metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊3SD16, Dic10⋊9D4, C42.75D10, C4.54(D4×D5), C4⋊2(D4.D5), C20⋊3C8⋊32C2, C5⋊4(C4⋊SD16), C4⋊1D4.5D5, C20.33(C2×D4), (C2×D4).58D10, (C2×C20).149D4, (C4×Dic10)⋊22C2, C20.77(C4○D4), D4⋊Dic5⋊23C2, C4.4(D4⋊2D5), C10.58(C2×SD16), C2.13(C20⋊2D4), C10.96(C8⋊C22), (C4×C20).123C22, (C2×C20).393C23, (D4×C10).74C22, C10.104(C4⋊D4), C4⋊Dic5.345C22, C2.17(D4.D10), (C2×Dic10).282C22, (C2×D4.D5)⋊14C2, (C5×C4⋊1D4).4C2, C2.12(C2×D4.D5), (C2×C10).524(C2×D4), (C2×C4).186(C5⋊D4), (C2×C4).491(C22×D5), C22.197(C2×C5⋊D4), (C2×C5⋊2C8).132C22, SmallGroup(320,702)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊9D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, ac=ca, dad=a11, bc=cb, dbd=a5b, dcd=c-1 >
Subgroups: 438 in 128 conjugacy classes, 45 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4 [×3], C2×C4 [×2], D4 [×8], Q8 [×3], C23 [×2], C10 [×3], C10 [×2], C42, C42, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C2×D4 [×2], C2×D4 [×2], C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], C20, C2×C10, C2×C10 [×6], D4⋊C4 [×2], C4⋊C8, C4×Q8, C4⋊1D4, C2×SD16 [×2], C5⋊2C8 [×2], Dic10 [×2], Dic10, C2×Dic5 [×2], C2×C20 [×3], C5×D4 [×8], C22×C10 [×2], C4⋊SD16, C2×C5⋊2C8 [×2], C4×Dic5, C10.D4, C4⋊Dic5, D4.D5 [×4], C4×C20, C2×Dic10, D4×C10 [×2], D4×C10 [×2], C20⋊3C8, D4⋊Dic5 [×2], C4×Dic10, C2×D4.D5 [×2], C5×C4⋊1D4, Dic10⋊9D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, SD16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×SD16, C8⋊C22, C5⋊D4 [×2], C22×D5, C4⋊SD16, D4.D5 [×2], D4×D5, D4⋊2D5, C2×C5⋊D4, D4.D10, C2×D4.D5, C20⋊2D4, Dic10⋊9D4
►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 68 11 78)(2 67 12 77)(3 66 13 76)(4 65 14 75)(5 64 15 74)(6 63 16 73)(7 62 17 72)(8 61 18 71)(9 80 19 70)(10 79 20 69)(21 149 31 159)(22 148 32 158)(23 147 33 157)(24 146 34 156)(25 145 35 155)(26 144 36 154)(27 143 37 153)(28 142 38 152)(29 141 39 151)(30 160 40 150)(41 113 51 103)(42 112 52 102)(43 111 53 101)(44 110 54 120)(45 109 55 119)(46 108 56 118)(47 107 57 117)(48 106 58 116)(49 105 59 115)(50 104 60 114)(81 126 91 136)(82 125 92 135)(83 124 93 134)(84 123 94 133)(85 122 95 132)(86 121 96 131)(87 140 97 130)(88 139 98 129)(89 138 99 128)(90 137 100 127)
(1 81 48 24)(2 82 49 25)(3 83 50 26)(4 84 51 27)(5 85 52 28)(6 86 53 29)(7 87 54 30)(8 88 55 31)(9 89 56 32)(10 90 57 33)(11 91 58 34)(12 92 59 35)(13 93 60 36)(14 94 41 37)(15 95 42 38)(16 96 43 39)(17 97 44 40)(18 98 45 21)(19 99 46 22)(20 100 47 23)(61 139 119 159)(62 140 120 160)(63 121 101 141)(64 122 102 142)(65 123 103 143)(66 124 104 144)(67 125 105 145)(68 126 106 146)(69 127 107 147)(70 128 108 148)(71 129 109 149)(72 130 110 150)(73 131 111 151)(74 132 112 152)(75 133 113 153)(76 134 114 154)(77 135 115 155)(78 136 116 156)(79 137 117 157)(80 138 118 158)
(2 12)(4 14)(6 16)(8 18)(10 20)(21 88)(22 99)(23 90)(24 81)(25 92)(26 83)(27 94)(28 85)(29 96)(30 87)(31 98)(32 89)(33 100)(34 91)(35 82)(36 93)(37 84)(38 95)(39 86)(40 97)(41 51)(43 53)(45 55)(47 57)(49 59)(61 66)(62 77)(63 68)(64 79)(65 70)(67 72)(69 74)(71 76)(73 78)(75 80)(101 106)(102 117)(103 108)(104 119)(105 110)(107 112)(109 114)(111 116)(113 118)(115 120)(121 146)(122 157)(123 148)(124 159)(125 150)(126 141)(127 152)(128 143)(129 154)(130 145)(131 156)(132 147)(133 158)(134 149)(135 160)(136 151)(137 142)(138 153)(139 144)(140 155)
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), (1,68,11,78)(2,67,12,77)(3,66,13,76)(4,65,14,75)(5,64,15,74)(6,63,16,73)(7,62,17,72)(8,61,18,71)(9,80,19,70)(10,79,20,69)(21,149,31,159)(22,148,32,158)(23,147,33,157)(24,146,34,156)(25,145,35,155)(26,144,36,154)(27,143,37,153)(28,142,38,152)(29,141,39,151)(30,160,40,150)(41,113,51,103)(42,112,52,102)(43,111,53,101)(44,110,54,120)(45,109,55,119)(46,108,56,118)(47,107,57,117)(48,106,58,116)(49,105,59,115)(50,104,60,114)(81,126,91,136)(82,125,92,135)(83,124,93,134)(84,123,94,133)(85,122,95,132)(86,121,96,131)(87,140,97,130)(88,139,98,129)(89,138,99,128)(90,137,100,127), (1,81,48,24)(2,82,49,25)(3,83,50,26)(4,84,51,27)(5,85,52,28)(6,86,53,29)(7,87,54,30)(8,88,55,31)(9,89,56,32)(10,90,57,33)(11,91,58,34)(12,92,59,35)(13,93,60,36)(14,94,41,37)(15,95,42,38)(16,96,43,39)(17,97,44,40)(18,98,45,21)(19,99,46,22)(20,100,47,23)(61,139,119,159)(62,140,120,160)(63,121,101,141)(64,122,102,142)(65,123,103,143)(66,124,104,144)(67,125,105,145)(68,126,106,146)(69,127,107,147)(70,128,108,148)(71,129,109,149)(72,130,110,150)(73,131,111,151)(74,132,112,152)(75,133,113,153)(76,134,114,154)(77,135,115,155)(78,136,116,156)(79,137,117,157)(80,138,118,158), (2,12)(4,14)(6,16)(8,18)(10,20)(21,88)(22,99)(23,90)(24,81)(25,92)(26,83)(27,94)(28,85)(29,96)(30,87)(31,98)(32,89)(33,100)(34,91)(35,82)(36,93)(37,84)(38,95)(39,86)(40,97)(41,51)(43,53)(45,55)(47,57)(49,59)(61,66)(62,77)(63,68)(64,79)(65,70)(67,72)(69,74)(71,76)(73,78)(75,80)(101,106)(102,117)(103,108)(104,119)(105,110)(107,112)(109,114)(111,116)(113,118)(115,120)(121,146)(122,157)(123,148)(124,159)(125,150)(126,141)(127,152)(128,143)(129,154)(130,145)(131,156)(132,147)(133,158)(134,149)(135,160)(136,151)(137,142)(138,153)(139,144)(140,155)>;
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), (1,68,11,78)(2,67,12,77)(3,66,13,76)(4,65,14,75)(5,64,15,74)(6,63,16,73)(7,62,17,72)(8,61,18,71)(9,80,19,70)(10,79,20,69)(21,149,31,159)(22,148,32,158)(23,147,33,157)(24,146,34,156)(25,145,35,155)(26,144,36,154)(27,143,37,153)(28,142,38,152)(29,141,39,151)(30,160,40,150)(41,113,51,103)(42,112,52,102)(43,111,53,101)(44,110,54,120)(45,109,55,119)(46,108,56,118)(47,107,57,117)(48,106,58,116)(49,105,59,115)(50,104,60,114)(81,126,91,136)(82,125,92,135)(83,124,93,134)(84,123,94,133)(85,122,95,132)(86,121,96,131)(87,140,97,130)(88,139,98,129)(89,138,99,128)(90,137,100,127), (1,81,48,24)(2,82,49,25)(3,83,50,26)(4,84,51,27)(5,85,52,28)(6,86,53,29)(7,87,54,30)(8,88,55,31)(9,89,56,32)(10,90,57,33)(11,91,58,34)(12,92,59,35)(13,93,60,36)(14,94,41,37)(15,95,42,38)(16,96,43,39)(17,97,44,40)(18,98,45,21)(19,99,46,22)(20,100,47,23)(61,139,119,159)(62,140,120,160)(63,121,101,141)(64,122,102,142)(65,123,103,143)(66,124,104,144)(67,125,105,145)(68,126,106,146)(69,127,107,147)(70,128,108,148)(71,129,109,149)(72,130,110,150)(73,131,111,151)(74,132,112,152)(75,133,113,153)(76,134,114,154)(77,135,115,155)(78,136,116,156)(79,137,117,157)(80,138,118,158), (2,12)(4,14)(6,16)(8,18)(10,20)(21,88)(22,99)(23,90)(24,81)(25,92)(26,83)(27,94)(28,85)(29,96)(30,87)(31,98)(32,89)(33,100)(34,91)(35,82)(36,93)(37,84)(38,95)(39,86)(40,97)(41,51)(43,53)(45,55)(47,57)(49,59)(61,66)(62,77)(63,68)(64,79)(65,70)(67,72)(69,74)(71,76)(73,78)(75,80)(101,106)(102,117)(103,108)(104,119)(105,110)(107,112)(109,114)(111,116)(113,118)(115,120)(121,146)(122,157)(123,148)(124,159)(125,150)(126,141)(127,152)(128,143)(129,154)(130,145)(131,156)(132,147)(133,158)(134,149)(135,160)(136,151)(137,142)(138,153)(139,144)(140,155) );
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)], [(1,68,11,78),(2,67,12,77),(3,66,13,76),(4,65,14,75),(5,64,15,74),(6,63,16,73),(7,62,17,72),(8,61,18,71),(9,80,19,70),(10,79,20,69),(21,149,31,159),(22,148,32,158),(23,147,33,157),(24,146,34,156),(25,145,35,155),(26,144,36,154),(27,143,37,153),(28,142,38,152),(29,141,39,151),(30,160,40,150),(41,113,51,103),(42,112,52,102),(43,111,53,101),(44,110,54,120),(45,109,55,119),(46,108,56,118),(47,107,57,117),(48,106,58,116),(49,105,59,115),(50,104,60,114),(81,126,91,136),(82,125,92,135),(83,124,93,134),(84,123,94,133),(85,122,95,132),(86,121,96,131),(87,140,97,130),(88,139,98,129),(89,138,99,128),(90,137,100,127)], [(1,81,48,24),(2,82,49,25),(3,83,50,26),(4,84,51,27),(5,85,52,28),(6,86,53,29),(7,87,54,30),(8,88,55,31),(9,89,56,32),(10,90,57,33),(11,91,58,34),(12,92,59,35),(13,93,60,36),(14,94,41,37),(15,95,42,38),(16,96,43,39),(17,97,44,40),(18,98,45,21),(19,99,46,22),(20,100,47,23),(61,139,119,159),(62,140,120,160),(63,121,101,141),(64,122,102,142),(65,123,103,143),(66,124,104,144),(67,125,105,145),(68,126,106,146),(69,127,107,147),(70,128,108,148),(71,129,109,149),(72,130,110,150),(73,131,111,151),(74,132,112,152),(75,133,113,153),(76,134,114,154),(77,135,115,155),(78,136,116,156),(79,137,117,157),(80,138,118,158)], [(2,12),(4,14),(6,16),(8,18),(10,20),(21,88),(22,99),(23,90),(24,81),(25,92),(26,83),(27,94),(28,85),(29,96),(30,87),(31,98),(32,89),(33,100),(34,91),(35,82),(36,93),(37,84),(38,95),(39,86),(40,97),(41,51),(43,53),(45,55),(47,57),(49,59),(61,66),(62,77),(63,68),(64,79),(65,70),(67,72),(69,74),(71,76),(73,78),(75,80),(101,106),(102,117),(103,108),(104,119),(105,110),(107,112),(109,114),(111,116),(113,118),(115,120),(121,146),(122,157),(123,148),(124,159),(125,150),(126,141),(127,152),(128,143),(129,154),(130,145),(131,156),(132,147),(133,158),(134,149),(135,160),(136,151),(137,142),(138,153),(139,144),(140,155)])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | SD16 | C4○D4 | D10 | D10 | C5⋊D4 | C8⋊C22 | D4.D5 | D4×D5 | D4⋊2D5 | D4.D10 |
| kernel | Dic10⋊9D4 | C20⋊3C8 | D4⋊Dic5 | C4×Dic10 | C2×D4.D5 | C5×C4⋊1D4 | Dic10 | C2×C20 | C4⋊1D4 | C20 | C20 | C42 | C2×D4 | C2×C4 | C10 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 1 | 4 | 2 | 2 | 4 |
Matrix representation of Dic10⋊9D4 ►in GL6(𝔽41)
| 1 | 37 | 0 | 0 | 0 | 0 |
| 21 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 0 | 0 |
| 0 | 0 | 9 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 22 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 25 | 0 | 0 |
| 0 | 0 | 2 | 22 | 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 | 16 | 18 |
| 0 | 0 | 0 | 0 | 29 | 25 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 21 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 28 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 21 | 40 |
G:=sub<GL(6,GF(41))| [1,21,0,0,0,0,37,40,0,0,0,0,0,0,31,9,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,13,0,0,0,0,22,0,0,0,0,0,0,0,19,2,0,0,0,0,25,22,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,16,29,0,0,0,0,18,25],[1,21,0,0,0,0,0,40,0,0,0,0,0,0,1,28,0,0,0,0,0,40,0,0,0,0,0,0,1,21,0,0,0,0,0,40] >;
Dic10⋊9D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_9D_4 % in TeX
G:=Group("Dic10:9D4"); // GroupNames label
G:=SmallGroup(320,702);
// by ID
G=gap.SmallGroup(320,702);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,254,219,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^11,b*c=c*b,d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations