metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊21D4, C10.762- 1+4, C5⋊3(D4×Q8), C5⋊D4⋊1Q8, C20⋊Q8⋊25C2, D10⋊6(C2×Q8), C22⋊Q8⋊9D5, C22⋊1(Q8×D5), Dic5⋊4(C2×Q8), C4.113(D4×D5), C4⋊C4.190D10, C20.236(C2×D4), D10⋊Q8⋊19C2, D10⋊2Q8⋊26C2, (C2×C20).55C23, (C2×Q8).127D10, C22⋊C4.58D10, Dic5.48(C2×D4), C10.78(C22×D4), Dic5⋊Q8⋊15C2, Dic5⋊3Q8⋊25C2, C10.35(C22×Q8), (C2×C10).176C24, Dic5⋊4D4.1C2, (C22×C4).238D10, (C22×Dic10)⋊17C2, C4⋊Dic5.216C22, (Q8×C10).108C22, C22.197(C23×D5), C23.190(C22×D5), Dic5.14D4⋊23C2, (C22×C10).204C23, (C22×C20).256C22, (C4×Dic5).114C22, (C2×Dic5).245C23, C10.D4.28C22, (C22×D5).208C23, C2.36(D4.10D10), C23.D5.117C22, D10⋊C4.107C22, (C2×Dic10).256C22, (C22×Dic5).118C22, (C2×Q8×D5)⋊7C2, C2.51(C2×D4×D5), C2.18(C2×Q8×D5), (C2×C10)⋊3(C2×Q8), (C4×C5⋊D4).7C2, (C5×C22⋊Q8)⋊12C2, (C2×C4×D5).105C22, (C2×C4).49(C22×D5), (C5×C4⋊C4).159C22, (C2×C5⋊D4).132C22, (C5×C22⋊C4).31C22, SmallGroup(320,1304)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊21D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, cac-1=a9, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 934 in 280 conjugacy classes, 115 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C4×D4, C4×Q8, C22⋊Q8, C22⋊Q8, C4⋊Q8, C22×Q8, Dic10, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×Q8, C22×D5, C22×C10, D4×Q8, C4×Dic5, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, Q8×D5, C22×Dic5, C2×C5⋊D4, C22×C20, Q8×C10, Dic5.14D4, Dic5⋊4D4, Dic5⋊3Q8, C20⋊Q8, D10⋊Q8, D10⋊2Q8, C4×C5⋊D4, Dic5⋊Q8, C5×C22⋊Q8, C22×Dic10, C2×Q8×D5, Dic10⋊21D4
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C24, D10, C22×D4, C22×Q8, 2- 1+4, C22×D5, D4×Q8, D4×D5, Q8×D5, C23×D5, C2×D4×D5, C2×Q8×D5, D4.10D10, Dic10⋊21D4
►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)
(1 121 11 131)(2 140 12 130)(3 139 13 129)(4 138 14 128)(5 137 15 127)(6 136 16 126)(7 135 17 125)(8 134 18 124)(9 133 19 123)(10 132 20 122)(21 119 31 109)(22 118 32 108)(23 117 33 107)(24 116 34 106)(25 115 35 105)(26 114 36 104)(27 113 37 103)(28 112 38 102)(29 111 39 101)(30 110 40 120)(41 96 51 86)(42 95 52 85)(43 94 53 84)(44 93 54 83)(45 92 55 82)(46 91 56 81)(47 90 57 100)(48 89 58 99)(49 88 59 98)(50 87 60 97)(61 155 71 145)(62 154 72 144)(63 153 73 143)(64 152 74 142)(65 151 75 141)(66 150 76 160)(67 149 77 159)(68 148 78 158)(69 147 79 157)(70 146 80 156)
(1 64 99 27)(2 73 100 36)(3 62 81 25)(4 71 82 34)(5 80 83 23)(6 69 84 32)(7 78 85 21)(8 67 86 30)(9 76 87 39)(10 65 88 28)(11 74 89 37)(12 63 90 26)(13 72 91 35)(14 61 92 24)(15 70 93 33)(16 79 94 22)(17 68 95 31)(18 77 96 40)(19 66 97 29)(20 75 98 38)(41 110 134 149)(42 119 135 158)(43 108 136 147)(44 117 137 156)(45 106 138 145)(46 115 139 154)(47 104 140 143)(48 113 121 152)(49 102 122 141)(50 111 123 150)(51 120 124 159)(52 109 125 148)(53 118 126 157)(54 107 127 146)(55 116 128 155)(56 105 129 144)(57 114 130 153)(58 103 131 142)(59 112 132 151)(60 101 133 160)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 21)(16 22)(17 23)(18 24)(19 25)(20 26)(41 145)(42 146)(43 147)(44 148)(45 149)(46 150)(47 151)(48 152)(49 153)(50 154)(51 155)(52 156)(53 157)(54 158)(55 159)(56 160)(57 141)(58 142)(59 143)(60 144)(61 96)(62 97)(63 98)(64 99)(65 100)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)(101 129)(102 130)(103 131)(104 132)(105 133)(106 134)(107 135)(108 136)(109 137)(110 138)(111 139)(112 140)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 127)(120 128)
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), (1,121,11,131)(2,140,12,130)(3,139,13,129)(4,138,14,128)(5,137,15,127)(6,136,16,126)(7,135,17,125)(8,134,18,124)(9,133,19,123)(10,132,20,122)(21,119,31,109)(22,118,32,108)(23,117,33,107)(24,116,34,106)(25,115,35,105)(26,114,36,104)(27,113,37,103)(28,112,38,102)(29,111,39,101)(30,110,40,120)(41,96,51,86)(42,95,52,85)(43,94,53,84)(44,93,54,83)(45,92,55,82)(46,91,56,81)(47,90,57,100)(48,89,58,99)(49,88,59,98)(50,87,60,97)(61,155,71,145)(62,154,72,144)(63,153,73,143)(64,152,74,142)(65,151,75,141)(66,150,76,160)(67,149,77,159)(68,148,78,158)(69,147,79,157)(70,146,80,156), (1,64,99,27)(2,73,100,36)(3,62,81,25)(4,71,82,34)(5,80,83,23)(6,69,84,32)(7,78,85,21)(8,67,86,30)(9,76,87,39)(10,65,88,28)(11,74,89,37)(12,63,90,26)(13,72,91,35)(14,61,92,24)(15,70,93,33)(16,79,94,22)(17,68,95,31)(18,77,96,40)(19,66,97,29)(20,75,98,38)(41,110,134,149)(42,119,135,158)(43,108,136,147)(44,117,137,156)(45,106,138,145)(46,115,139,154)(47,104,140,143)(48,113,121,152)(49,102,122,141)(50,111,123,150)(51,120,124,159)(52,109,125,148)(53,118,126,157)(54,107,127,146)(55,116,128,155)(56,105,129,144)(57,114,130,153)(58,103,131,142)(59,112,132,151)(60,101,133,160), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,21)(16,22)(17,23)(18,24)(19,25)(20,26)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,155)(52,156)(53,157)(54,158)(55,159)(56,160)(57,141)(58,142)(59,143)(60,144)(61,96)(62,97)(63,98)(64,99)(65,100)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(101,129)(102,130)(103,131)(104,132)(105,133)(106,134)(107,135)(108,136)(109,137)(110,138)(111,139)(112,140)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)>;
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), (1,121,11,131)(2,140,12,130)(3,139,13,129)(4,138,14,128)(5,137,15,127)(6,136,16,126)(7,135,17,125)(8,134,18,124)(9,133,19,123)(10,132,20,122)(21,119,31,109)(22,118,32,108)(23,117,33,107)(24,116,34,106)(25,115,35,105)(26,114,36,104)(27,113,37,103)(28,112,38,102)(29,111,39,101)(30,110,40,120)(41,96,51,86)(42,95,52,85)(43,94,53,84)(44,93,54,83)(45,92,55,82)(46,91,56,81)(47,90,57,100)(48,89,58,99)(49,88,59,98)(50,87,60,97)(61,155,71,145)(62,154,72,144)(63,153,73,143)(64,152,74,142)(65,151,75,141)(66,150,76,160)(67,149,77,159)(68,148,78,158)(69,147,79,157)(70,146,80,156), (1,64,99,27)(2,73,100,36)(3,62,81,25)(4,71,82,34)(5,80,83,23)(6,69,84,32)(7,78,85,21)(8,67,86,30)(9,76,87,39)(10,65,88,28)(11,74,89,37)(12,63,90,26)(13,72,91,35)(14,61,92,24)(15,70,93,33)(16,79,94,22)(17,68,95,31)(18,77,96,40)(19,66,97,29)(20,75,98,38)(41,110,134,149)(42,119,135,158)(43,108,136,147)(44,117,137,156)(45,106,138,145)(46,115,139,154)(47,104,140,143)(48,113,121,152)(49,102,122,141)(50,111,123,150)(51,120,124,159)(52,109,125,148)(53,118,126,157)(54,107,127,146)(55,116,128,155)(56,105,129,144)(57,114,130,153)(58,103,131,142)(59,112,132,151)(60,101,133,160), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,21)(16,22)(17,23)(18,24)(19,25)(20,26)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,155)(52,156)(53,157)(54,158)(55,159)(56,160)(57,141)(58,142)(59,143)(60,144)(61,96)(62,97)(63,98)(64,99)(65,100)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(101,129)(102,130)(103,131)(104,132)(105,133)(106,134)(107,135)(108,136)(109,137)(110,138)(111,139)(112,140)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128) );
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)], [(1,121,11,131),(2,140,12,130),(3,139,13,129),(4,138,14,128),(5,137,15,127),(6,136,16,126),(7,135,17,125),(8,134,18,124),(9,133,19,123),(10,132,20,122),(21,119,31,109),(22,118,32,108),(23,117,33,107),(24,116,34,106),(25,115,35,105),(26,114,36,104),(27,113,37,103),(28,112,38,102),(29,111,39,101),(30,110,40,120),(41,96,51,86),(42,95,52,85),(43,94,53,84),(44,93,54,83),(45,92,55,82),(46,91,56,81),(47,90,57,100),(48,89,58,99),(49,88,59,98),(50,87,60,97),(61,155,71,145),(62,154,72,144),(63,153,73,143),(64,152,74,142),(65,151,75,141),(66,150,76,160),(67,149,77,159),(68,148,78,158),(69,147,79,157),(70,146,80,156)], [(1,64,99,27),(2,73,100,36),(3,62,81,25),(4,71,82,34),(5,80,83,23),(6,69,84,32),(7,78,85,21),(8,67,86,30),(9,76,87,39),(10,65,88,28),(11,74,89,37),(12,63,90,26),(13,72,91,35),(14,61,92,24),(15,70,93,33),(16,79,94,22),(17,68,95,31),(18,77,96,40),(19,66,97,29),(20,75,98,38),(41,110,134,149),(42,119,135,158),(43,108,136,147),(44,117,137,156),(45,106,138,145),(46,115,139,154),(47,104,140,143),(48,113,121,152),(49,102,122,141),(50,111,123,150),(51,120,124,159),(52,109,125,148),(53,118,126,157),(54,107,127,146),(55,116,128,155),(56,105,129,144),(57,114,130,153),(58,103,131,142),(59,112,132,151),(60,101,133,160)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,37),(12,38),(13,39),(14,40),(15,21),(16,22),(17,23),(18,24),(19,25),(20,26),(41,145),(42,146),(43,147),(44,148),(45,149),(46,150),(47,151),(48,152),(49,153),(50,154),(51,155),(52,156),(53,157),(54,158),(55,159),(56,160),(57,141),(58,142),(59,143),(60,144),(61,96),(62,97),(63,98),(64,99),(65,100),(66,81),(67,82),(68,83),(69,84),(70,85),(71,86),(72,87),(73,88),(74,89),(75,90),(76,91),(77,92),(78,93),(79,94),(80,95),(101,129),(102,130),(103,131),(104,132),(105,133),(106,134),(107,135),(108,136),(109,137),(110,138),(111,139),(112,140),(113,121),(114,122),(115,123),(116,124),(117,125),(118,126),(119,127),(120,128)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4G | 4H | ··· | 4M | 4N | 4O | 4P | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20P |
| order | 1 | 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 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 2 | 2 | 4 | ··· | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 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 | Q8 | D5 | D10 | D10 | D10 | D10 | 2- 1+4 | D4×D5 | Q8×D5 | D4.10D10 |
| kernel | Dic10⋊21D4 | Dic5.14D4 | Dic5⋊4D4 | Dic5⋊3Q8 | C20⋊Q8 | D10⋊Q8 | D10⋊2Q8 | C4×C5⋊D4 | Dic5⋊Q8 | C5×C22⋊Q8 | C22×Dic10 | C2×Q8×D5 | Dic10 | C5⋊D4 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C10 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 4 | 6 | 2 | 2 | 1 | 4 | 4 | 4 |
Matrix representation of Dic10⋊21D4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 5 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 30 |
| 0 | 0 | 0 | 0 | 30 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 9 | 5 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 40 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 9 | 5 | 0 | 0 | 0 | 0 |
| 25 | 32 | 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 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,30,0,0,0,0,30,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,6,6,0,0,0,0,1,35,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[9,0,0,0,0,0,5,32,0,0,0,0,0,0,35,35,0,0,0,0,40,6,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,25,0,0,0,0,5,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;
Dic10⋊21D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_{21}D_4 % in TeX
G:=Group("Dic10:21D4"); // GroupNames label
G:=SmallGroup(320,1304);
// by ID
G=gap.SmallGroup(320,1304);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,100,570,185,80,12550]);
// 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^9,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations