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## G = D6⋊Dic7order 336 = 24·3·7

### The semidirect product of D6 and 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 — C2×C42 — C6×Dic7 — D6⋊Dic7
 Lower central C21 — C42 — D6⋊Dic7
 Upper central C1 — C22

Generators and relations for D6⋊Dic7
G = < a,b,c,d | a6=b2=c14=1, d2=c7, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 284 in 68 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C7, C2×C4, C23, Dic3, C12, D6, D6, C2×C6, C14, C14, C22⋊C4, C21, C2×Dic3, C2×C12, C22×S3, Dic7, C2×C14, C2×C14, S3×C7, C42, D6⋊C4, C2×Dic7, C2×Dic7, C22×C14, C3×Dic7, Dic21, S3×C14, S3×C14, C2×C42, C23.D7, C6×Dic7, C2×Dic21, S3×C2×C14, D6⋊Dic7
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, D7, C22⋊C4, C4×S3, D12, C3⋊D4, Dic7, D14, D6⋊C4, C2×Dic7, C7⋊D4, S3×D7, C23.D7, S3×Dic7, C21⋊D4, C7⋊D12, D6⋊Dic7

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

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6A 6B 6C 7A 7B 7C 12A 12B 12C 12D 14A ··· 14I 14J ··· 14U 21A 21B 21C 42A ··· 42I order 1 2 2 2 2 2 3 4 4 4 4 6 6 6 7 7 7 12 12 12 12 14 ··· 14 14 ··· 14 21 21 21 42 ··· 42 size 1 1 1 1 6 6 2 14 14 42 42 2 2 2 2 2 2 14 14 14 14 2 ··· 2 6 ··· 6 4 4 4 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + + - - + image C1 C2 C2 C2 C4 S3 D4 D6 D7 C4×S3 D12 C3⋊D4 Dic7 D14 C7⋊D4 S3×D7 S3×Dic7 C21⋊D4 C7⋊D12 kernel D6⋊Dic7 C6×Dic7 C2×Dic21 S3×C2×C14 S3×C14 C2×Dic7 C42 C2×C14 C22×S3 C14 C14 C14 D6 C2×C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 4 1 2 1 3 2 2 2 6 3 12 3 3 3 3

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

 336 0 0 0 0 336 0 0 0 0 2 325 0 0 253 336
,
 336 0 0 0 266 1 0 0 0 0 335 12 0 0 84 2
,
 52 0 0 0 183 175 0 0 0 0 336 0 0 0 0 336
,
 71 335 0 0 161 266 0 0 0 0 292 23 0 0 161 45
G:=sub<GL(4,GF(337))| [336,0,0,0,0,336,0,0,0,0,2,253,0,0,325,336],[336,266,0,0,0,1,0,0,0,0,335,84,0,0,12,2],[52,183,0,0,0,175,0,0,0,0,336,0,0,0,0,336],[71,161,0,0,335,266,0,0,0,0,292,161,0,0,23,45] >;

D6⋊Dic7 in GAP, Magma, Sage, TeX

D_6\rtimes {\rm Dic}_7
% in TeX

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

G:=SmallGroup(336,43);
// by ID

G=gap.SmallGroup(336,43);
# by ID

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

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

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