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## G = C21⋊SD16order 336 = 24·3·7

### 4th semidirect product of C21 and SD16 acting via SD16/C4=C22

Aliases: C42.7D4, C6.8D28, C214SD16, D84.3C2, C28.24D6, C12.6D14, Dic141S3, C84.10C22, C3⋊C83D7, C4.3(S3×D7), C32(C56⋊C2), C71(Q82S3), (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

 Derived series C1 — C84 — C21⋊SD16
 Chief series C1 — C7 — C21 — C42 — C84 — C3×Dic14 — C21⋊SD16
 Lower central C21 — C42 — C84 — C21⋊SD16
 Upper central C1 — C2 — C4

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

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

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

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

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

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