metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.7D4, C6.8D28, C21⋊4SD16, D84.3C2, C28.24D6, C12.6D14, Dic14⋊1S3, C84.10C22, C3⋊C8⋊3D7, C4.3(S3×D7), C3⋊2(C56⋊C2), C7⋊1(Q8⋊2S3), (C3×Dic14)⋊1C2, C14.3(C3⋊D4), C2.6(C3⋊D28), (C7×C3⋊C8)⋊3C2, SmallGroup(336,35)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C21⋊SD16
G = < a,b,c | a21=b8=c2=1, bab-1=a8, cac=a-1, cbc=b3 >
Smallest permutation representation of C21⋊SD16
►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)
(1 110 51 168 29 97 66 127)(2 118 52 155 30 105 67 135)(3 126 53 163 31 92 68 143)(4 113 54 150 32 100 69 130)(5 121 55 158 33 87 70 138)(6 108 56 166 34 95 71 146)(7 116 57 153 35 103 72 133)(8 124 58 161 36 90 73 141)(9 111 59 148 37 98 74 128)(10 119 60 156 38 85 75 136)(11 106 61 164 39 93 76 144)(12 114 62 151 40 101 77 131)(13 122 63 159 41 88 78 139)(14 109 43 167 42 96 79 147)(15 117 44 154 22 104 80 134)(16 125 45 162 23 91 81 142)(17 112 46 149 24 99 82 129)(18 120 47 157 25 86 83 137)(19 107 48 165 26 94 84 145)(20 115 49 152 27 102 64 132)(21 123 50 160 28 89 65 140)
(1 51)(2 50)(3 49)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 63)(11 62)(12 61)(13 60)(14 59)(15 58)(16 57)(17 56)(18 55)(19 54)(20 53)(21 52)(22 73)(23 72)(24 71)(25 70)(26 69)(27 68)(28 67)(29 66)(30 65)(31 64)(32 84)(33 83)(34 82)(35 81)(36 80)(37 79)(38 78)(39 77)(40 76)(41 75)(42 74)(85 122)(86 121)(87 120)(88 119)(89 118)(90 117)(91 116)(92 115)(93 114)(94 113)(95 112)(96 111)(97 110)(98 109)(99 108)(100 107)(101 106)(102 126)(103 125)(104 124)(105 123)(128 147)(129 146)(130 145)(131 144)(132 143)(133 142)(134 141)(135 140)(136 139)(137 138)(148 167)(149 166)(150 165)(151 164)(152 163)(153 162)(154 161)(155 160)(156 159)(157 158)
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), (1,110,51,168,29,97,66,127)(2,118,52,155,30,105,67,135)(3,126,53,163,31,92,68,143)(4,113,54,150,32,100,69,130)(5,121,55,158,33,87,70,138)(6,108,56,166,34,95,71,146)(7,116,57,153,35,103,72,133)(8,124,58,161,36,90,73,141)(9,111,59,148,37,98,74,128)(10,119,60,156,38,85,75,136)(11,106,61,164,39,93,76,144)(12,114,62,151,40,101,77,131)(13,122,63,159,41,88,78,139)(14,109,43,167,42,96,79,147)(15,117,44,154,22,104,80,134)(16,125,45,162,23,91,81,142)(17,112,46,149,24,99,82,129)(18,120,47,157,25,86,83,137)(19,107,48,165,26,94,84,145)(20,115,49,152,27,102,64,132)(21,123,50,160,28,89,65,140), (1,51)(2,50)(3,49)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,63)(11,62)(12,61)(13,60)(14,59)(15,58)(16,57)(17,56)(18,55)(19,54)(20,53)(21,52)(22,73)(23,72)(24,71)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,84)(33,83)(34,82)(35,81)(36,80)(37,79)(38,78)(39,77)(40,76)(41,75)(42,74)(85,122)(86,121)(87,120)(88,119)(89,118)(90,117)(91,116)(92,115)(93,114)(94,113)(95,112)(96,111)(97,110)(98,109)(99,108)(100,107)(101,106)(102,126)(103,125)(104,124)(105,123)(128,147)(129,146)(130,145)(131,144)(132,143)(133,142)(134,141)(135,140)(136,139)(137,138)(148,167)(149,166)(150,165)(151,164)(152,163)(153,162)(154,161)(155,160)(156,159)(157,158)>;
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), (1,110,51,168,29,97,66,127)(2,118,52,155,30,105,67,135)(3,126,53,163,31,92,68,143)(4,113,54,150,32,100,69,130)(5,121,55,158,33,87,70,138)(6,108,56,166,34,95,71,146)(7,116,57,153,35,103,72,133)(8,124,58,161,36,90,73,141)(9,111,59,148,37,98,74,128)(10,119,60,156,38,85,75,136)(11,106,61,164,39,93,76,144)(12,114,62,151,40,101,77,131)(13,122,63,159,41,88,78,139)(14,109,43,167,42,96,79,147)(15,117,44,154,22,104,80,134)(16,125,45,162,23,91,81,142)(17,112,46,149,24,99,82,129)(18,120,47,157,25,86,83,137)(19,107,48,165,26,94,84,145)(20,115,49,152,27,102,64,132)(21,123,50,160,28,89,65,140), (1,51)(2,50)(3,49)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,63)(11,62)(12,61)(13,60)(14,59)(15,58)(16,57)(17,56)(18,55)(19,54)(20,53)(21,52)(22,73)(23,72)(24,71)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,84)(33,83)(34,82)(35,81)(36,80)(37,79)(38,78)(39,77)(40,76)(41,75)(42,74)(85,122)(86,121)(87,120)(88,119)(89,118)(90,117)(91,116)(92,115)(93,114)(94,113)(95,112)(96,111)(97,110)(98,109)(99,108)(100,107)(101,106)(102,126)(103,125)(104,124)(105,123)(128,147)(129,146)(130,145)(131,144)(132,143)(133,142)(134,141)(135,140)(136,139)(137,138)(148,167)(149,166)(150,165)(151,164)(152,163)(153,162)(154,161)(155,160)(156,159)(157,158) );
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)], [(1,110,51,168,29,97,66,127),(2,118,52,155,30,105,67,135),(3,126,53,163,31,92,68,143),(4,113,54,150,32,100,69,130),(5,121,55,158,33,87,70,138),(6,108,56,166,34,95,71,146),(7,116,57,153,35,103,72,133),(8,124,58,161,36,90,73,141),(9,111,59,148,37,98,74,128),(10,119,60,156,38,85,75,136),(11,106,61,164,39,93,76,144),(12,114,62,151,40,101,77,131),(13,122,63,159,41,88,78,139),(14,109,43,167,42,96,79,147),(15,117,44,154,22,104,80,134),(16,125,45,162,23,91,81,142),(17,112,46,149,24,99,82,129),(18,120,47,157,25,86,83,137),(19,107,48,165,26,94,84,145),(20,115,49,152,27,102,64,132),(21,123,50,160,28,89,65,140)], [(1,51),(2,50),(3,49),(4,48),(5,47),(6,46),(7,45),(8,44),(9,43),(10,63),(11,62),(12,61),(13,60),(14,59),(15,58),(16,57),(17,56),(18,55),(19,54),(20,53),(21,52),(22,73),(23,72),(24,71),(25,70),(26,69),(27,68),(28,67),(29,66),(30,65),(31,64),(32,84),(33,83),(34,82),(35,81),(36,80),(37,79),(38,78),(39,77),(40,76),(41,75),(42,74),(85,122),(86,121),(87,120),(88,119),(89,118),(90,117),(91,116),(92,115),(93,114),(94,113),(95,112),(96,111),(97,110),(98,109),(99,108),(100,107),(101,106),(102,126),(103,125),(104,124),(105,123),(128,147),(129,146),(130,145),(131,144),(132,143),(133,142),(134,141),(135,140),(136,139),(137,138),(148,167),(149,166),(150,165),(151,164),(152,163),(153,162),(154,161),(155,160),(156,159),(157,158)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 6 | 7A | 7B | 7C | 8A | 8B | 12A | 12B | 12C | 14A | 14B | 14C | 21A | 21B | 21C | 28A | ··· | 28F | 42A | 42B | 42C | 56A | ··· | 56L | 84A | ··· | 84F |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 6 | 7 | 7 | 7 | 8 | 8 | 12 | 12 | 12 | 14 | 14 | 14 | 21 | 21 | 21 | 28 | ··· | 28 | 42 | 42 | 42 | 56 | ··· | 56 | 84 | ··· | 84 |
| size | 1 | 1 | 84 | 2 | 2 | 28 | 2 | 2 | 2 | 2 | 6 | 6 | 4 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D7 | SD16 | C3⋊D4 | D14 | D28 | C56⋊C2 | Q8⋊2S3 | S3×D7 | C3⋊D28 | C21⋊SD16 |
| kernel | C21⋊SD16 | C7×C3⋊C8 | C3×Dic14 | D84 | Dic14 | C42 | C28 | C3⋊C8 | C21 | C14 | C12 | C6 | C3 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 12 | 1 | 3 | 3 | 6 |
Matrix representation of C21⋊SD16 ►in GL4(𝔽337) generated by
| 0 | 1 | 0 | 0 |
| 336 | 336 | 0 | 0 |
| 0 | 0 | 193 | 159 |
| 0 | 0 | 143 | 34 |
| 139 | 278 | 0 | 0 |
| 139 | 198 | 0 | 0 |
| 0 | 0 | 197 | 198 |
| 0 | 0 | 159 | 336 |
| 336 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 281 | 219 |
| 0 | 0 | 18 | 56 |
G:=sub<GL(4,GF(337))| [0,336,0,0,1,336,0,0,0,0,193,143,0,0,159,34],[139,139,0,0,278,198,0,0,0,0,197,159,0,0,198,336],[336,1,0,0,0,1,0,0,0,0,281,18,0,0,219,56] >;
C21⋊SD16 in GAP, Magma, Sage, TeX
C_{21}\rtimes {\rm SD}_{16} % in TeX
G:=Group("C21:SD16"); // GroupNames label
G:=SmallGroup(336,35);
// by ID
G=gap.SmallGroup(336,35);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,48,73,31,218,50,490,10373]);
// Polycyclic
G:=Group<a,b,c|a^21=b^8=c^2=1,b*a*b^-1=a^8,c*a*c=a^-1,c*b*c=b^3>;
// generators/relations
Export