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## G = S3×C7⋊C8order 336 = 24·3·7

### Direct product of S3 and C7⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — S3×C7⋊C8
 Chief series C1 — C7 — C21 — C42 — C84 — C3×C7⋊C8 — S3×C7⋊C8
 Lower central C21 — S3×C7⋊C8
 Upper central C1 — C4

Generators and relations for S3×C7⋊C8
G = < a,b,c,d | a3=b2=c7=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Smallest permutation representation of S3×C7⋊C8
On 168 points
Generators in S168
(1 37 82)(2 38 83)(3 39 84)(4 40 85)(5 33 86)(6 34 87)(7 35 88)(8 36 81)(9 23 155)(10 24 156)(11 17 157)(12 18 158)(13 19 159)(14 20 160)(15 21 153)(16 22 154)(25 112 119)(26 105 120)(27 106 113)(28 107 114)(29 108 115)(30 109 116)(31 110 117)(32 111 118)(41 56 92)(42 49 93)(43 50 94)(44 51 95)(45 52 96)(46 53 89)(47 54 90)(48 55 91)(57 79 66)(58 80 67)(59 73 68)(60 74 69)(61 75 70)(62 76 71)(63 77 72)(64 78 65)(97 145 163)(98 146 164)(99 147 165)(100 148 166)(101 149 167)(102 150 168)(103 151 161)(104 152 162)(121 139 134)(122 140 135)(123 141 136)(124 142 129)(125 143 130)(126 144 131)(127 137 132)(128 138 133)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 153)(18 154)(19 155)(20 156)(21 157)(22 158)(23 159)(24 160)(25 29)(26 30)(27 31)(28 32)(33 82)(34 83)(35 84)(36 85)(37 86)(38 87)(39 88)(40 81)(41 45)(42 46)(43 47)(44 48)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(57 61)(58 62)(59 63)(60 64)(65 74)(66 75)(67 76)(68 77)(69 78)(70 79)(71 80)(72 73)(97 149)(98 150)(99 151)(100 152)(101 145)(102 146)(103 147)(104 148)(105 116)(106 117)(107 118)(108 119)(109 120)(110 113)(111 114)(112 115)(121 130)(122 131)(123 132)(124 133)(125 134)(126 135)(127 136)(128 129)(137 141)(138 142)(139 143)(140 144)(161 165)(162 166)(163 167)(164 168)
(1 46 30 140 63 15 165)(2 166 16 64 141 31 47)(3 48 32 142 57 9 167)(4 168 10 58 143 25 41)(5 42 26 144 59 11 161)(6 162 12 60 137 27 43)(7 44 28 138 61 13 163)(8 164 14 62 139 29 45)(17 103 33 49 105 131 73)(18 74 132 106 50 34 104)(19 97 35 51 107 133 75)(20 76 134 108 52 36 98)(21 99 37 53 109 135 77)(22 78 136 110 54 38 100)(23 101 39 55 111 129 79)(24 80 130 112 56 40 102)(65 123 117 90 83 148 154)(66 155 149 84 91 118 124)(67 125 119 92 85 150 156)(68 157 151 86 93 120 126)(69 127 113 94 87 152 158)(70 159 145 88 95 114 128)(71 121 115 96 81 146 160)(72 153 147 82 89 116 122)
(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)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 14A 14B 14C 14D ··· 14I 21A 21B 21C 24A 24B 24C 24D 28A ··· 28F 28G ··· 28L 42A 42B 42C 84A ··· 84F order 1 2 2 2 3 4 4 4 4 6 7 7 7 8 8 8 8 8 8 8 8 12 12 14 14 14 14 ··· 14 21 21 21 24 24 24 24 28 ··· 28 28 ··· 28 42 42 42 84 ··· 84 size 1 1 3 3 2 1 1 3 3 2 2 2 2 7 7 7 7 21 21 21 21 2 2 2 2 2 6 ··· 6 4 4 4 14 14 14 14 2 ··· 2 6 ··· 6 4 4 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + - + - + - image C1 C2 C2 C2 C4 C4 C8 S3 D6 D7 C4×S3 Dic7 D14 Dic7 S3×C8 C7⋊C8 S3×D7 S3×Dic7 S3×C7⋊C8 kernel S3×C7⋊C8 C3×C7⋊C8 C21⋊C8 S3×C28 C7×Dic3 S3×C14 S3×C7 C7⋊C8 C28 C4×S3 C14 Dic3 C12 D6 C7 S3 C4 C2 C1 # reps 1 1 1 1 2 2 8 1 1 3 2 3 3 3 4 12 3 3 6

Matrix representation of S3×C7⋊C8 in GL4(𝔽337) generated by

 0 336 0 0 1 336 0 0 0 0 1 0 0 0 0 1
,
 0 336 0 0 336 0 0 0 0 0 336 0 0 0 0 336
,
 1 0 0 0 0 1 0 0 0 0 228 336 0 0 229 336
,
 111 0 0 0 0 111 0 0 0 0 304 169 0 0 140 33
G:=sub<GL(4,GF(337))| [0,1,0,0,336,336,0,0,0,0,1,0,0,0,0,1],[0,336,0,0,336,0,0,0,0,0,336,0,0,0,0,336],[1,0,0,0,0,1,0,0,0,0,228,229,0,0,336,336],[111,0,0,0,0,111,0,0,0,0,304,140,0,0,169,33] >;

S3×C7⋊C8 in GAP, Magma, Sage, TeX

S_3\times C_7\rtimes C_8
% in TeX

G:=Group("S3xC7:C8");
// GroupNames label

G:=SmallGroup(336,24);
// by ID

G=gap.SmallGroup(336,24);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,24,50,490,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^7=d^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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