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## G = D7×Dic6order 336 = 24·3·7

### Direct product of D7 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — D7×Dic6
 Chief series C1 — C7 — C21 — C42 — C6×D7 — Dic3×D7 — D7×Dic6
 Lower central C21 — C42 — D7×Dic6
 Upper central C1 — C2 — C4

Generators and relations for D7×Dic6
G = < a,b,c,d | a7=b2=c12=1, d2=c6, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 372 in 76 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C7, C2×C4, Q8, Dic3, Dic3, C12, C12, C2×C6, D7, C14, C2×Q8, C21, Dic6, Dic6, C2×Dic3, C2×C12, Dic7, Dic7, C28, C28, D14, C3×D7, C42, C2×Dic6, Dic14, C4×D7, C4×D7, C7×Q8, C7×Dic3, C3×Dic7, Dic21, C84, C6×D7, Q8×D7, Dic3×D7, C21⋊Q8, C12×D7, C7×Dic6, Dic42, D7×Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, D7, C2×Q8, Dic6, C22×S3, D14, C2×Dic6, C22×D7, S3×D7, Q8×D7, C2×S3×D7, D7×Dic6

Smallest permutation representation of D7×Dic6
On 168 points
Generators in S168
(1 91 21 113 74 36 70)(2 92 22 114 75 25 71)(3 93 23 115 76 26 72)(4 94 24 116 77 27 61)(5 95 13 117 78 28 62)(6 96 14 118 79 29 63)(7 85 15 119 80 30 64)(8 86 16 120 81 31 65)(9 87 17 109 82 32 66)(10 88 18 110 83 33 67)(11 89 19 111 84 34 68)(12 90 20 112 73 35 69)(37 163 148 103 143 56 129)(38 164 149 104 144 57 130)(39 165 150 105 133 58 131)(40 166 151 106 134 59 132)(41 167 152 107 135 60 121)(42 168 153 108 136 49 122)(43 157 154 97 137 50 123)(44 158 155 98 138 51 124)(45 159 156 99 139 52 125)(46 160 145 100 140 53 126)(47 161 146 101 141 54 127)(48 162 147 102 142 55 128)
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 61)(11 62)(12 63)(13 84)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(25 86)(26 87)(27 88)(28 89)(29 90)(30 91)(31 92)(32 93)(33 94)(34 95)(35 96)(36 85)(37 123)(38 124)(39 125)(40 126)(41 127)(42 128)(43 129)(44 130)(45 131)(46 132)(47 121)(48 122)(49 162)(50 163)(51 164)(52 165)(53 166)(54 167)(55 168)(56 157)(57 158)(58 159)(59 160)(60 161)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)(133 156)(134 145)(135 146)(136 147)(137 148)(138 149)(139 150)(140 151)(141 152)(142 153)(143 154)(144 155)
(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 45 7 39)(2 44 8 38)(3 43 9 37)(4 42 10 48)(5 41 11 47)(6 40 12 46)(13 152 19 146)(14 151 20 145)(15 150 21 156)(16 149 22 155)(17 148 23 154)(18 147 24 153)(25 51 31 57)(26 50 32 56)(27 49 33 55)(28 60 34 54)(29 59 35 53)(30 58 36 52)(61 122 67 128)(62 121 68 127)(63 132 69 126)(64 131 70 125)(65 130 71 124)(66 129 72 123)(73 140 79 134)(74 139 80 133)(75 138 81 144)(76 137 82 143)(77 136 83 142)(78 135 84 141)(85 165 91 159)(86 164 92 158)(87 163 93 157)(88 162 94 168)(89 161 95 167)(90 160 96 166)(97 109 103 115)(98 120 104 114)(99 119 105 113)(100 118 106 112)(101 117 107 111)(102 116 108 110)

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + + + + - + + + - + - image C1 C2 C2 C2 C2 C2 S3 Q8 D6 D6 D6 D7 Dic6 D14 D14 S3×D7 Q8×D7 C2×S3×D7 D7×Dic6 kernel D7×Dic6 Dic3×D7 C21⋊Q8 C12×D7 C7×Dic6 Dic42 C4×D7 C3×D7 Dic7 C28 D14 Dic6 D7 Dic3 C12 C4 C3 C2 C1 # reps 1 2 2 1 1 1 1 2 1 1 1 3 4 6 3 3 3 3 6

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

 1 0 0 0 0 1 0 0 0 0 194 1 0 0 251 109
,
 336 0 0 0 0 336 0 0 0 0 109 336 0 0 85 228
,
 30 15 0 0 322 15 0 0 0 0 336 0 0 0 0 336
,
 0 189 0 0 189 0 0 0 0 0 336 0 0 0 0 336
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,194,251,0,0,1,109],[336,0,0,0,0,336,0,0,0,0,109,85,0,0,336,228],[30,322,0,0,15,15,0,0,0,0,336,0,0,0,0,336],[0,189,0,0,189,0,0,0,0,0,336,0,0,0,0,336] >;

D7×Dic6 in GAP, Magma, Sage, TeX

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

G:=Group("D7xDic6");
// GroupNames label

G:=SmallGroup(336,137);
// by ID

G=gap.SmallGroup(336,137);
# by ID

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

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

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