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## G = D6.Dic7order 336 = 24·3·7

### The non-split extension by D6 of Dic7 acting via Dic7/C14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — D6.Dic7
 Chief series C1 — C7 — C21 — C42 — C84 — C3×C7⋊C8 — D6.Dic7
 Lower central C21 — C42 — D6.Dic7
 Upper central C1 — C4

Generators and relations for D6.Dic7
G = < a,b,c,d | a6=b2=1, c14=a3, d2=a3c7, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c13 >

Smallest permutation representation of D6.Dic7
On 168 points
Generators in S168
(1 34 130 15 48 116)(2 35 131 16 49 117)(3 36 132 17 50 118)(4 37 133 18 51 119)(5 38 134 19 52 120)(6 39 135 20 53 121)(7 40 136 21 54 122)(8 41 137 22 55 123)(9 42 138 23 56 124)(10 43 139 24 29 125)(11 44 140 25 30 126)(12 45 113 26 31 127)(13 46 114 27 32 128)(14 47 115 28 33 129)(57 110 163 71 96 149)(58 111 164 72 97 150)(59 112 165 73 98 151)(60 85 166 74 99 152)(61 86 167 75 100 153)(62 87 168 76 101 154)(63 88 141 77 102 155)(64 89 142 78 103 156)(65 90 143 79 104 157)(66 91 144 80 105 158)(67 92 145 81 106 159)(68 93 146 82 107 160)(69 94 147 83 108 161)(70 95 148 84 109 162)
(1 116)(2 117)(3 118)(4 119)(5 120)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 127)(13 128)(14 129)(15 130)(16 131)(17 132)(18 133)(19 134)(20 135)(21 136)(22 137)(23 138)(24 139)(25 140)(26 113)(27 114)(28 115)(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)(85 152)(86 153)(87 154)(88 155)(89 156)(90 157)(91 158)(92 159)(93 160)(94 161)(95 162)(96 163)(97 164)(98 165)(99 166)(100 167)(101 168)(102 141)(103 142)(104 143)(105 144)(106 145)(107 146)(108 147)(109 148)(110 149)(111 150)(112 151)
(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)
(1 148 22 141 15 162 8 155)(2 161 23 154 16 147 9 168)(3 146 24 167 17 160 10 153)(4 159 25 152 18 145 11 166)(5 144 26 165 19 158 12 151)(6 157 27 150 20 143 13 164)(7 142 28 163 21 156 14 149)(29 75 50 68 43 61 36 82)(30 60 51 81 44 74 37 67)(31 73 52 66 45 59 38 80)(32 58 53 79 46 72 39 65)(33 71 54 64 47 57 40 78)(34 84 55 77 48 70 41 63)(35 69 56 62 49 83 42 76)(85 119 106 140 99 133 92 126)(86 132 107 125 100 118 93 139)(87 117 108 138 101 131 94 124)(88 130 109 123 102 116 95 137)(89 115 110 136 103 129 96 122)(90 128 111 121 104 114 97 135)(91 113 112 134 105 127 98 120)

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + - + - + - image C1 C2 C2 C2 C4 C4 S3 D6 D7 M4(2) C4×S3 Dic7 D14 Dic7 C8⋊S3 C4.Dic7 S3×D7 S3×Dic7 D6.Dic7 kernel D6.Dic7 C3×C7⋊C8 C21⋊C8 S3×C28 C7×Dic3 S3×C14 C7⋊C8 C28 C4×S3 C21 C14 Dic3 C12 D6 C7 C3 C4 C2 C1 # reps 1 1 1 1 2 2 1 1 3 2 2 3 3 3 4 12 3 3 6

Matrix representation of D6.Dic7 in GL4(𝔽337) generated by

 336 336 0 0 1 0 0 0 0 0 336 0 0 0 0 336
,
 336 336 0 0 0 1 0 0 0 0 336 0 0 0 11 1
,
 336 0 0 0 0 336 0 0 0 0 164 0 0 0 260 150
,
 148 0 0 0 0 148 0 0 0 0 155 212 0 0 250 182
G:=sub<GL(4,GF(337))| [336,1,0,0,336,0,0,0,0,0,336,0,0,0,0,336],[336,0,0,0,336,1,0,0,0,0,336,11,0,0,0,1],[336,0,0,0,0,336,0,0,0,0,164,260,0,0,0,150],[148,0,0,0,0,148,0,0,0,0,155,250,0,0,212,182] >;

D6.Dic7 in GAP, Magma, Sage, TeX

D_6.{\rm Dic}_7
% in TeX

G:=Group("D6.Dic7");
// GroupNames label

G:=SmallGroup(336,27);
// by ID

G=gap.SmallGroup(336,27);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,24,121,50,490,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=1,c^14=a^3,d^2=a^3*c^7,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^13>;
// generators/relations

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