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## G = C3×C4.Dic7order 336 = 24·3·7

### Direct product of C3 and C4.Dic7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C3×C4.Dic7
 Chief series C1 — C7 — C14 — C28 — C84 — C3×C7⋊C8 — C3×C4.Dic7
 Lower central C7 — C14 — C3×C4.Dic7
 Upper central C1 — C12 — C2×C12

Generators and relations for C3×C4.Dic7
G = < a,b,c,d | a3=b4=1, c14=b2, d2=b2c7, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c13 >

Smallest permutation representation of C3×C4.Dic7
On 168 points
Generators in S168
(1 57 50)(2 58 51)(3 59 52)(4 60 53)(5 61 54)(6 62 55)(7 63 56)(8 64 29)(9 65 30)(10 66 31)(11 67 32)(12 68 33)(13 69 34)(14 70 35)(15 71 36)(16 72 37)(17 73 38)(18 74 39)(19 75 40)(20 76 41)(21 77 42)(22 78 43)(23 79 44)(24 80 45)(25 81 46)(26 82 47)(27 83 48)(28 84 49)(85 163 139)(86 164 140)(87 165 113)(88 166 114)(89 167 115)(90 168 116)(91 141 117)(92 142 118)(93 143 119)(94 144 120)(95 145 121)(96 146 122)(97 147 123)(98 148 124)(99 149 125)(100 150 126)(101 151 127)(102 152 128)(103 153 129)(104 154 130)(105 155 131)(106 156 132)(107 157 133)(108 158 134)(109 159 135)(110 160 136)(111 161 137)(112 162 138)
(1 22 15 8)(2 23 16 9)(3 24 17 10)(4 25 18 11)(5 26 19 12)(6 27 20 13)(7 28 21 14)(29 50 43 36)(30 51 44 37)(31 52 45 38)(32 53 46 39)(33 54 47 40)(34 55 48 41)(35 56 49 42)(57 78 71 64)(58 79 72 65)(59 80 73 66)(60 81 74 67)(61 82 75 68)(62 83 76 69)(63 84 77 70)(85 92 99 106)(86 93 100 107)(87 94 101 108)(88 95 102 109)(89 96 103 110)(90 97 104 111)(91 98 105 112)(113 120 127 134)(114 121 128 135)(115 122 129 136)(116 123 130 137)(117 124 131 138)(118 125 132 139)(119 126 133 140)(141 148 155 162)(142 149 156 163)(143 150 157 164)(144 151 158 165)(145 152 159 166)(146 153 160 167)(147 154 161 168)
(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)
(1 105 22 98 15 91 8 112)(2 90 23 111 16 104 9 97)(3 103 24 96 17 89 10 110)(4 88 25 109 18 102 11 95)(5 101 26 94 19 87 12 108)(6 86 27 107 20 100 13 93)(7 99 28 92 21 85 14 106)(29 138 50 131 43 124 36 117)(30 123 51 116 44 137 37 130)(31 136 52 129 45 122 38 115)(32 121 53 114 46 135 39 128)(33 134 54 127 47 120 40 113)(34 119 55 140 48 133 41 126)(35 132 56 125 49 118 42 139)(57 155 78 148 71 141 64 162)(58 168 79 161 72 154 65 147)(59 153 80 146 73 167 66 160)(60 166 81 159 74 152 67 145)(61 151 82 144 75 165 68 158)(62 164 83 157 76 150 69 143)(63 149 84 142 77 163 70 156)

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

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

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

102 conjugacy classes

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

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 D7 M4(2) Dic7 D14 Dic7 C3×D7 C3×M4(2) C3×Dic7 C6×D7 C3×Dic7 C4.Dic7 C3×C4.Dic7 kernel C3×C4.Dic7 C3×C7⋊C8 C2×C84 C4.Dic7 C84 C2×C42 C7⋊C8 C2×C28 C28 C2×C14 C2×C12 C21 C12 C12 C2×C6 C2×C4 C7 C4 C4 C22 C3 C1 # reps 1 2 1 2 2 2 4 2 4 4 3 2 3 3 3 6 4 6 6 6 12 24

Matrix representation of C3×C4.Dic7 in GL2(𝔽337) generated by

 208 0 0 208
,
 189 0 213 148
,
 103 0 81 301
,
 183 335 137 154
G:=sub<GL(2,GF(337))| [208,0,0,208],[189,213,0,148],[103,81,0,301],[183,137,335,154] >;

C3×C4.Dic7 in GAP, Magma, Sage, TeX

C_3\times C_4.{\rm Dic}_7
% in TeX

G:=Group("C3xC4.Dic7");
// GroupNames label

G:=SmallGroup(336,64);
// by ID

G=gap.SmallGroup(336,64);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,72,313,69,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=1,c^14=b^2,d^2=b^2*c^7,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^13>;
// generators/relations

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