,36,75)(27,64,37,74)(28,63,38,73)(29,62,39,72)(30,61,40,71)(41,139,51,129)(42,138,52,128)(43,137,53,127)(44,136,54,126)(45,135,55,125)(46,134,56,124)(47,133,57,123)(48,132,58,122)(49,131,59,121)(50,130,60,140)(81,103,91,113)(82,102,92,112)(83,101,93,111)(84,120,94,110)(85,119,95,109)(86,118,96,108)(87,117,97,107)(88,116,98,106)(89,115,99,105)(90,114,100,104), (1,85,24,55)(2,86,25,56)(3,87,26,57)(4,88,27,58)(5,89,28,59)(6,90,29,60)(7,91,30,41)(8,92,31,42)(9,93,32,43)(10,94,33,44)(11,95,34,45)(12,96,35,46)(13,97,36,47)(14,98,37,48)(15,99,38,49)(16,100,39,50)(17,81,40,51)(18,82,21,52)(19,83,22,53)(20,84,23,54)(61,129,150,103)(62,130,151,104)(63,131,152,105)(64,132,153,106)(65,133,154,107)(66,134,155,108)(67,135,156,109)(68,136,157,110)(69,137,158,111)(70,138,159,112)(71,139,160,113)(72,140,141,114)(73,121,142,115)(74,122,143,116)(75,123,144,117)(76,124,145,118)(77,125,146,119)(78,126,147,120)(79,127,148,101)(80,128,149,102), (1,24)(2,33)(3,22)(4,31)(5,40)(6,29)(7,38)(8,27)(9,36)(10,25)(11,34)(12,23)(13,32)(14,21)(15,30)(16,39)(17,28)(18,37)(19,26)(20,35)(41,49)(42,58)(43,47)(44,56)(46,54)(48,52)(51,59)(53,57)(61,152)(62,141)(63,150)(64,159)(65,148)(66,157)(67,146)(68,155)(69,144)(70,153)(71,142)(72,151)(73,160)(74,149)(75,158)(76,147)(77,156)(78,145)(79,154)(80,143)(81,89)(82,98)(83,87)(84,96)(86,94)(88,92)(91,99)(93,97)(101,107)(102,116)(103,105)(104,114)(106,112)(108,110)(109,119)(111,117)(113,115)(118,120)(121,139)(122,128)(123,137)(124,126)(125,135)(127,133)(129,131)(130,140)(132,138)(134,136) );
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)], [(1,156,11,146),(2,155,12,145),(3,154,13,144),(4,153,14,143),(5,152,15,142),(6,151,16,141),(7,150,17,160),(8,149,18,159),(9,148,19,158),(10,147,20,157),(21,70,31,80),(22,69,32,79),(23,68,33,78),(24,67,34,77),(25,66,35,76),(26,65,36,75),(27,64,37,74),(28,63,38,73),(29,62,39,72),(30,61,40,71),(41,139,51,129),(42,138,52,128),(43,137,53,127),(44,136,54,126),(45,135,55,125),(46,134,56,124),(47,133,57,123),(48,132,58,122),(49,131,59,121),(50,130,60,140),(81,103,91,113),(82,102,92,112),(83,101,93,111),(84,120,94,110),(85,119,95,109),(86,118,96,108),(87,117,97,107),(88,116,98,106),(89,115,99,105),(90,114,100,104)], [(1,85,24,55),(2,86,25,56),(3,87,26,57),(4,88,27,58),(5,89,28,59),(6,90,29,60),(7,91,30,41),(8,92,31,42),(9,93,32,43),(10,94,33,44),(11,95,34,45),(12,96,35,46),(13,97,36,47),(14,98,37,48),(15,99,38,49),(16,100,39,50),(17,81,40,51),(18,82,21,52),(19,83,22,53),(20,84,23,54),(61,129,150,103),(62,130,151,104),(63,131,152,105),(64,132,153,106),(65,133,154,107),(66,134,155,108),(67,135,156,109),(68,136,157,110),(69,137,158,111),(70,138,159,112),(71,139,160,113),(72,140,141,114),(73,121,142,115),(74,122,143,116),(75,123,144,117),(76,124,145,118),(77,125,146,119),(78,126,147,120),(79,127,148,101),(80,128,149,102)], [(1,24),(2,33),(3,22),(4,31),(5,40),(6,29),(7,38),(8,27),(9,36),(10,25),(11,34),(12,23),(13,32),(14,21),(15,30),(16,39),(17,28),(18,37),(19,26),(20,35),(41,49),(42,58),(43,47),(44,56),(46,54),(48,52),(51,59),(53,57),(61,152),(62,141),(63,150),(64,159),(65,148),(66,157),(67,146),(68,155),(69,144),(70,153),(71,142),(72,151),(73,160),(74,149),(75,158),(76,147),(77,156),(78,145),(79,154),(80,143),(81,89),(82,98),(83,87),(84,96),(86,94),(88,92),(91,99),(93,97),(101,107),(102,116),(103,105),(104,114),(106,112),(108,110),(109,119),(111,117),(113,115),(118,120),(121,139),(122,128),(123,137),(124,126),(125,135),(127,133),(129,131),(130,140),(132,138),(134,136)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 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 | D5×C4○D4 | D4.10D10 |
| kernel | Dic10⋊10D4 | C4×Dic10 | C4×D20 | Dic5.14D4 | Dic5⋊4D4 | D10⋊D4 | Dic5.5D4 | C20⋊2D4 | D10⋊3Q8 | C5×C4.4D4 | C2×D4⋊2D5 | C2×Q8×D5 | Dic10 | C4.4D4 | D10 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 2 | 8 | 2 | 2 | 1 | 4 | 4 | 4 |
Matrix representation of Dic10⋊10D4 ►in GL6(𝔽41)
| 7 | 40 | 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 | 9 | 0 |
| 0 | 0 | 0 | 0 | 32 | 32 |
| 34 | 7 | 0 | 0 | 0 | 0 |
| 40 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 |
| 34 | 7 | 0 | 0 | 0 | 0 |
| 40 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 |
G:=sub<GL(6,GF(41))| [7,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,32,0,0,0,0,0,32],[34,40,0,0,0,0,7,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[34,40,0,0,0,0,7,7,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,40,0,0,0,0,0,40] >;
Dic10⋊10D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_{10}D_4 % in TeX
G:=Group("Dic10:10D4"); // GroupNames label
G:=SmallGroup(320,1349);
// by ID
G=gap.SmallGroup(320,1349);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,100,1571,297,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^9,c*b*c^-1=d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations