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## G = C7×Dic6order 168 = 23·3·7

### Direct product of C7 and Dic6

Aliases: C7×Dic6, C214Q8, C84.5C2, C28.3S3, C12.1C14, C14.13D6, Dic3.C14, C42.18C22, C3⋊(C7×Q8), C4.(S3×C7), C2.3(S3×C14), C6.1(C2×C14), (C7×Dic3).2C2, SmallGroup(168,29)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C7×Dic6
 Chief series C1 — C3 — C6 — C42 — C7×Dic3 — C7×Dic6
 Lower central C3 — C6 — C7×Dic6
 Upper central C1 — C14 — C28

Generators and relations for C7×Dic6
G = < a,b,c | a7=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C7×Dic6
Regular action on 168 points
Generators in S168
(1 161 68 87 81 19 151)(2 162 69 88 82 20 152)(3 163 70 89 83 21 153)(4 164 71 90 84 22 154)(5 165 72 91 73 23 155)(6 166 61 92 74 24 156)(7 167 62 93 75 13 145)(8 168 63 94 76 14 146)(9 157 64 95 77 15 147)(10 158 65 96 78 16 148)(11 159 66 85 79 17 149)(12 160 67 86 80 18 150)(25 99 136 45 54 126 113)(26 100 137 46 55 127 114)(27 101 138 47 56 128 115)(28 102 139 48 57 129 116)(29 103 140 37 58 130 117)(30 104 141 38 59 131 118)(31 105 142 39 60 132 119)(32 106 143 40 49 121 120)(33 107 144 41 50 122 109)(34 108 133 42 51 123 110)(35 97 134 43 52 124 111)(36 98 135 44 53 125 112)
(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 46 7 40)(2 45 8 39)(3 44 9 38)(4 43 10 37)(5 42 11 48)(6 41 12 47)(13 106 19 100)(14 105 20 99)(15 104 21 98)(16 103 22 97)(17 102 23 108)(18 101 24 107)(25 76 31 82)(26 75 32 81)(27 74 33 80)(28 73 34 79)(29 84 35 78)(30 83 36 77)(49 161 55 167)(50 160 56 166)(51 159 57 165)(52 158 58 164)(53 157 59 163)(54 168 60 162)(61 122 67 128)(62 121 68 127)(63 132 69 126)(64 131 70 125)(65 130 71 124)(66 129 72 123)(85 116 91 110)(86 115 92 109)(87 114 93 120)(88 113 94 119)(89 112 95 118)(90 111 96 117)(133 149 139 155)(134 148 140 154)(135 147 141 153)(136 146 142 152)(137 145 143 151)(138 156 144 150)

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

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

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

C7×Dic6 is a maximal subgroup of
C28.D6  Dic6⋊D7  C21⋊Q16  C7⋊Dic12  D28⋊S3  D21⋊Q8  D14.D6  S3×C7×Q8

63 conjugacy classes

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

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C7 C14 C14 S3 Q8 D6 Dic6 S3×C7 C7×Q8 S3×C14 C7×Dic6 kernel C7×Dic6 C7×Dic3 C84 Dic6 Dic3 C12 C28 C21 C14 C7 C4 C3 C2 C1 # reps 1 2 1 6 12 6 1 1 1 2 6 6 6 12

Matrix representation of C7×Dic6 in GL4(𝔽337) generated by

 79 0 0 0 0 79 0 0 0 0 79 0 0 0 0 79
,
 0 336 0 0 1 1 0 0 0 0 1 224 0 0 3 336
,
 296 146 0 0 187 41 0 0 0 0 302 240 0 0 301 35
G:=sub<GL(4,GF(337))| [79,0,0,0,0,79,0,0,0,0,79,0,0,0,0,79],[0,1,0,0,336,1,0,0,0,0,1,3,0,0,224,336],[296,187,0,0,146,41,0,0,0,0,302,301,0,0,240,35] >;

C7×Dic6 in GAP, Magma, Sage, TeX

C_7\times {\rm Dic}_6
% in TeX

G:=Group("C7xDic6");
// GroupNames label

G:=SmallGroup(168,29);
// by ID

G=gap.SmallGroup(168,29);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-3,140,301,146,2804]);
// Polycyclic

G:=Group<a,b,c|a^7=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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