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## G = Dic6⋊D7order 336 = 24·3·7

### 1st semidirect product of Dic6 and D7 acting via D7/C7=C2

Aliases: C28.6D6, C42.9D4, C216SD16, Dic61D7, D84.4C2, C14.8D12, C12.24D14, C84.17C22, C7⋊C83S3, C31(Q8⋊D7), C72(C24⋊C2), C4.10(S3×D7), (C7×Dic6)⋊1C2, C6.3(C7⋊D4), C2.6(C7⋊D12), (C3×C7⋊C8)⋊3C2, SmallGroup(336,37)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C84 — Dic6⋊D7
 Chief series C1 — C7 — C21 — C42 — C84 — C3×C7⋊C8 — Dic6⋊D7
 Lower central C21 — C42 — C84 — Dic6⋊D7
 Upper central C1 — C2 — C4

Generators and relations for Dic6⋊D7
G = < a,b,c,d | a12=c7=d2=1, b2=a6, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a3b, dcd=c-1 >

Smallest permutation representation of Dic6⋊D7
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)
(1 154 7 148)(2 153 8 147)(3 152 9 146)(4 151 10 145)(5 150 11 156)(6 149 12 155)(13 51 19 57)(14 50 20 56)(15 49 21 55)(16 60 22 54)(17 59 23 53)(18 58 24 52)(25 161 31 167)(26 160 32 166)(27 159 33 165)(28 158 34 164)(29 157 35 163)(30 168 36 162)(37 104 43 98)(38 103 44 97)(39 102 45 108)(40 101 46 107)(41 100 47 106)(42 99 48 105)(61 133 67 139)(62 144 68 138)(63 143 69 137)(64 142 70 136)(65 141 71 135)(66 140 72 134)(73 86 79 92)(74 85 80 91)(75 96 81 90)(76 95 82 89)(77 94 83 88)(78 93 84 87)(109 129 115 123)(110 128 116 122)(111 127 117 121)(112 126 118 132)(113 125 119 131)(114 124 120 130)
(1 19 30 106 75 66 126)(2 20 31 107 76 67 127)(3 21 32 108 77 68 128)(4 22 33 97 78 69 129)(5 23 34 98 79 70 130)(6 24 35 99 80 71 131)(7 13 36 100 81 72 132)(8 14 25 101 82 61 121)(9 15 26 102 83 62 122)(10 16 27 103 84 63 123)(11 17 28 104 73 64 124)(12 18 29 105 74 65 125)(37 92 136 114 150 53 164)(38 93 137 115 151 54 165)(39 94 138 116 152 55 166)(40 95 139 117 153 56 167)(41 96 140 118 154 57 168)(42 85 141 119 155 58 157)(43 86 142 120 156 59 158)(44 87 143 109 145 60 159)(45 88 144 110 146 49 160)(46 89 133 111 147 50 161)(47 90 134 112 148 51 162)(48 91 135 113 149 52 163)
(1 126)(2 125)(3 124)(4 123)(5 122)(6 121)(7 132)(8 131)(9 130)(10 129)(11 128)(12 127)(13 72)(14 71)(15 70)(16 69)(17 68)(18 67)(19 66)(20 65)(21 64)(22 63)(23 62)(24 61)(25 80)(26 79)(27 78)(28 77)(29 76)(30 75)(31 74)(32 73)(33 84)(34 83)(35 82)(36 81)(37 42)(38 41)(39 40)(43 48)(44 47)(45 46)(49 133)(50 144)(51 143)(52 142)(53 141)(54 140)(55 139)(56 138)(57 137)(58 136)(59 135)(60 134)(85 164)(86 163)(87 162)(88 161)(89 160)(90 159)(91 158)(92 157)(93 168)(94 167)(95 166)(96 165)(97 103)(98 102)(99 101)(104 108)(105 107)(109 148)(110 147)(111 146)(112 145)(113 156)(114 155)(115 154)(116 153)(117 152)(118 151)(119 150)(120 149)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 D4 D6 D7 SD16 D12 D14 C24⋊C2 C7⋊D4 S3×D7 Q8⋊D7 C7⋊D12 Dic6⋊D7 kernel Dic6⋊D7 C3×C7⋊C8 C7×Dic6 D84 C7⋊C8 C42 C28 Dic6 C21 C14 C12 C7 C6 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 3 2 2 3 4 6 3 3 3 6

Matrix representation of Dic6⋊D7 in GL4(𝔽337) generated by

 322 322 0 0 15 307 0 0 0 0 336 0 0 0 0 336
,
 117 313 0 0 93 220 0 0 0 0 95 63 0 0 274 242
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 336 227
,
 1 336 0 0 0 336 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(337))| [322,15,0,0,322,307,0,0,0,0,336,0,0,0,0,336],[117,93,0,0,313,220,0,0,0,0,95,274,0,0,63,242],[1,0,0,0,0,1,0,0,0,0,0,336,0,0,1,227],[1,0,0,0,336,336,0,0,0,0,0,1,0,0,1,0] >;

Dic6⋊D7 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes D_7
% in TeX

G:=Group("Dic6:D7");
// GroupNames label

G:=SmallGroup(336,37);
// by ID

G=gap.SmallGroup(336,37);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,24,169,55,116,50,490,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^7=d^2=1,b^2=a^6,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations

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