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## G = C7×C8⋊S3order 336 = 24·3·7

### Direct product of C7 and C8⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C7×C8⋊S3
 Chief series C1 — C3 — C6 — C12 — C84 — S3×C28 — C7×C8⋊S3
 Lower central C3 — C6 — C7×C8⋊S3
 Upper central C1 — C28 — C56

Generators and relations for C7×C8⋊S3
G = < a,b,c,d | a7=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

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

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

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

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

126 conjugacy classes

 class 1 2A 2B 3 4A 4B 4C 6 7A ··· 7F 8A 8B 8C 8D 12A 12B 14A ··· 14F 14G ··· 14L 21A ··· 21F 24A 24B 24C 24D 28A ··· 28L 28M ··· 28R 42A ··· 42F 56A ··· 56L 56M ··· 56X 84A ··· 84L 168A ··· 168X order 1 2 2 3 4 4 4 6 7 ··· 7 8 8 8 8 12 12 14 ··· 14 14 ··· 14 21 ··· 21 24 24 24 24 28 ··· 28 28 ··· 28 42 ··· 42 56 ··· 56 56 ··· 56 84 ··· 84 168 ··· 168 size 1 1 6 2 1 1 6 2 1 ··· 1 2 2 6 6 2 2 1 ··· 1 6 ··· 6 2 ··· 2 2 2 2 2 1 ··· 1 6 ··· 6 2 ··· 2 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 C7 C14 C14 C14 C28 C28 S3 D6 M4(2) C4×S3 S3×C7 C8⋊S3 S3×C14 C7×M4(2) S3×C28 C7×C8⋊S3 kernel C7×C8⋊S3 C7×C3⋊C8 C168 S3×C28 C7×Dic3 S3×C14 C8⋊S3 C3⋊C8 C24 C4×S3 Dic3 D6 C56 C28 C21 C14 C8 C7 C4 C3 C2 C1 # reps 1 1 1 1 2 2 6 6 6 6 12 12 1 1 2 2 6 4 6 12 12 24

Matrix representation of C7×C8⋊S3 in GL2(𝔽29) generated by

 20 0 0 20
,
 23 4 17 6
,
 27 1 26 1
,
 1 28 0 28
G:=sub<GL(2,GF(29))| [20,0,0,20],[23,17,4,6],[27,26,1,1],[1,0,28,28] >;

C7×C8⋊S3 in GAP, Magma, Sage, TeX

C_7\times C_8\rtimes S_3
% in TeX

G:=Group("C7xC8:S3");
// GroupNames label

G:=SmallGroup(336,75);
// by ID

G=gap.SmallGroup(336,75);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-3,697,175,69,8069]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^8=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations

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