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## G = C7×D4.S3order 336 = 24·3·7

### Direct product of C7 and D4.S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C7×D4.S3
 Chief series C1 — C3 — C6 — C12 — C84 — C7×Dic6 — C7×D4.S3
 Lower central C3 — C6 — C12 — C7×D4.S3
 Upper central C1 — C14 — C28 — C7×D4

Generators and relations for C7×D4.S3
G = < a,b,c,d,e | a7=b4=c2=d3=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Smallest permutation representation of C7×D4.S3
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)
(1 135 55 19)(2 136 56 20)(3 137 50 21)(4 138 51 15)(5 139 52 16)(6 140 53 17)(7 134 54 18)(8 106 144 45)(9 107 145 46)(10 108 146 47)(11 109 147 48)(12 110 141 49)(13 111 142 43)(14 112 143 44)(22 60 158 121)(23 61 159 122)(24 62 160 123)(25 63 161 124)(26 57 155 125)(27 58 156 126)(28 59 157 120)(29 68 149 99)(30 69 150 100)(31 70 151 101)(32 64 152 102)(33 65 153 103)(34 66 154 104)(35 67 148 105)(36 127 165 73)(37 128 166 74)(38 129 167 75)(39 130 168 76)(40 131 162 77)(41 132 163 71)(42 133 164 72)(78 86 116 93)(79 87 117 94)(80 88 118 95)(81 89 119 96)(82 90 113 97)(83 91 114 98)(84 85 115 92)
(1 19)(2 20)(3 21)(4 15)(5 16)(6 17)(7 18)(8 144)(9 145)(10 146)(11 147)(12 141)(13 142)(14 143)(29 99)(30 100)(31 101)(32 102)(33 103)(34 104)(35 105)(36 165)(37 166)(38 167)(39 168)(40 162)(41 163)(42 164)(50 137)(51 138)(52 139)(53 140)(54 134)(55 135)(56 136)(57 125)(58 126)(59 120)(60 121)(61 122)(62 123)(63 124)(64 152)(65 153)(66 154)(67 148)(68 149)(69 150)(70 151)(78 93)(79 94)(80 95)(81 96)(82 97)(83 98)(84 92)(85 115)(86 116)(87 117)(88 118)(89 119)(90 113)(91 114)
(1 118 35)(2 119 29)(3 113 30)(4 114 31)(5 115 32)(6 116 33)(7 117 34)(8 126 40)(9 120 41)(10 121 42)(11 122 36)(12 123 37)(13 124 38)(14 125 39)(15 91 101)(16 85 102)(17 86 103)(18 87 104)(19 88 105)(20 89 99)(21 90 100)(22 133 108)(23 127 109)(24 128 110)(25 129 111)(26 130 112)(27 131 106)(28 132 107)(43 161 75)(44 155 76)(45 156 77)(46 157 71)(47 158 72)(48 159 73)(49 160 74)(50 82 150)(51 83 151)(52 84 152)(53 78 153)(54 79 154)(55 80 148)(56 81 149)(57 168 143)(58 162 144)(59 163 145)(60 164 146)(61 165 147)(62 166 141)(63 167 142)(64 139 92)(65 140 93)(66 134 94)(67 135 95)(68 136 96)(69 137 97)(70 138 98)
(1 146 55 10)(2 147 56 11)(3 141 50 12)(4 142 51 13)(5 143 52 14)(6 144 53 8)(7 145 54 9)(15 43 138 111)(16 44 139 112)(17 45 140 106)(18 46 134 107)(19 47 135 108)(20 48 136 109)(21 49 137 110)(22 105 158 67)(23 99 159 68)(24 100 160 69)(25 101 161 70)(26 102 155 64)(27 103 156 65)(28 104 157 66)(29 61 149 122)(30 62 150 123)(31 63 151 124)(32 57 152 125)(33 58 153 126)(34 59 154 120)(35 60 148 121)(36 119 165 81)(37 113 166 82)(38 114 167 83)(39 115 168 84)(40 116 162 78)(41 117 163 79)(42 118 164 80)(71 94 132 87)(72 95 133 88)(73 96 127 89)(74 97 128 90)(75 98 129 91)(76 92 130 85)(77 93 131 86)

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + - image C1 C2 C2 C2 C7 C14 C14 C14 S3 D4 D6 SD16 C3⋊D4 S3×C7 C7×D4 S3×C14 C7×SD16 C7×C3⋊D4 D4.S3 C7×D4.S3 kernel C7×D4.S3 C7×C3⋊C8 C7×Dic6 D4×C21 D4.S3 C3⋊C8 Dic6 C3×D4 C7×D4 C42 C28 C21 C14 D4 C6 C4 C3 C2 C7 C1 # reps 1 1 1 1 6 6 6 6 1 1 1 2 2 6 6 6 12 12 1 6

Matrix representation of C7×D4.S3 in GL4(𝔽337) generated by

 295 0 0 0 0 295 0 0 0 0 79 0 0 0 0 79
,
 336 0 0 0 0 336 0 0 0 0 1 199 0 0 127 336
,
 336 0 0 0 0 1 0 0 0 0 336 0 0 0 210 1
,
 128 0 0 0 0 208 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 196 293 0 0 314 141
G:=sub<GL(4,GF(337))| [295,0,0,0,0,295,0,0,0,0,79,0,0,0,0,79],[336,0,0,0,0,336,0,0,0,0,1,127,0,0,199,336],[336,0,0,0,0,1,0,0,0,0,336,210,0,0,0,1],[128,0,0,0,0,208,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,196,314,0,0,293,141] >;

C7×D4.S3 in GAP, Magma, Sage, TeX

C_7\times D_4.S_3
% in TeX

G:=Group("C7xD4.S3");
// GroupNames label

G:=SmallGroup(336,86);
// by ID

G=gap.SmallGroup(336,86);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-3,336,361,2019,1017,69,8069]);
// Polycyclic

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

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