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## G = SD16×C21order 336 = 24·3·7

### Direct product of C21 and SD16

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — SD16×C21
 Chief series C1 — C2 — C4 — C28 — C84 — Q8×C21 — SD16×C21
 Lower central C1 — C2 — C4 — SD16×C21
 Upper central C1 — C42 — C84 — SD16×C21

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)
(1 136 74 117 88 38 49 152)(2 137 75 118 89 39 50 153)(3 138 76 119 90 40 51 154)(4 139 77 120 91 41 52 155)(5 140 78 121 92 42 53 156)(6 141 79 122 93 22 54 157)(7 142 80 123 94 23 55 158)(8 143 81 124 95 24 56 159)(9 144 82 125 96 25 57 160)(10 145 83 126 97 26 58 161)(11 146 84 106 98 27 59 162)(12 147 64 107 99 28 60 163)(13 127 65 108 100 29 61 164)(14 128 66 109 101 30 62 165)(15 129 67 110 102 31 63 166)(16 130 68 111 103 32 43 167)(17 131 69 112 104 33 44 168)(18 132 70 113 105 34 45 148)(19 133 71 114 85 35 46 149)(20 134 72 115 86 36 47 150)(21 135 73 116 87 37 48 151)
(22 157)(23 158)(24 159)(25 160)(26 161)(27 162)(28 163)(29 164)(30 165)(31 166)(32 167)(33 168)(34 148)(35 149)(36 150)(37 151)(38 152)(39 153)(40 154)(41 155)(42 156)(43 68)(44 69)(45 70)(46 71)(47 72)(48 73)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 81)(57 82)(58 83)(59 84)(60 64)(61 65)(62 66)(63 67)(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), (1,136,74,117,88,38,49,152)(2,137,75,118,89,39,50,153)(3,138,76,119,90,40,51,154)(4,139,77,120,91,41,52,155)(5,140,78,121,92,42,53,156)(6,141,79,122,93,22,54,157)(7,142,80,123,94,23,55,158)(8,143,81,124,95,24,56,159)(9,144,82,125,96,25,57,160)(10,145,83,126,97,26,58,161)(11,146,84,106,98,27,59,162)(12,147,64,107,99,28,60,163)(13,127,65,108,100,29,61,164)(14,128,66,109,101,30,62,165)(15,129,67,110,102,31,63,166)(16,130,68,111,103,32,43,167)(17,131,69,112,104,33,44,168)(18,132,70,113,105,34,45,148)(19,133,71,114,85,35,46,149)(20,134,72,115,86,36,47,150)(21,135,73,116,87,37,48,151), (22,157)(23,158)(24,159)(25,160)(26,161)(27,162)(28,163)(29,164)(30,165)(31,166)(32,167)(33,168)(34,148)(35,149)(36,150)(37,151)(38,152)(39,153)(40,154)(41,155)(42,156)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,81)(57,82)(58,83)(59,84)(60,64)(61,65)(62,66)(63,67)(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), (1,136,74,117,88,38,49,152)(2,137,75,118,89,39,50,153)(3,138,76,119,90,40,51,154)(4,139,77,120,91,41,52,155)(5,140,78,121,92,42,53,156)(6,141,79,122,93,22,54,157)(7,142,80,123,94,23,55,158)(8,143,81,124,95,24,56,159)(9,144,82,125,96,25,57,160)(10,145,83,126,97,26,58,161)(11,146,84,106,98,27,59,162)(12,147,64,107,99,28,60,163)(13,127,65,108,100,29,61,164)(14,128,66,109,101,30,62,165)(15,129,67,110,102,31,63,166)(16,130,68,111,103,32,43,167)(17,131,69,112,104,33,44,168)(18,132,70,113,105,34,45,148)(19,133,71,114,85,35,46,149)(20,134,72,115,86,36,47,150)(21,135,73,116,87,37,48,151), (22,157)(23,158)(24,159)(25,160)(26,161)(27,162)(28,163)(29,164)(30,165)(31,166)(32,167)(33,168)(34,148)(35,149)(36,150)(37,151)(38,152)(39,153)(40,154)(41,155)(42,156)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,81)(57,82)(58,83)(59,84)(60,64)(61,65)(62,66)(63,67)(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)], [(1,136,74,117,88,38,49,152),(2,137,75,118,89,39,50,153),(3,138,76,119,90,40,51,154),(4,139,77,120,91,41,52,155),(5,140,78,121,92,42,53,156),(6,141,79,122,93,22,54,157),(7,142,80,123,94,23,55,158),(8,143,81,124,95,24,56,159),(9,144,82,125,96,25,57,160),(10,145,83,126,97,26,58,161),(11,146,84,106,98,27,59,162),(12,147,64,107,99,28,60,163),(13,127,65,108,100,29,61,164),(14,128,66,109,101,30,62,165),(15,129,67,110,102,31,63,166),(16,130,68,111,103,32,43,167),(17,131,69,112,104,33,44,168),(18,132,70,113,105,34,45,148),(19,133,71,114,85,35,46,149),(20,134,72,115,86,36,47,150),(21,135,73,116,87,37,48,151)], [(22,157),(23,158),(24,159),(25,160),(26,161),(27,162),(28,163),(29,164),(30,165),(31,166),(32,167),(33,168),(34,148),(35,149),(36,150),(37,151),(38,152),(39,153),(40,154),(41,155),(42,156),(43,68),(44,69),(45,70),(46,71),(47,72),(48,73),(49,74),(50,75),(51,76),(52,77),(53,78),(54,79),(55,80),(56,81),(57,82),(58,83),(59,84),(60,64),(61,65),(62,66),(63,67),(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

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