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## G = C7×Q8⋊3S3order 336 = 24·3·7

### Direct product of C7 and Q8⋊3S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C7×Q8⋊3S3
 Chief series C1 — C3 — C6 — C42 — S3×C14 — S3×C28 — C7×Q8⋊3S3
 Lower central C3 — C6 — C7×Q8⋊3S3
 Upper central C1 — C14 — C7×Q8

Generators and relations for C7×Q83S3
G = < a,b,c,d,e | a7=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 160 in 80 conjugacy classes, 46 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C7, C2×C4, D4, Q8, Dic3, C12, D6, C14, C14, C4○D4, C21, C4×S3, D12, C3×Q8, C28, C28, C2×C14, S3×C7, C42, Q83S3, C2×C28, C7×D4, C7×Q8, C7×Dic3, C84, S3×C14, C7×C4○D4, S3×C28, C7×D12, Q8×C21, C7×Q83S3
Quotients: C1, C2, C22, S3, C7, C23, D6, C14, C4○D4, C22×S3, C2×C14, S3×C7, Q83S3, C22×C14, S3×C14, C7×C4○D4, S3×C2×C14, C7×Q83S3

Smallest permutation representation of C7×Q83S3
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 93 51 78)(2 94 52 79)(3 95 53 80)(4 96 54 81)(5 97 55 82)(6 98 56 83)(7 92 50 84)(8 142 42 152)(9 143 36 153)(10 144 37 154)(11 145 38 148)(12 146 39 149)(13 147 40 150)(14 141 41 151)(15 132 25 159)(16 133 26 160)(17 127 27 161)(18 128 28 155)(19 129 22 156)(20 130 23 157)(21 131 24 158)(29 121 163 136)(30 122 164 137)(31 123 165 138)(32 124 166 139)(33 125 167 140)(34 126 168 134)(35 120 162 135)(43 119 77 85)(44 113 71 86)(45 114 72 87)(46 115 73 88)(47 116 74 89)(48 117 75 90)(49 118 76 91)(57 109 67 99)(58 110 68 100)(59 111 69 101)(60 112 70 102)(61 106 64 103)(62 107 65 104)(63 108 66 105)
(1 135 51 120)(2 136 52 121)(3 137 53 122)(4 138 54 123)(5 139 55 124)(6 140 56 125)(7 134 50 126)(8 110 42 100)(9 111 36 101)(10 112 37 102)(11 106 38 103)(12 107 39 104)(13 108 40 105)(14 109 41 99)(15 117 25 90)(16 118 26 91)(17 119 27 85)(18 113 28 86)(19 114 22 87)(20 115 23 88)(21 116 24 89)(29 94 163 79)(30 95 164 80)(31 96 165 81)(32 97 166 82)(33 98 167 83)(34 92 168 84)(35 93 162 78)(43 161 77 127)(44 155 71 128)(45 156 72 129)(46 157 73 130)(47 158 74 131)(48 159 75 132)(49 160 76 133)(57 151 67 141)(58 152 68 142)(59 153 69 143)(60 154 70 144)(61 148 64 145)(62 149 65 146)(63 150 66 147)
(1 69 43)(2 70 44)(3 64 45)(4 65 46)(5 66 47)(6 67 48)(7 68 49)(8 26 168)(9 27 162)(10 28 163)(11 22 164)(12 23 165)(13 24 166)(14 25 167)(15 33 41)(16 34 42)(17 35 36)(18 29 37)(19 30 38)(20 31 39)(21 32 40)(50 58 76)(51 59 77)(52 60 71)(53 61 72)(54 62 73)(55 63 74)(56 57 75)(78 111 85)(79 112 86)(80 106 87)(81 107 88)(82 108 89)(83 109 90)(84 110 91)(92 100 118)(93 101 119)(94 102 113)(95 103 114)(96 104 115)(97 105 116)(98 99 117)(120 153 127)(121 154 128)(122 148 129)(123 149 130)(124 150 131)(125 151 132)(126 152 133)(134 142 160)(135 143 161)(136 144 155)(137 145 156)(138 146 157)(139 147 158)(140 141 159)
(8 16)(9 17)(10 18)(11 19)(12 20)(13 21)(14 15)(22 38)(23 39)(24 40)(25 41)(26 42)(27 36)(28 37)(29 163)(30 164)(31 165)(32 166)(33 167)(34 168)(35 162)(43 69)(44 70)(45 64)(46 65)(47 66)(48 67)(49 68)(57 75)(58 76)(59 77)(60 71)(61 72)(62 73)(63 74)(78 93)(79 94)(80 95)(81 96)(82 97)(83 98)(84 92)(85 101)(86 102)(87 103)(88 104)(89 105)(90 99)(91 100)(106 114)(107 115)(108 116)(109 117)(110 118)(111 119)(112 113)(127 153)(128 154)(129 148)(130 149)(131 150)(132 151)(133 152)(141 159)(142 160)(143 161)(144 155)(145 156)(146 157)(147 158)

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

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

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

105 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6 7A ··· 7F 12A 12B 12C 14A ··· 14F 14G ··· 14X 21A ··· 21F 28A ··· 28R 28S ··· 28AD 42A ··· 42F 84A ··· 84R order 1 2 2 2 2 3 4 4 4 4 4 6 7 ··· 7 12 12 12 14 ··· 14 14 ··· 14 21 ··· 21 28 ··· 28 28 ··· 28 42 ··· 42 84 ··· 84 size 1 1 6 6 6 2 2 2 2 3 3 2 1 ··· 1 4 4 4 1 ··· 1 6 ··· 6 2 ··· 2 2 ··· 2 3 ··· 3 2 ··· 2 4 ··· 4

105 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C7 C14 C14 C14 S3 D6 C4○D4 S3×C7 S3×C14 C7×C4○D4 Q8⋊3S3 C7×Q8⋊3S3 kernel C7×Q8⋊3S3 S3×C28 C7×D12 Q8×C21 Q8⋊3S3 C4×S3 D12 C3×Q8 C7×Q8 C28 C21 Q8 C4 C3 C7 C1 # reps 1 3 3 1 6 18 18 6 1 3 2 6 18 12 1 6

Matrix representation of C7×Q83S3 in GL4(𝔽337) generated by

 79 0 0 0 0 79 0 0 0 0 295 0 0 0 0 295
,
 336 0 0 0 0 336 0 0 0 0 0 1 0 0 336 0
,
 1 0 0 0 0 1 0 0 0 0 189 0 0 0 0 148
,
 336 1 0 0 336 0 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 336
G:=sub<GL(4,GF(337))| [79,0,0,0,0,79,0,0,0,0,295,0,0,0,0,295],[336,0,0,0,0,336,0,0,0,0,0,336,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,189,0,0,0,0,148],[336,336,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,336] >;

C7×Q83S3 in GAP, Magma, Sage, TeX

C_7\times Q_8\rtimes_3S_3
% in TeX

G:=Group("C7xQ8:3S3");
// GroupNames label

G:=SmallGroup(336,191);
// by ID

G=gap.SmallGroup(336,191);
# by ID

G:=PCGroup([6,-2,-2,-2,-7,-2,-3,343,1082,548,266,8069]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^4=d^3=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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