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## G = S3×C7×Q8order 336 = 24·3·7

### Direct product of C7, S3 and Q8

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×C7×Q8
G = < a,b,c,d,e | a7=b3=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 128 in 76 conjugacy classes, 50 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C7, C2×C4, Q8, Q8, Dic3, C12, D6, C14, C14, C2×Q8, C21, Dic6, C4×S3, C3×Q8, C28, C28, C2×C14, S3×C7, C42, S3×Q8, C2×C28, C7×Q8, C7×Q8, C7×Dic3, C84, S3×C14, Q8×C14, C7×Dic6, S3×C28, Q8×C21, S3×C7×Q8
Quotients: C1, C2, C22, S3, C7, Q8, C23, D6, C14, C2×Q8, C22×S3, C2×C14, S3×C7, S3×Q8, C7×Q8, C22×C14, S3×C14, Q8×C14, S3×C2×C14, S3×C7×Q8

Smallest permutation representation of S3×C7×Q8
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 43 50)(2 44 51)(3 45 52)(4 46 53)(5 47 54)(6 48 55)(7 49 56)(8 15 162)(9 16 163)(10 17 164)(11 18 165)(12 19 166)(13 20 167)(14 21 168)(22 29 36)(23 30 37)(24 31 38)(25 32 39)(26 33 40)(27 34 41)(28 35 42)(57 64 71)(58 65 72)(59 66 73)(60 67 74)(61 68 75)(62 69 76)(63 70 77)(78 85 92)(79 86 93)(80 87 94)(81 88 95)(82 89 96)(83 90 97)(84 91 98)(99 106 113)(100 107 114)(101 108 115)(102 109 116)(103 110 117)(104 111 118)(105 112 119)(120 127 134)(121 128 135)(122 129 136)(123 130 137)(124 131 138)(125 132 139)(126 133 140)(141 148 155)(142 149 156)(143 150 157)(144 151 158)(145 152 159)(146 153 160)(147 154 161)
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 162)(23 163)(24 164)(25 165)(26 166)(27 167)(28 168)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)(85 113)(86 114)(87 115)(88 116)(89 117)(90 118)(91 119)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(120 141)(121 142)(122 143)(123 144)(124 145)(125 146)(126 147)(127 155)(128 156)(129 157)(130 158)(131 159)(132 160)(133 161)(134 148)(135 149)(136 150)(137 151)(138 152)(139 153)(140 154)
(1 99 57 78)(2 100 58 79)(3 101 59 80)(4 102 60 81)(5 103 61 82)(6 104 62 83)(7 105 63 84)(8 148 29 127)(9 149 30 128)(10 150 31 129)(11 151 32 130)(12 152 33 131)(13 153 34 132)(14 154 35 133)(15 155 36 134)(16 156 37 135)(17 157 38 136)(18 158 39 137)(19 159 40 138)(20 160 41 139)(21 161 42 140)(22 120 162 141)(23 121 163 142)(24 122 164 143)(25 123 165 144)(26 124 166 145)(27 125 167 146)(28 126 168 147)(43 106 64 85)(44 107 65 86)(45 108 66 87)(46 109 67 88)(47 110 68 89)(48 111 69 90)(49 112 70 91)(50 113 71 92)(51 114 72 93)(52 115 73 94)(53 116 74 95)(54 117 75 96)(55 118 76 97)(56 119 77 98)
(1 141 57 120)(2 142 58 121)(3 143 59 122)(4 144 60 123)(5 145 61 124)(6 146 62 125)(7 147 63 126)(8 85 29 106)(9 86 30 107)(10 87 31 108)(11 88 32 109)(12 89 33 110)(13 90 34 111)(14 91 35 112)(15 92 36 113)(16 93 37 114)(17 94 38 115)(18 95 39 116)(19 96 40 117)(20 97 41 118)(21 98 42 119)(22 99 162 78)(23 100 163 79)(24 101 164 80)(25 102 165 81)(26 103 166 82)(27 104 167 83)(28 105 168 84)(43 148 64 127)(44 149 65 128)(45 150 66 129)(46 151 67 130)(47 152 68 131)(48 153 69 132)(49 154 70 133)(50 155 71 134)(51 156 72 135)(52 157 73 136)(53 158 74 137)(54 159 75 138)(55 160 76 139)(56 161 77 140)

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

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

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

105 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6 7A ··· 7F 12A 12B 12C 14A ··· 14F 14G ··· 14R 21A ··· 21F 28A ··· 28R 28S ··· 28AJ 42A ··· 42F 84A ··· 84R order 1 2 2 2 3 4 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 3 3 2 2 2 2 6 6 6 2 1 ··· 1 4 4 4 1 ··· 1 3 ··· 3 2 ··· 2 2 ··· 2 6 ··· 6 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 Q8 D6 S3×C7 C7×Q8 S3×C14 S3×Q8 S3×C7×Q8 kernel S3×C7×Q8 C7×Dic6 S3×C28 Q8×C21 S3×Q8 Dic6 C4×S3 C3×Q8 C7×Q8 S3×C7 C28 Q8 S3 C4 C7 C1 # reps 1 3 3 1 6 18 18 6 1 2 3 6 12 18 1 6

Matrix representation of S3×C7×Q8 in GL4(𝔽337) generated by

 64 0 0 0 0 64 0 0 0 0 1 0 0 0 0 1
,
 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 0 336 0 0 0 0 336
,
 336 0 0 0 0 336 0 0 0 0 0 1 0 0 336 0
,
 336 0 0 0 0 336 0 0 0 0 18 90 0 0 90 319
G:=sub<GL(4,GF(337))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[336,1,0,0,336,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,336,0,0,0,0,336],[336,0,0,0,0,336,0,0,0,0,0,336,0,0,1,0],[336,0,0,0,0,336,0,0,0,0,18,90,0,0,90,319] >;

S3×C7×Q8 in GAP, Magma, Sage, TeX

S_3\times C_7\times Q_8
% in TeX

G:=Group("S3xC7xQ8");
// GroupNames label

G:=SmallGroup(336,190);
// by ID

G=gap.SmallGroup(336,190);
# by ID

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

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

׿
×
𝔽