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## G = Dic3×C14order 168 = 23·3·7

### Direct product of C14 and Dic3

Aliases: Dic3×C14, C6⋊C28, C423C4, C14.16D6, C42.21C22, C219(C2×C4), C32(C2×C28), (C2×C6).C14, C22.(S3×C7), C2.2(S3×C14), (C2×C14).2S3, C6.4(C2×C14), (C2×C42).3C2, SmallGroup(168,32)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C14
 Chief series C1 — C3 — C6 — C42 — C7×Dic3 — Dic3×C14
 Lower central C3 — Dic3×C14
 Upper central C1 — C2×C14

Generators and relations for Dic3×C14
G = < a,b,c | a14=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of Dic3×C14
Regular action 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 78 20 142 37 163)(2 79 21 143 38 164)(3 80 22 144 39 165)(4 81 23 145 40 166)(5 82 24 146 41 167)(6 83 25 147 42 168)(7 84 26 148 29 155)(8 71 27 149 30 156)(9 72 28 150 31 157)(10 73 15 151 32 158)(11 74 16 152 33 159)(12 75 17 153 34 160)(13 76 18 154 35 161)(14 77 19 141 36 162)(43 104 114 65 92 131)(44 105 115 66 93 132)(45 106 116 67 94 133)(46 107 117 68 95 134)(47 108 118 69 96 135)(48 109 119 70 97 136)(49 110 120 57 98 137)(50 111 121 58 85 138)(51 112 122 59 86 139)(52 99 123 60 87 140)(53 100 124 61 88 127)(54 101 125 62 89 128)(55 102 126 63 90 129)(56 103 113 64 91 130)
(1 95 142 107)(2 96 143 108)(3 97 144 109)(4 98 145 110)(5 85 146 111)(6 86 147 112)(7 87 148 99)(8 88 149 100)(9 89 150 101)(10 90 151 102)(11 91 152 103)(12 92 153 104)(13 93 154 105)(14 94 141 106)(15 126 158 129)(16 113 159 130)(17 114 160 131)(18 115 161 132)(19 116 162 133)(20 117 163 134)(21 118 164 135)(22 119 165 136)(23 120 166 137)(24 121 167 138)(25 122 168 139)(26 123 155 140)(27 124 156 127)(28 125 157 128)(29 52 84 60)(30 53 71 61)(31 54 72 62)(32 55 73 63)(33 56 74 64)(34 43 75 65)(35 44 76 66)(36 45 77 67)(37 46 78 68)(38 47 79 69)(39 48 80 70)(40 49 81 57)(41 50 82 58)(42 51 83 59)

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

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

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

Dic3×C14 is a maximal subgroup of   D14⋊Dic3  D42⋊C4  C42.Q8  Dic21⋊C4  C14.Dic6  Dic7.D6  S3×C2×C28

84 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6A 6B 6C 7A ··· 7F 14A ··· 14R 21A ··· 21F 28A ··· 28X 42A ··· 42R order 1 2 2 2 3 4 4 4 4 6 6 6 7 ··· 7 14 ··· 14 21 ··· 21 28 ··· 28 42 ··· 42 size 1 1 1 1 2 3 3 3 3 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 3 ··· 3 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C7 C14 C14 C28 S3 Dic3 D6 S3×C7 C7×Dic3 S3×C14 kernel Dic3×C14 C7×Dic3 C2×C42 C42 C2×Dic3 Dic3 C2×C6 C6 C2×C14 C14 C14 C22 C2 C2 # reps 1 2 1 4 6 12 6 24 1 2 1 6 12 6

Matrix representation of Dic3×C14 in GL3(𝔽337) generated by

 336 0 0 0 42 0 0 0 42
,
 336 0 0 0 336 1 0 336 0
,
 148 0 0 0 54 271 0 325 283
G:=sub<GL(3,GF(337))| [336,0,0,0,42,0,0,0,42],[336,0,0,0,336,336,0,1,0],[148,0,0,0,54,325,0,271,283] >;

Dic3×C14 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{14}
% in TeX

G:=Group("Dic3xC14");
// GroupNames label

G:=SmallGroup(168,32);
// by ID

G=gap.SmallGroup(168,32);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-3,140,2804]);
// Polycyclic

G:=Group<a,b,c|a^14=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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