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## G = C3×D4.D7order 336 = 24·3·7

### Direct product of C3 and D4.D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C3×D4.D7
 Chief series C1 — C7 — C14 — C28 — C84 — C3×Dic14 — C3×D4.D7
 Lower central C7 — C14 — C28 — C3×D4.D7
 Upper central C1 — C6 — C12 — C3×D4

Generators and relations for C3×D4.D7
G = < a,b,c,d,e | a3=b4=c2=d7=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 C3×D4.D7
On 168 points
Generators in S168
(1 57 29)(2 58 30)(3 59 31)(4 60 32)(5 61 33)(6 62 34)(7 63 35)(8 64 36)(9 65 37)(10 66 38)(11 67 39)(12 68 40)(13 69 41)(14 70 42)(15 71 43)(16 72 44)(17 73 45)(18 74 46)(19 75 47)(20 76 48)(21 77 49)(22 78 50)(23 79 51)(24 80 52)(25 81 53)(26 82 54)(27 83 55)(28 84 56)(85 141 113)(86 142 114)(87 143 115)(88 144 116)(89 145 117)(90 146 118)(91 147 119)(92 148 120)(93 149 121)(94 150 122)(95 151 123)(96 152 124)(97 153 125)(98 154 126)(99 155 127)(100 156 128)(101 157 129)(102 158 130)(103 159 131)(104 160 132)(105 161 133)(106 162 134)(107 163 135)(108 164 136)(109 165 137)(110 166 138)(111 167 139)(112 168 140)
(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)(29 50 36 43)(30 51 37 44)(31 52 38 45)(32 53 39 46)(33 54 40 47)(34 55 41 48)(35 56 42 49)(57 78 64 71)(58 79 65 72)(59 80 66 73)(60 81 67 74)(61 82 68 75)(62 83 69 76)(63 84 70 77)(85 99 92 106)(86 100 93 107)(87 101 94 108)(88 102 95 109)(89 103 96 110)(90 104 97 111)(91 105 98 112)(113 127 120 134)(114 128 121 135)(115 129 122 136)(116 130 123 137)(117 131 124 138)(118 132 125 139)(119 133 126 140)(141 155 148 162)(142 156 149 163)(143 157 150 164)(144 158 151 165)(145 159 152 166)(146 160 153 167)(147 161 154 168)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(113 120)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(141 148)(142 149)(143 150)(144 151)(145 152)(146 153)(147 154)
(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 96 8 89)(2 95 9 88)(3 94 10 87)(4 93 11 86)(5 92 12 85)(6 98 13 91)(7 97 14 90)(15 110 22 103)(16 109 23 102)(17 108 24 101)(18 107 25 100)(19 106 26 99)(20 112 27 105)(21 111 28 104)(29 124 36 117)(30 123 37 116)(31 122 38 115)(32 121 39 114)(33 120 40 113)(34 126 41 119)(35 125 42 118)(43 138 50 131)(44 137 51 130)(45 136 52 129)(46 135 53 128)(47 134 54 127)(48 140 55 133)(49 139 56 132)(57 152 64 145)(58 151 65 144)(59 150 66 143)(60 149 67 142)(61 148 68 141)(62 154 69 147)(63 153 70 146)(71 166 78 159)(72 165 79 158)(73 164 80 157)(74 163 81 156)(75 162 82 155)(76 168 83 161)(77 167 84 160)

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

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

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

66 conjugacy classes

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

66 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 C3 C6 C6 C6 D4 D7 SD16 C3×D4 D14 C3×D7 C3×SD16 C7⋊D4 C6×D7 C3×C7⋊D4 D4.D7 C3×D4.D7 kernel C3×D4.D7 C3×C7⋊C8 C3×Dic14 D4×C21 D4.D7 C7⋊C8 Dic14 C7×D4 C42 C3×D4 C21 C14 C12 D4 C7 C6 C4 C2 C3 C1 # reps 1 1 1 1 2 2 2 2 1 3 2 2 3 6 4 6 6 12 3 6

Matrix representation of C3×D4.D7 in GL4(𝔽337) generated by

 208 0 0 0 0 208 0 0 0 0 1 0 0 0 0 1
,
 336 0 0 0 0 336 0 0 0 0 336 308 0 0 186 1
,
 336 0 0 0 109 1 0 0 0 0 336 308 0 0 0 1
,
 52 0 0 0 132 175 0 0 0 0 1 0 0 0 0 1
,
 72 159 0 0 315 265 0 0 0 0 196 146 0 0 307 141
G:=sub<GL(4,GF(337))| [208,0,0,0,0,208,0,0,0,0,1,0,0,0,0,1],[336,0,0,0,0,336,0,0,0,0,336,186,0,0,308,1],[336,109,0,0,0,1,0,0,0,0,336,0,0,0,308,1],[52,132,0,0,0,175,0,0,0,0,1,0,0,0,0,1],[72,315,0,0,159,265,0,0,0,0,196,307,0,0,146,141] >;

C3×D4.D7 in GAP, Magma, Sage, TeX

C_3\times D_4.D_7
% in TeX

G:=Group("C3xD4.D7");
// GroupNames label

G:=SmallGroup(336,70);
// by ID

G=gap.SmallGroup(336,70);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,144,169,867,441,69,10373]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^7=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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