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## G = D7×C2×C12order 336 = 24·3·7

### Direct product of C2×C12 and D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — D7×C2×C12
 Chief series C1 — C7 — C14 — C42 — C6×D7 — C2×C6×D7 — D7×C2×C12
 Lower central C7 — D7×C2×C12
 Upper central C1 — C2×C12

Generators and relations for D7×C2×C12
G = < a,b,c,d | a2=b12=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 336 in 108 conjugacy classes, 70 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C7, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, D7, C14, C14, C22×C4, C21, C2×C12, C2×C12, C22×C6, Dic7, C28, D14, C2×C14, C3×D7, C42, C42, C22×C12, C4×D7, C2×Dic7, C2×C28, C22×D7, C3×Dic7, C84, C6×D7, C2×C42, C2×C4×D7, C12×D7, C6×Dic7, C2×C84, C2×C6×D7, D7×C2×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, D7, C22×C4, C2×C12, C22×C6, D14, C3×D7, C22×C12, C4×D7, C22×D7, C6×D7, C2×C4×D7, C12×D7, C2×C6×D7, D7×C2×C12

Smallest permutation representation of D7×C2×C12
On 168 points
Generators in S168
(1 126)(2 127)(3 128)(4 129)(5 130)(6 131)(7 132)(8 121)(9 122)(10 123)(11 124)(12 125)(13 72)(14 61)(15 62)(16 63)(17 64)(18 65)(19 66)(20 67)(21 68)(22 69)(23 70)(24 71)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 49)(36 50)(37 108)(38 97)(39 98)(40 99)(41 100)(42 101)(43 102)(44 103)(45 104)(46 105)(47 106)(48 107)(73 158)(74 159)(75 160)(76 161)(77 162)(78 163)(79 164)(80 165)(81 166)(82 167)(83 168)(84 157)(85 146)(86 147)(87 148)(88 149)(89 150)(90 151)(91 152)(92 153)(93 154)(94 155)(95 156)(96 145)(109 138)(110 139)(111 140)(112 141)(113 142)(114 143)(115 144)(116 133)(117 134)(118 135)(119 136)(120 137)
(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)
(1 115 15 104 161 152 33)(2 116 16 105 162 153 34)(3 117 17 106 163 154 35)(4 118 18 107 164 155 36)(5 119 19 108 165 156 25)(6 120 20 97 166 145 26)(7 109 21 98 167 146 27)(8 110 22 99 168 147 28)(9 111 23 100 157 148 29)(10 112 24 101 158 149 30)(11 113 13 102 159 150 31)(12 114 14 103 160 151 32)(37 80 95 51 130 136 66)(38 81 96 52 131 137 67)(39 82 85 53 132 138 68)(40 83 86 54 121 139 69)(41 84 87 55 122 140 70)(42 73 88 56 123 141 71)(43 74 89 57 124 142 72)(44 75 90 58 125 143 61)(45 76 91 59 126 144 62)(46 77 92 60 127 133 63)(47 78 93 49 128 134 64)(48 79 94 50 129 135 65)
(1 33)(2 34)(3 35)(4 36)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 159)(14 160)(15 161)(16 162)(17 163)(18 164)(19 165)(20 166)(21 167)(22 168)(23 157)(24 158)(49 128)(50 129)(51 130)(52 131)(53 132)(54 121)(55 122)(56 123)(57 124)(58 125)(59 126)(60 127)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(71 73)(72 74)(85 138)(86 139)(87 140)(88 141)(89 142)(90 143)(91 144)(92 133)(93 134)(94 135)(95 136)(96 137)(109 146)(110 147)(111 148)(112 149)(113 150)(114 151)(115 152)(116 153)(117 154)(118 155)(119 156)(120 145)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G ··· 6N 7A 7B 7C 12A ··· 12H 12I ··· 12P 14A ··· 14I 21A ··· 21F 28A ··· 28L 42A ··· 42R 84A ··· 84X order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 7 7 7 12 ··· 12 12 ··· 12 14 ··· 14 21 ··· 21 28 ··· 28 42 ··· 42 84 ··· 84 size 1 1 1 1 7 7 7 7 1 1 1 1 1 1 7 7 7 7 1 ··· 1 7 ··· 7 2 2 2 1 ··· 1 7 ··· 7 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

120 irreducible representations

 dim 1 1 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 C4 C6 C6 C6 C6 C12 D7 D14 D14 C3×D7 C4×D7 C6×D7 C6×D7 C12×D7 kernel D7×C2×C12 C12×D7 C6×Dic7 C2×C84 C2×C6×D7 C2×C4×D7 C6×D7 C4×D7 C2×Dic7 C2×C28 C22×D7 D14 C2×C12 C12 C2×C6 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 2 8 8 2 2 2 16 3 6 3 6 12 12 6 24

Matrix representation of D7×C2×C12 in GL3(𝔽337) generated by

 336 0 0 0 1 0 0 0 1
,
 208 0 0 0 189 0 0 0 189
,
 1 0 0 0 0 1 0 336 227
,
 336 0 0 0 304 110 0 33 33
G:=sub<GL(3,GF(337))| [336,0,0,0,1,0,0,0,1],[208,0,0,0,189,0,0,0,189],[1,0,0,0,0,336,0,1,227],[336,0,0,0,304,33,0,110,33] >;

D7×C2×C12 in GAP, Magma, Sage, TeX

D_7\times C_2\times C_{12}
% in TeX

G:=Group("D7xC2xC12");
// GroupNames label

G:=SmallGroup(336,175);
// by ID

G=gap.SmallGroup(336,175);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-7,122,10373]);
// Polycyclic

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

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