direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: SD16×C21, C8⋊2C42, D4.C42, C56⋊14C6, C24⋊6C14, Q8⋊2C42, C168⋊14C2, C42.55D4, C84.78C22, C4.2(C2×C42), (C7×Q8)⋊14C6, (C3×Q8)⋊4C14, (C7×D4).4C6, C6.15(C7×D4), C2.4(D4×C21), C28.41(C2×C6), (Q8×C21)⋊10C2, (C3×D4).2C14, (D4×C21).4C2, C14.31(C3×D4), C12.18(C2×C14), SmallGroup(336,112)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16×C21
G = < a,b,c | a21=b8=c2=1, ab=ba, ac=ca, cbc=b3 >
Smallest permutation representation of SD16×C21
►On 168 points
Generators in S168
(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)(148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)
(1 22 131 160 93 60 112 70)(2 23 132 161 94 61 113 71)(3 24 133 162 95 62 114 72)(4 25 134 163 96 63 115 73)(5 26 135 164 97 43 116 74)(6 27 136 165 98 44 117 75)(7 28 137 166 99 45 118 76)(8 29 138 167 100 46 119 77)(9 30 139 168 101 47 120 78)(10 31 140 148 102 48 121 79)(11 32 141 149 103 49 122 80)(12 33 142 150 104 50 123 81)(13 34 143 151 105 51 124 82)(14 35 144 152 85 52 125 83)(15 36 145 153 86 53 126 84)(16 37 146 154 87 54 106 64)(17 38 147 155 88 55 107 65)(18 39 127 156 89 56 108 66)(19 40 128 157 90 57 109 67)(20 41 129 158 91 58 110 68)(21 42 130 159 92 59 111 69)
(22 160)(23 161)(24 162)(25 163)(26 164)(27 165)(28 166)(29 167)(30 168)(31 148)(32 149)(33 150)(34 151)(35 152)(36 153)(37 154)(38 155)(39 156)(40 157)(41 158)(42 159)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 80)(50 81)(51 82)(52 83)(53 84)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(106 146)(107 147)(108 127)(109 128)(110 129)(111 130)(112 131)(113 132)(114 133)(115 134)(116 135)(117 136)(118 137)(119 138)(120 139)(121 140)(122 141)(123 142)(124 143)(125 144)(126 145)
G:=sub<Sym(168)| (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)(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,22,131,160,93,60,112,70)(2,23,132,161,94,61,113,71)(3,24,133,162,95,62,114,72)(4,25,134,163,96,63,115,73)(5,26,135,164,97,43,116,74)(6,27,136,165,98,44,117,75)(7,28,137,166,99,45,118,76)(8,29,138,167,100,46,119,77)(9,30,139,168,101,47,120,78)(10,31,140,148,102,48,121,79)(11,32,141,149,103,49,122,80)(12,33,142,150,104,50,123,81)(13,34,143,151,105,51,124,82)(14,35,144,152,85,52,125,83)(15,36,145,153,86,53,126,84)(16,37,146,154,87,54,106,64)(17,38,147,155,88,55,107,65)(18,39,127,156,89,56,108,66)(19,40,128,157,90,57,109,67)(20,41,129,158,91,58,110,68)(21,42,130,159,92,59,111,69), (22,160)(23,161)(24,162)(25,163)(26,164)(27,165)(28,166)(29,167)(30,168)(31,148)(32,149)(33,150)(34,151)(35,152)(36,153)(37,154)(38,155)(39,156)(40,157)(41,158)(42,159)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,81)(51,82)(52,83)(53,84)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,73)(106,146)(107,147)(108,127)(109,128)(110,129)(111,130)(112,131)(113,132)(114,133)(115,134)(116,135)(117,136)(118,137)(119,138)(120,139)(121,140)(122,141)(123,142)(124,143)(125,144)(126,145)>;
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)(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,22,131,160,93,60,112,70)(2,23,132,161,94,61,113,71)(3,24,133,162,95,62,114,72)(4,25,134,163,96,63,115,73)(5,26,135,164,97,43,116,74)(6,27,136,165,98,44,117,75)(7,28,137,166,99,45,118,76)(8,29,138,167,100,46,119,77)(9,30,139,168,101,47,120,78)(10,31,140,148,102,48,121,79)(11,32,141,149,103,49,122,80)(12,33,142,150,104,50,123,81)(13,34,143,151,105,51,124,82)(14,35,144,152,85,52,125,83)(15,36,145,153,86,53,126,84)(16,37,146,154,87,54,106,64)(17,38,147,155,88,55,107,65)(18,39,127,156,89,56,108,66)(19,40,128,157,90,57,109,67)(20,41,129,158,91,58,110,68)(21,42,130,159,92,59,111,69), (22,160)(23,161)(24,162)(25,163)(26,164)(27,165)(28,166)(29,167)(30,168)(31,148)(32,149)(33,150)(34,151)(35,152)(36,153)(37,154)(38,155)(39,156)(40,157)(41,158)(42,159)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,81)(51,82)(52,83)(53,84)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,73)(106,146)(107,147)(108,127)(109,128)(110,129)(111,130)(112,131)(113,132)(114,133)(115,134)(116,135)(117,136)(118,137)(119,138)(120,139)(121,140)(122,141)(123,142)(124,143)(125,144)(126,145) );
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),(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)], [(1,22,131,160,93,60,112,70),(2,23,132,161,94,61,113,71),(3,24,133,162,95,62,114,72),(4,25,134,163,96,63,115,73),(5,26,135,164,97,43,116,74),(6,27,136,165,98,44,117,75),(7,28,137,166,99,45,118,76),(8,29,138,167,100,46,119,77),(9,30,139,168,101,47,120,78),(10,31,140,148,102,48,121,79),(11,32,141,149,103,49,122,80),(12,33,142,150,104,50,123,81),(13,34,143,151,105,51,124,82),(14,35,144,152,85,52,125,83),(15,36,145,153,86,53,126,84),(16,37,146,154,87,54,106,64),(17,38,147,155,88,55,107,65),(18,39,127,156,89,56,108,66),(19,40,128,157,90,57,109,67),(20,41,129,158,91,58,110,68),(21,42,130,159,92,59,111,69)], [(22,160),(23,161),(24,162),(25,163),(26,164),(27,165),(28,166),(29,167),(30,168),(31,148),(32,149),(33,150),(34,151),(35,152),(36,153),(37,154),(38,155),(39,156),(40,157),(41,158),(42,159),(43,74),(44,75),(45,76),(46,77),(47,78),(48,79),(49,80),(50,81),(51,82),(52,83),(53,84),(54,64),(55,65),(56,66),(57,67),(58,68),(59,69),(60,70),(61,71),(62,72),(63,73),(106,146),(107,147),(108,127),(109,128),(110,129),(111,130),(112,131),(113,132),(114,133),(115,134),(116,135),(117,136),(118,137),(119,138),(120,139),(121,140),(122,141),(123,142),(124,143),(125,144),(126,145)]])
147 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 6A | 6B | 6C | 6D | 7A | ··· | 7F | 8A | 8B | 12A | 12B | 12C | 12D | 14A | ··· | 14F | 14G | ··· | 14L | 21A | ··· | 21L | 24A | 24B | 24C | 24D | 28A | ··· | 28F | 28G | ··· | 28L | 42A | ··· | 42L | 42M | ··· | 42X | 56A | ··· | 56L | 84A | ··· | 84L | 84M | ··· | 84X | 168A | ··· | 168X |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 7 | ··· | 7 | 8 | 8 | 12 | 12 | 12 | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 24 | 24 | 24 | 24 | 28 | ··· | 28 | 28 | ··· | 28 | 42 | ··· | 42 | 42 | ··· | 42 | 56 | ··· | 56 | 84 | ··· | 84 | 84 | ··· | 84 | 168 | ··· | 168 |
| size | 1 | 1 | 4 | 1 | 1 | 2 | 4 | 1 | 1 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
147 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C7 | C14 | C14 | C14 | C21 | C42 | C42 | C42 | D4 | SD16 | C3×D4 | C3×SD16 | C7×D4 | C7×SD16 | D4×C21 | SD16×C21 |
| kernel | SD16×C21 | C168 | D4×C21 | Q8×C21 | C7×SD16 | C56 | C7×D4 | C7×Q8 | C3×SD16 | C24 | C3×D4 | C3×Q8 | SD16 | C8 | D4 | Q8 | C42 | C21 | C14 | C7 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 1 | 2 | 2 | 4 | 6 | 12 | 12 | 24 |
Matrix representation of SD16×C21 ►in GL2(𝔽43) generated by
| 23 | 0 |
| 0 | 23 |
| 0 | 37 |
| 7 | 16 |
| 42 | 0 |
| 17 | 1 |
G:=sub<GL(2,GF(43))| [23,0,0,23],[0,7,37,16],[42,17,0,1] >;
SD16×C21 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\times C_{21} % in TeX
G:=Group("SD16xC21"); // GroupNames label
G:=SmallGroup(336,112);
// by ID
G=gap.SmallGroup(336,112);
# by ID
G:=PCGroup([6,-2,-2,-3,-7,-2,-2,1008,1033,7564,3790,88]);
// Polycyclic
G:=Group<a,b,c|a^21=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations
Export