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## G = C7×C6.D4order 336 = 24·3·7

### Direct product of C7 and C6.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C7×C6.D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C42 — Dic3×C14 — C7×C6.D4
 Lower central C3 — C6 — C7×C6.D4
 Upper central C1 — C2×C14 — C22×C14

Generators and relations for C7×C6.D4
G = < a,b,c,d | a7=b6=c4=1, d2=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b3c-1 >

Subgroups: 120 in 68 conjugacy classes, 38 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C7, C2×C4, C23, Dic3, C2×C6, C2×C6, C2×C6, C14, C14, C14, C22⋊C4, C21, C2×Dic3, C22×C6, C28, C2×C14, C2×C14, C2×C14, C42, C42, C42, C6.D4, C2×C28, C22×C14, C7×Dic3, C2×C42, C2×C42, C2×C42, C7×C22⋊C4, Dic3×C14, C22×C42, C7×C6.D4
Quotients: C1, C2, C4, C22, S3, C7, C2×C4, D4, Dic3, D6, C14, C22⋊C4, C2×Dic3, C3⋊D4, C28, C2×C14, S3×C7, C6.D4, C2×C28, C7×D4, C7×Dic3, S3×C14, C7×C22⋊C4, Dic3×C14, C7×C3⋊D4, C7×C6.D4

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

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

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

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

126 conjugacy classes

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

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C4 C7 C14 C14 C28 S3 D4 Dic3 D6 C3⋊D4 S3×C7 C7×D4 C7×Dic3 S3×C14 C7×C3⋊D4 kernel C7×C6.D4 Dic3×C14 C22×C42 C2×C42 C6.D4 C2×Dic3 C22×C6 C2×C6 C22×C14 C42 C2×C14 C2×C14 C14 C23 C6 C22 C22 C2 # reps 1 2 1 4 6 12 6 24 1 2 2 1 4 6 12 12 6 24

Matrix representation of C7×C6.D4 in GL5(𝔽337)

 1 0 0 0 0 0 295 0 0 0 0 0 295 0 0 0 0 0 1 0 0 0 0 0 1
,
 336 0 0 0 0 0 208 0 0 0 0 138 128 0 0 0 0 0 128 0 0 0 0 92 208
,
 148 0 0 0 0 0 290 164 0 0 0 266 47 0 0 0 0 0 212 199 0 0 0 123 125
,
 189 0 0 0 0 0 290 164 0 0 0 266 47 0 0 0 0 0 212 199 0 0 0 333 125

G:=sub<GL(5,GF(337))| [1,0,0,0,0,0,295,0,0,0,0,0,295,0,0,0,0,0,1,0,0,0,0,0,1],[336,0,0,0,0,0,208,138,0,0,0,0,128,0,0,0,0,0,128,92,0,0,0,0,208],[148,0,0,0,0,0,290,266,0,0,0,164,47,0,0,0,0,0,212,123,0,0,0,199,125],[189,0,0,0,0,0,290,266,0,0,0,164,47,0,0,0,0,0,212,333,0,0,0,199,125] >;

C7×C6.D4 in GAP, Magma, Sage, TeX

C_7\times C_6.D_4
% in TeX

G:=Group("C7xC6.D4");
// GroupNames label

G:=SmallGroup(336,89);
// by ID

G=gap.SmallGroup(336,89);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-3,168,697,8069]);
// Polycyclic

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

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