metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊11D4, C42.167D10, C10.782+ 1+4, C4⋊1D4⋊7D5, C4.72(D4×D5), (C4×D20)⋊50C2, C20⋊5(C4○D4), C5⋊4(Q8⋊6D4), C20.67(C2×D4), C20⋊2D4⋊37C2, C4⋊1(D4⋊2D5), C20⋊D4⋊27C2, (D4×Dic5)⋊35C2, (C4×Dic10)⋊51C2, (C2×D4).115D10, Dic5.53(C2×D4), C10.96(C22×D4), Dic5⋊D4⋊37C2, (C2×C10).262C24, (C2×C20).636C23, (C4×C20).204C22, C2.82(D4⋊6D10), C23.68(C22×D5), (C2×D20).278C22, (D4×C10).214C22, C4⋊Dic5.381C22, (C22×C10).76C23, C22.283(C23×D5), C23.D5.73C22, (C2×Dic5).279C23, (C4×Dic5).163C22, (C22×D5).116C23, D10⋊C4.149C22, (C2×Dic10).309C22, C10.D4.164C22, (C22×Dic5).158C22, C2.69(C2×D4×D5), (C5×C4⋊1D4)⋊9C2, C10.97(C2×C4○D4), (C2×D4⋊2D5)⋊22C2, C2.61(C2×D4⋊2D5), (C2×C4×D5).148C22, (C2×C4).598(C22×D5), (C2×C5⋊D4).78C22, SmallGroup(320,1390)
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=a-1, ac=ca, dad=a11, bc=cb, dbd=a10b, dcd=c-1 >
Subgroups: 1158 in 312 conjugacy classes, 107 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C4×D4, C4×Q8, C4⋊D4, C4⋊1D4, C4⋊1D4, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, Q8⋊6D4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C2×Dic10, C2×C4×D5, C2×D20, D4⋊2D5, C22×Dic5, C2×C5⋊D4, D4×C10, C4×Dic10, C4×D20, D4×Dic5, C20⋊2D4, Dic5⋊D4, C20⋊D4, C5×C4⋊1D4, C2×D4⋊2D5, Dic10⋊11D4
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, D4⋊2D5, C23×D5, C2×D4×D5, C2×D4⋊2D5, D4⋊6D10, 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 53 11 43)(2 52 12 42)(3 51 13 41)(4 50 14 60)(5 49 15 59)(6 48 16 58)(7 47 17 57)(8 46 18 56)(9 45 19 55)(10 44 20 54)(21 153 31 143)(22 152 32 142)(23 151 33 141)(24 150 34 160)(25 149 35 159)(26 148 36 158)(27 147 37 157)(28 146 38 156)(29 145 39 155)(30 144 40 154)(61 114 71 104)(62 113 72 103)(63 112 73 102)(64 111 74 101)(65 110 75 120)(66 109 76 119)(67 108 77 118)(68 107 78 117)(69 106 79 116)(70 105 80 115)(81 131 91 121)(82 130 92 140)(83 129 93 139)(84 128 94 138)(85 127 95 137)(86 126 96 136)(87 125 97 135)(88 124 98 134)(89 123 99 133)(90 122 100 132)
(1 135 40 76)(2 136 21 77)(3 137 22 78)(4 138 23 79)(5 139 24 80)(6 140 25 61)(7 121 26 62)(8 122 27 63)(9 123 28 64)(10 124 29 65)(11 125 30 66)(12 126 31 67)(13 127 32 68)(14 128 33 69)(15 129 34 70)(16 130 35 71)(17 131 36 72)(18 132 37 73)(19 133 38 74)(20 134 39 75)(41 95 142 107)(42 96 143 108)(43 97 144 109)(44 98 145 110)(45 99 146 111)(46 100 147 112)(47 81 148 113)(48 82 149 114)(49 83 150 115)(50 84 151 116)(51 85 152 117)(52 86 153 118)(53 87 154 119)(54 88 155 120)(55 89 156 101)(56 90 157 102)(57 91 158 103)(58 92 159 104)(59 93 160 105)(60 94 141 106)
(1 119)(2 110)(3 101)(4 112)(5 103)(6 114)(7 105)(8 116)(9 107)(10 118)(11 109)(12 120)(13 111)(14 102)(15 113)(16 104)(17 115)(18 106)(19 117)(20 108)(21 98)(22 89)(23 100)(24 91)(25 82)(26 93)(27 84)(28 95)(29 86)(30 97)(31 88)(32 99)(33 90)(34 81)(35 92)(36 83)(37 94)(38 85)(39 96)(40 87)(41 64)(42 75)(43 66)(44 77)(45 68)(46 79)(47 70)(48 61)(49 72)(50 63)(51 74)(52 65)(53 76)(54 67)(55 78)(56 69)(57 80)(58 71)(59 62)(60 73)(121 160)(122 151)(123 142)(124 153)(125 144)(126 155)(127 146)(128 157)(129 148)(130 159)(131 150)(132 141)(133 152)(134 143)(135 154)(136 145)(137 156)(138 147)(139 158)(140 149)
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), (1,53,11,43)(2,52,12,42)(3,51,13,41)(4,50,14,60)(5,49,15,59)(6,48,16,58)(7,47,17,57)(8,46,18,56)(9,45,19,55)(10,44,20,54)(21,153,31,143)(22,152,32,142)(23,151,33,141)(24,150,34,160)(25,149,35,159)(26,148,36,158)(27,147,37,157)(28,146,38,156)(29,145,39,155)(30,144,40,154)(61,114,71,104)(62,113,72,103)(63,112,73,102)(64,111,74,101)(65,110,75,120)(66,109,76,119)(67,108,77,118)(68,107,78,117)(69,106,79,116)(70,105,80,115)(81,131,91,121)(82,130,92,140)(83,129,93,139)(84,128,94,138)(85,127,95,137)(86,126,96,136)(87,125,97,135)(88,124,98,134)(89,123,99,133)(90,122,100,132), (1,135,40,76)(2,136,21,77)(3,137,22,78)(4,138,23,79)(5,139,24,80)(6,140,25,61)(7,121,26,62)(8,122,27,63)(9,123,28,64)(10,124,29,65)(11,125,30,66)(12,126,31,67)(13,127,32,68)(14,128,33,69)(15,129,34,70)(16,130,35,71)(17,131,36,72)(18,132,37,73)(19,133,38,74)(20,134,39,75)(41,95,142,107)(42,96,143,108)(43,97,144,109)(44,98,145,110)(45,99,146,111)(46,100,147,112)(47,81,148,113)(48,82,149,114)(49,83,150,115)(50,84,151,116)(51,85,152,117)(52,86,153,118)(53,87,154,119)(54,88,155,120)(55,89,156,101)(56,90,157,102)(57,91,158,103)(58,92,159,104)(59,93,160,105)(60,94,141,106), (1,119)(2,110)(3,101)(4,112)(5,103)(6,114)(7,105)(8,116)(9,107)(10,118)(11,109)(12,120)(13,111)(14,102)(15,113)(16,104)(17,115)(18,106)(19,117)(20,108)(21,98)(22,89)(23,100)(24,91)(25,82)(26,93)(27,84)(28,95)(29,86)(30,97)(31,88)(32,99)(33,90)(34,81)(35,92)(36,83)(37,94)(38,85)(39,96)(40,87)(41,64)(42,75)(43,66)(44,77)(45,68)(46,79)(47,70)(48,61)(49,72)(50,63)(51,74)(52,65)(53,76)(54,67)(55,78)(56,69)(57,80)(58,71)(59,62)(60,73)(121,160)(122,151)(123,142)(124,153)(125,144)(126,155)(127,146)(128,157)(129,148)(130,159)(131,150)(132,141)(133,152)(134,143)(135,154)(136,145)(137,156)(138,147)(139,158)(140,149)>;
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), (1,53,11,43)(2,52,12,42)(3,51,13,41)(4,50,14,60)(5,49,15,59)(6,48,16,58)(7,47,17,57)(8,46,18,56)(9,45,19,55)(10,44,20,54)(21,153,31,143)(22,152,32,142)(23,151,33,141)(24,150,34,160)(25,149,35,159)(26,148,36,158)(27,147,37,157)(28,146,38,156)(29,145,39,155)(30,144,40,154)(61,114,71,104)(62,113,72,103)(63,112,73,102)(64,111,74,101)(65,110,75,120)(66,109,76,119)(67,108,77,118)(68,107,78,117)(69,106,79,116)(70,105,80,115)(81,131,91,121)(82,130,92,140)(83,129,93,139)(84,128,94,138)(85,127,95,137)(86,126,96,136)(87,125,97,135)(88,124,98,134)(89,123,99,133)(90,122,100,132), (1,135,40,76)(2,136,21,77)(3,137,22,78)(4,138,23,79)(5,139,24,80)(6,140,25,61)(7,121,26,62)(8,122,27,63)(9,123,28,64)(10,124,29,65)(11,125,30,66)(12,126,31,67)(13,127,32,68)(14,128,33,69)(15,129,34,70)(16,130,35,71)(17,131,36,72)(18,132,37,73)(19,133,38,74)(20,134,39,75)(41,95,142,107)(42,96,143,108)(43,97,144,109)(44,98,145,110)(45,99,146,111)(46,100,147,112)(47,81,148,113)(48,82,149,114)(49,83,150,115)(50,84,151,116)(51,85,152,117)(52,86,153,118)(53,87,154,119)(54,88,155,120)(55,89,156,101)(56,90,157,102)(57,91,158,103)(58,92,159,104)(59,93,160,105)(60,94,141,106), (1,119)(2,110)(3,101)(4,112)(5,103)(6,114)(7,105)(8,116)(9,107)(10,118)(11,109)(12,120)(13,111)(14,102)(15,113)(16,104)(17,115)(18,106)(19,117)(20,108)(21,98)(22,89)(23,100)(24,91)(25,82)(26,93)(27,84)(28,95)(29,86)(30,97)(31,88)(32,99)(33,90)(34,81)(35,92)(36,83)(37,94)(38,85)(39,96)(40,87)(41,64)(42,75)(43,66)(44,77)(45,68)(46,79)(47,70)(48,61)(49,72)(50,63)(51,74)(52,65)(53,76)(54,67)(55,78)(56,69)(57,80)(58,71)(59,62)(60,73)(121,160)(122,151)(123,142)(124,153)(125,144)(126,155)(127,146)(128,157)(129,148)(130,159)(131,150)(132,141)(133,152)(134,143)(135,154)(136,145)(137,156)(138,147)(139,158)(140,149) );
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)], [(1,53,11,43),(2,52,12,42),(3,51,13,41),(4,50,14,60),(5,49,15,59),(6,48,16,58),(7,47,17,57),(8,46,18,56),(9,45,19,55),(10,44,20,54),(21,153,31,143),(22,152,32,142),(23,151,33,141),(24,150,34,160),(25,149,35,159),(26,148,36,158),(27,147,37,157),(28,146,38,156),(29,145,39,155),(30,144,40,154),(61,114,71,104),(62,113,72,103),(63,112,73,102),(64,111,74,101),(65,110,75,120),(66,109,76,119),(67,108,77,118),(68,107,78,117),(69,106,79,116),(70,105,80,115),(81,131,91,121),(82,130,92,140),(83,129,93,139),(84,128,94,138),(85,127,95,137),(86,126,96,136),(87,125,97,135),(88,124,98,134),(89,123,99,133),(90,122,100,132)], [(1,135,40,76),(2,136,21,77),(3,137,22,78),(4,138,23,79),(5,139,24,80),(6,140,25,61),(7,121,26,62),(8,122,27,63),(9,123,28,64),(10,124,29,65),(11,125,30,66),(12,126,31,67),(13,127,32,68),(14,128,33,69),(15,129,34,70),(16,130,35,71),(17,131,36,72),(18,132,37,73),(19,133,38,74),(20,134,39,75),(41,95,142,107),(42,96,143,108),(43,97,144,109),(44,98,145,110),(45,99,146,111),(46,100,147,112),(47,81,148,113),(48,82,149,114),(49,83,150,115),(50,84,151,116),(51,85,152,117),(52,86,153,118),(53,87,154,119),(54,88,155,120),(55,89,156,101),(56,90,157,102),(57,91,158,103),(58,92,159,104),(59,93,160,105),(60,94,141,106)], [(1,119),(2,110),(3,101),(4,112),(5,103),(6,114),(7,105),(8,116),(9,107),(10,118),(11,109),(12,120),(13,111),(14,102),(15,113),(16,104),(17,115),(18,106),(19,117),(20,108),(21,98),(22,89),(23,100),(24,91),(25,82),(26,93),(27,84),(28,95),(29,86),(30,97),(31,88),(32,99),(33,90),(34,81),(35,92),(36,83),(37,94),(38,85),(39,96),(40,87),(41,64),(42,75),(43,66),(44,77),(45,68),(46,79),(47,70),(48,61),(49,72),(50,63),(51,74),(52,65),(53,76),(54,67),(55,78),(56,69),(57,80),(58,71),(59,62),(60,73),(121,160),(122,151),(123,142),(124,153),(125,144),(126,155),(127,146),(128,157),(129,148),(130,159),(131,150),(132,141),(133,152),(134,143),(135,154),(136,145),(137,156),(138,147),(139,158),(140,149)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 4N | 4O | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 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 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | 2+ 1+4 | D4×D5 | D4⋊2D5 | D4⋊6D10 |
| kernel | Dic10⋊11D4 | C4×Dic10 | C4×D20 | D4×Dic5 | C20⋊2D4 | Dic5⋊D4 | C20⋊D4 | C5×C4⋊1D4 | C2×D4⋊2D5 | Dic10 | C4⋊1D4 | C20 | C42 | C2×D4 | C10 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 2 | 4 | 2 | 4 | 2 | 12 | 1 | 4 | 4 | 4 |
Matrix representation of Dic10⋊11D4 ►in GL6(𝔽41)
| 1 | 40 | 0 | 0 | 0 | 0 |
| 36 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 32 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 35 | 40 | 0 | 0 | 0 | 0 |
| 35 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 2 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 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 | 40 | 5 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 36 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [1,36,0,0,0,0,40,6,0,0,0,0,0,0,32,32,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[35,35,0,0,0,0,40,6,0,0,0,0,0,0,40,40,0,0,0,0,2,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,16,0,0,0,0,5,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,36,40] >;
Dic10⋊11D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_{11}D_4 % in TeX
G:=Group("Dic10:11D4"); // GroupNames label
G:=SmallGroup(320,1390);
// by ID
G=gap.SmallGroup(320,1390);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,100,675,570,185,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^11,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations