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## G = C6×C7⋊D4order 336 = 24·3·7

### Direct product of C6 and C7⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C6×C7⋊D4
 Chief series C1 — C7 — C14 — C42 — C6×D7 — C2×C6×D7 — C6×C7⋊D4
 Lower central C7 — C14 — C6×C7⋊D4
 Upper central C1 — C2×C6 — C22×C6

Generators and relations for C6×C7⋊D4
G = < a,b,c,d | a6=b7=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 368 in 108 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C7, C2×C4, D4, C23, C23, C12, C2×C6, C2×C6, C2×C6, D7, C14, C14, C14, C2×D4, C21, C2×C12, C3×D4, C22×C6, C22×C6, Dic7, D14, D14, C2×C14, C2×C14, C2×C14, C3×D7, C42, C42, C42, C6×D4, C2×Dic7, C7⋊D4, C22×D7, C22×C14, C3×Dic7, C6×D7, C6×D7, C2×C42, C2×C42, C2×C42, C2×C7⋊D4, C6×Dic7, C3×C7⋊D4, C2×C6×D7, C22×C42, C6×C7⋊D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, D7, C2×D4, C3×D4, C22×C6, D14, C3×D7, C6×D4, C7⋊D4, C22×D7, C6×D7, C2×C7⋊D4, C3×C7⋊D4, C2×C6×D7, C6×C7⋊D4

Smallest permutation representation of C6×C7⋊D4
On 168 points
Generators in S168
(1 106 29 92 15 120)(2 107 30 93 16 121)(3 108 31 94 17 122)(4 109 32 95 18 123)(5 110 33 96 19 124)(6 111 34 97 20 125)(7 112 35 98 21 126)(8 99 36 85 22 113)(9 100 37 86 23 114)(10 101 38 87 24 115)(11 102 39 88 25 116)(12 103 40 89 26 117)(13 104 41 90 27 118)(14 105 42 91 28 119)(43 148 71 134 57 162)(44 149 72 135 58 163)(45 150 73 136 59 164)(46 151 74 137 60 165)(47 152 75 138 61 166)(48 153 76 139 62 167)(49 154 77 140 63 168)(50 141 78 127 64 155)(51 142 79 128 65 156)(52 143 80 129 66 157)(53 144 81 130 67 158)(54 145 82 131 68 159)(55 146 83 132 69 160)(56 147 84 133 70 161)
(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 43 8 50)(2 49 9 56)(3 48 10 55)(4 47 11 54)(5 46 12 53)(6 45 13 52)(7 44 14 51)(15 57 22 64)(16 63 23 70)(17 62 24 69)(18 61 25 68)(19 60 26 67)(20 59 27 66)(21 58 28 65)(29 71 36 78)(30 77 37 84)(31 76 38 83)(32 75 39 82)(33 74 40 81)(34 73 41 80)(35 72 42 79)(85 127 92 134)(86 133 93 140)(87 132 94 139)(88 131 95 138)(89 130 96 137)(90 129 97 136)(91 128 98 135)(99 141 106 148)(100 147 107 154)(101 146 108 153)(102 145 109 152)(103 144 110 151)(104 143 111 150)(105 142 112 149)(113 155 120 162)(114 161 121 168)(115 160 122 167)(116 159 123 166)(117 158 124 165)(118 157 125 164)(119 156 126 163)
(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(16 21)(17 20)(18 19)(23 28)(24 27)(25 26)(30 35)(31 34)(32 33)(37 42)(38 41)(39 40)(43 50)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(57 64)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(71 78)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(86 91)(87 90)(88 89)(93 98)(94 97)(95 96)(100 105)(101 104)(102 103)(107 112)(108 111)(109 110)(114 119)(115 118)(116 117)(121 126)(122 125)(123 124)(127 134)(128 140)(129 139)(130 138)(131 137)(132 136)(133 135)(141 148)(142 154)(143 153)(144 152)(145 151)(146 150)(147 149)(155 162)(156 168)(157 167)(158 166)(159 165)(160 164)(161 163)

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

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

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

102 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 6N 7A 7B 7C 12A 12B 12C 12D 14A ··· 14U 21A ··· 21F 42A ··· 42AP order 1 2 2 2 2 2 2 2 3 3 4 4 6 ··· 6 6 6 6 6 6 6 6 6 7 7 7 12 12 12 12 14 ··· 14 21 ··· 21 42 ··· 42 size 1 1 1 1 2 2 14 14 1 1 14 14 1 ··· 1 2 2 2 2 14 14 14 14 2 2 2 14 14 14 14 2 ··· 2 2 ··· 2 2 ··· 2

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 D7 C3×D4 D14 C3×D7 C7⋊D4 C6×D7 C3×C7⋊D4 kernel C6×C7⋊D4 C6×Dic7 C3×C7⋊D4 C2×C6×D7 C22×C42 C2×C7⋊D4 C2×Dic7 C7⋊D4 C22×D7 C22×C14 C42 C22×C6 C14 C2×C6 C23 C6 C22 C2 # reps 1 1 4 1 1 2 2 8 2 2 2 3 4 9 6 12 18 24

Matrix representation of C6×C7⋊D4 in GL3(𝔽337) generated by

 336 0 0 0 208 0 0 0 208
,
 1 0 0 0 110 336 0 78 33
,
 336 0 0 0 95 182 0 280 242
,
 336 0 0 0 1 228 0 0 336
G:=sub<GL(3,GF(337))| [336,0,0,0,208,0,0,0,208],[1,0,0,0,110,78,0,336,33],[336,0,0,0,95,280,0,182,242],[336,0,0,0,1,0,0,228,336] >;

C6×C7⋊D4 in GAP, Magma, Sage, TeX

C_6\times C_7\rtimes D_4
% in TeX

G:=Group("C6xC7:D4");
// GroupNames label

G:=SmallGroup(336,183);
// by ID

G=gap.SmallGroup(336,183);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-7,506,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^7=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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