metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊17D4, C4⋊C4.60D10, C5⋊3(Q8⋊D4), (C2×C10)⋊3SD16, C4⋊D4.5D5, C4.100(D4×D5), (C2×C20).73D4, (C2×D4).40D10, C20.149(C2×D4), C10.46C22≀C2, C10.Q16⋊34C2, C22⋊2(D4.D5), C10.55(C2×SD16), (C22×C10).86D4, C20.55D4⋊12C2, (C2×C20).359C23, (D4×C10).56C22, (C22×C4).122D10, C23.59(C5⋊D4), C2.14(C23⋊D10), (C22×Dic10)⋊13C2, C2.12(D4.9D10), C10.114(C8.C22), (C22×C20).163C22, (C2×Dic10).277C22, (C2×D4.D5)⋊9C2, C2.9(C2×D4.D5), (C5×C4⋊D4).4C2, (C2×C10).490(C2×D4), (C2×C4).51(C5⋊D4), (C5×C4⋊C4).107C22, (C2×C4).459(C22×D5), C22.165(C2×C5⋊D4), (C2×C5⋊2C8).110C22, SmallGroup(320,667)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊17D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, cac-1=a11, ad=da, cbc-1=a5b, bd=db, dcd=c-1 >
Subgroups: 574 in 158 conjugacy classes, 47 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C10, C10, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, Q8⋊C4, C4⋊D4, C2×SD16, C22×Q8, C5⋊2C8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, Q8⋊D4, C2×C5⋊2C8, D4.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, D4×C10, D4×C10, C10.Q16, C20.55D4, C2×D4.D5, C5×C4⋊D4, C22×Dic10, Dic10⋊17D4
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, D10, C22≀C2, C2×SD16, C8.C22, C5⋊D4, C22×D5, Q8⋊D4, D4.D5, D4×D5, C2×C5⋊D4, C2×D4.D5, C23⋊D10, D4.9D10, Dic10⋊17D4
►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 35 11 25)(2 34 12 24)(3 33 13 23)(4 32 14 22)(5 31 15 21)(6 30 16 40)(7 29 17 39)(8 28 18 38)(9 27 19 37)(10 26 20 36)(41 129 51 139)(42 128 52 138)(43 127 53 137)(44 126 54 136)(45 125 55 135)(46 124 56 134)(47 123 57 133)(48 122 58 132)(49 121 59 131)(50 140 60 130)(61 148 71 158)(62 147 72 157)(63 146 73 156)(64 145 74 155)(65 144 75 154)(66 143 76 153)(67 142 77 152)(68 141 78 151)(69 160 79 150)(70 159 80 149)(81 109 91 119)(82 108 92 118)(83 107 93 117)(84 106 94 116)(85 105 95 115)(86 104 96 114)(87 103 97 113)(88 102 98 112)(89 101 99 111)(90 120 100 110)
(1 103 137 156)(2 114 138 147)(3 105 139 158)(4 116 140 149)(5 107 121 160)(6 118 122 151)(7 109 123 142)(8 120 124 153)(9 111 125 144)(10 102 126 155)(11 113 127 146)(12 104 128 157)(13 115 129 148)(14 106 130 159)(15 117 131 150)(16 108 132 141)(17 119 133 152)(18 110 134 143)(19 101 135 154)(20 112 136 145)(21 98 49 64)(22 89 50 75)(23 100 51 66)(24 91 52 77)(25 82 53 68)(26 93 54 79)(27 84 55 70)(28 95 56 61)(29 86 57 72)(30 97 58 63)(31 88 59 74)(32 99 60 65)(33 90 41 76)(34 81 42 67)(35 92 43 78)(36 83 44 69)(37 94 45 80)(38 85 46 71)(39 96 47 62)(40 87 48 73)
(1 156)(2 157)(3 158)(4 159)(5 160)(6 141)(7 142)(8 143)(9 144)(10 145)(11 146)(12 147)(13 148)(14 149)(15 150)(16 151)(17 152)(18 153)(19 154)(20 155)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)(41 95)(42 96)(43 97)(44 98)(45 99)(46 100)(47 81)(48 82)(49 83)(50 84)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)(57 91)(58 92)(59 93)(60 94)(101 135)(102 136)(103 137)(104 138)(105 139)(106 140)(107 121)(108 122)(109 123)(110 124)(111 125)(112 126)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)
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), (1,35,11,25)(2,34,12,24)(3,33,13,23)(4,32,14,22)(5,31,15,21)(6,30,16,40)(7,29,17,39)(8,28,18,38)(9,27,19,37)(10,26,20,36)(41,129,51,139)(42,128,52,138)(43,127,53,137)(44,126,54,136)(45,125,55,135)(46,124,56,134)(47,123,57,133)(48,122,58,132)(49,121,59,131)(50,140,60,130)(61,148,71,158)(62,147,72,157)(63,146,73,156)(64,145,74,155)(65,144,75,154)(66,143,76,153)(67,142,77,152)(68,141,78,151)(69,160,79,150)(70,159,80,149)(81,109,91,119)(82,108,92,118)(83,107,93,117)(84,106,94,116)(85,105,95,115)(86,104,96,114)(87,103,97,113)(88,102,98,112)(89,101,99,111)(90,120,100,110), (1,103,137,156)(2,114,138,147)(3,105,139,158)(4,116,140,149)(5,107,121,160)(6,118,122,151)(7,109,123,142)(8,120,124,153)(9,111,125,144)(10,102,126,155)(11,113,127,146)(12,104,128,157)(13,115,129,148)(14,106,130,159)(15,117,131,150)(16,108,132,141)(17,119,133,152)(18,110,134,143)(19,101,135,154)(20,112,136,145)(21,98,49,64)(22,89,50,75)(23,100,51,66)(24,91,52,77)(25,82,53,68)(26,93,54,79)(27,84,55,70)(28,95,56,61)(29,86,57,72)(30,97,58,63)(31,88,59,74)(32,99,60,65)(33,90,41,76)(34,81,42,67)(35,92,43,78)(36,83,44,69)(37,94,45,80)(38,85,46,71)(39,96,47,62)(40,87,48,73), (1,156)(2,157)(3,158)(4,159)(5,160)(6,141)(7,142)(8,143)(9,144)(10,145)(11,146)(12,147)(13,148)(14,149)(15,150)(16,151)(17,152)(18,153)(19,154)(20,155)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,81)(48,82)(49,83)(50,84)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94)(101,135)(102,136)(103,137)(104,138)(105,139)(106,140)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)>;
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), (1,35,11,25)(2,34,12,24)(3,33,13,23)(4,32,14,22)(5,31,15,21)(6,30,16,40)(7,29,17,39)(8,28,18,38)(9,27,19,37)(10,26,20,36)(41,129,51,139)(42,128,52,138)(43,127,53,137)(44,126,54,136)(45,125,55,135)(46,124,56,134)(47,123,57,133)(48,122,58,132)(49,121,59,131)(50,140,60,130)(61,148,71,158)(62,147,72,157)(63,146,73,156)(64,145,74,155)(65,144,75,154)(66,143,76,153)(67,142,77,152)(68,141,78,151)(69,160,79,150)(70,159,80,149)(81,109,91,119)(82,108,92,118)(83,107,93,117)(84,106,94,116)(85,105,95,115)(86,104,96,114)(87,103,97,113)(88,102,98,112)(89,101,99,111)(90,120,100,110), (1,103,137,156)(2,114,138,147)(3,105,139,158)(4,116,140,149)(5,107,121,160)(6,118,122,151)(7,109,123,142)(8,120,124,153)(9,111,125,144)(10,102,126,155)(11,113,127,146)(12,104,128,157)(13,115,129,148)(14,106,130,159)(15,117,131,150)(16,108,132,141)(17,119,133,152)(18,110,134,143)(19,101,135,154)(20,112,136,145)(21,98,49,64)(22,89,50,75)(23,100,51,66)(24,91,52,77)(25,82,53,68)(26,93,54,79)(27,84,55,70)(28,95,56,61)(29,86,57,72)(30,97,58,63)(31,88,59,74)(32,99,60,65)(33,90,41,76)(34,81,42,67)(35,92,43,78)(36,83,44,69)(37,94,45,80)(38,85,46,71)(39,96,47,62)(40,87,48,73), (1,156)(2,157)(3,158)(4,159)(5,160)(6,141)(7,142)(8,143)(9,144)(10,145)(11,146)(12,147)(13,148)(14,149)(15,150)(16,151)(17,152)(18,153)(19,154)(20,155)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,81)(48,82)(49,83)(50,84)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94)(101,135)(102,136)(103,137)(104,138)(105,139)(106,140)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134) );
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)], [(1,35,11,25),(2,34,12,24),(3,33,13,23),(4,32,14,22),(5,31,15,21),(6,30,16,40),(7,29,17,39),(8,28,18,38),(9,27,19,37),(10,26,20,36),(41,129,51,139),(42,128,52,138),(43,127,53,137),(44,126,54,136),(45,125,55,135),(46,124,56,134),(47,123,57,133),(48,122,58,132),(49,121,59,131),(50,140,60,130),(61,148,71,158),(62,147,72,157),(63,146,73,156),(64,145,74,155),(65,144,75,154),(66,143,76,153),(67,142,77,152),(68,141,78,151),(69,160,79,150),(70,159,80,149),(81,109,91,119),(82,108,92,118),(83,107,93,117),(84,106,94,116),(85,105,95,115),(86,104,96,114),(87,103,97,113),(88,102,98,112),(89,101,99,111),(90,120,100,110)], [(1,103,137,156),(2,114,138,147),(3,105,139,158),(4,116,140,149),(5,107,121,160),(6,118,122,151),(7,109,123,142),(8,120,124,153),(9,111,125,144),(10,102,126,155),(11,113,127,146),(12,104,128,157),(13,115,129,148),(14,106,130,159),(15,117,131,150),(16,108,132,141),(17,119,133,152),(18,110,134,143),(19,101,135,154),(20,112,136,145),(21,98,49,64),(22,89,50,75),(23,100,51,66),(24,91,52,77),(25,82,53,68),(26,93,54,79),(27,84,55,70),(28,95,56,61),(29,86,57,72),(30,97,58,63),(31,88,59,74),(32,99,60,65),(33,90,41,76),(34,81,42,67),(35,92,43,78),(36,83,44,69),(37,94,45,80),(38,85,46,71),(39,96,47,62),(40,87,48,73)], [(1,156),(2,157),(3,158),(4,159),(5,160),(6,141),(7,142),(8,143),(9,144),(10,145),(11,146),(12,147),(13,148),(14,149),(15,150),(16,151),(17,152),(18,153),(19,154),(20,155),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,61),(34,62),(35,63),(36,64),(37,65),(38,66),(39,67),(40,68),(41,95),(42,96),(43,97),(44,98),(45,99),(46,100),(47,81),(48,82),(49,83),(50,84),(51,85),(52,86),(53,87),(54,88),(55,89),(56,90),(57,91),(58,92),(59,93),(60,94),(101,135),(102,136),(103,137),(104,138),(105,139),(106,140),(107,121),(108,122),(109,123),(110,124),(111,125),(112,126),(113,127),(114,128),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 20A | ··· | 20H | 20I | 20J | 20K | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 4 | 8 | 20 | 20 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | SD16 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | C8.C22 | D4×D5 | D4.D5 | D4.9D10 |
| kernel | Dic10⋊17D4 | C10.Q16 | C20.55D4 | C2×D4.D5 | C5×C4⋊D4 | C22×Dic10 | Dic10 | C2×C20 | C22×C10 | C4⋊D4 | C2×C10 | C4⋊C4 | C22×C4 | C2×D4 | C2×C4 | C23 | C10 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 1 | 4 | 4 | 4 |
Matrix representation of Dic10⋊17D4 ►in GL6(𝔽41)
| 40 | 32 | 0 | 0 | 0 | 0 |
| 23 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 5 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 11 | 29 | 0 | 0 | 0 | 0 |
| 17 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 6 | 0 | 0 |
| 0 | 0 | 26 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 18 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [40,23,0,0,0,0,32,1,0,0,0,0,0,0,40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,17,0,0,0,0,29,30,0,0,0,0,0,0,38,26,0,0,0,0,6,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,18,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,39,1] >;
Dic10⋊17D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_{17}D_4 % in TeX
G:=Group("Dic10:17D4"); // GroupNames label
G:=SmallGroup(320,667);
// by ID
G=gap.SmallGroup(320,667);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,254,219,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations