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## G = C7×D6⋊C4order 336 = 24·3·7

### Direct product of C7 and D6⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C7×D6⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C42 — S3×C2×C14 — C7×D6⋊C4
 Lower central C3 — C6 — C7×D6⋊C4
 Upper central C1 — C2×C14 — C2×C28

Generators and relations for C7×D6⋊C4
G = < a,b,c,d | a7=b6=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >

Subgroups: 152 in 68 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C7, C2×C4, C2×C4, C23, Dic3, C12, D6, D6, C2×C6, C14, C14, C22⋊C4, C21, C2×Dic3, C2×C12, C22×S3, C28, C2×C14, C2×C14, S3×C7, C42, D6⋊C4, C2×C28, C2×C28, C22×C14, C7×Dic3, C84, S3×C14, S3×C14, C2×C42, C7×C22⋊C4, Dic3×C14, C2×C84, S3×C2×C14, C7×D6⋊C4
Quotients: C1, C2, C4, C22, S3, C7, C2×C4, D4, D6, C14, C22⋊C4, C4×S3, D12, C3⋊D4, C28, C2×C14, S3×C7, D6⋊C4, C2×C28, C7×D4, S3×C14, C7×C22⋊C4, S3×C28, C7×D12, C7×C3⋊D4, C7×D6⋊C4

Smallest permutation representation of C7×D6⋊C4
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 80 35 55 118 148)(2 81 29 56 119 149)(3 82 30 50 113 150)(4 83 31 51 114 151)(5 84 32 52 115 152)(6 78 33 53 116 153)(7 79 34 54 117 154)(8 162 126 144 40 58)(9 163 120 145 41 59)(10 164 121 146 42 60)(11 165 122 147 36 61)(12 166 123 141 37 62)(13 167 124 142 38 63)(14 168 125 143 39 57)(15 98 101 138 91 70)(16 92 102 139 85 64)(17 93 103 140 86 65)(18 94 104 134 87 66)(19 95 105 135 88 67)(20 96 99 136 89 68)(21 97 100 137 90 69)(22 47 133 158 108 72)(23 48 127 159 109 73)(24 49 128 160 110 74)(25 43 129 161 111 75)(26 44 130 155 112 76)(27 45 131 156 106 77)(28 46 132 157 107 71)
(1 148)(2 149)(3 150)(4 151)(5 152)(6 153)(7 154)(8 126)(9 120)(10 121)(11 122)(12 123)(13 124)(14 125)(15 70)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 108)(23 109)(24 110)(25 111)(26 112)(27 106)(28 107)(29 56)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(43 161)(44 155)(45 156)(46 157)(47 158)(48 159)(49 160)(57 143)(58 144)(59 145)(60 146)(61 147)(62 141)(63 142)(78 116)(79 117)(80 118)(81 119)(82 113)(83 114)(84 115)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 136)(100 137)(101 138)(102 139)(103 140)(104 134)(105 135)
(1 108 19 10)(2 109 20 11)(3 110 21 12)(4 111 15 13)(5 112 16 14)(6 106 17 8)(7 107 18 9)(22 105 121 35)(23 99 122 29)(24 100 123 30)(25 101 124 31)(26 102 125 32)(27 103 126 33)(28 104 120 34)(36 119 127 89)(37 113 128 90)(38 114 129 91)(39 115 130 85)(40 116 131 86)(41 117 132 87)(42 118 133 88)(43 138 142 51)(44 139 143 52)(45 140 144 53)(46 134 145 54)(47 135 146 55)(48 136 147 56)(49 137 141 50)(57 152 155 64)(58 153 156 65)(59 154 157 66)(60 148 158 67)(61 149 159 68)(62 150 160 69)(63 151 161 70)(71 94 163 79)(72 95 164 80)(73 96 165 81)(74 97 166 82)(75 98 167 83)(76 92 168 84)(77 93 162 78)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6A 6B 6C 7A ··· 7F 12A 12B 12C 12D 14A ··· 14R 14S ··· 14AD 21A ··· 21F 28A ··· 28L 28M ··· 28X 42A ··· 42R 84A ··· 84X order 1 2 2 2 2 2 3 4 4 4 4 6 6 6 7 ··· 7 12 12 12 12 14 ··· 14 14 ··· 14 21 ··· 21 28 ··· 28 28 ··· 28 42 ··· 42 84 ··· 84 size 1 1 1 1 6 6 2 2 2 6 6 2 2 2 1 ··· 1 2 2 2 2 1 ··· 1 6 ··· 6 2 ··· 2 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C4 C7 C14 C14 C14 C28 S3 D4 D6 C4×S3 D12 C3⋊D4 S3×C7 C7×D4 S3×C14 S3×C28 C7×D12 C7×C3⋊D4 kernel C7×D6⋊C4 Dic3×C14 C2×C84 S3×C2×C14 S3×C14 D6⋊C4 C2×Dic3 C2×C12 C22×S3 D6 C2×C28 C42 C2×C14 C14 C14 C14 C2×C4 C6 C22 C2 C2 C2 # reps 1 1 1 1 4 6 6 6 6 24 1 2 1 2 2 2 6 12 6 12 12 12

Matrix representation of C7×D6⋊C4 in GL3(𝔽337) generated by

 1 0 0 0 8 0 0 0 8
,
 1 0 0 0 0 336 0 1 1
,
 336 0 0 0 0 336 0 336 0
,
 148 0 0 0 139 278 0 59 198
G:=sub<GL(3,GF(337))| [1,0,0,0,8,0,0,0,8],[1,0,0,0,0,1,0,336,1],[336,0,0,0,0,336,0,336,0],[148,0,0,0,139,59,0,278,198] >;

C7×D6⋊C4 in GAP, Magma, Sage, TeX

C_7\times D_6\rtimes C_4
% in TeX

G:=Group("C7xD6:C4");
// GroupNames label

G:=SmallGroup(336,84);
// by ID

G=gap.SmallGroup(336,84);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-3,697,175,8069]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^6=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations

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