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## G = C14×C3⋊D4order 336 = 24·3·7

### Direct product of C14 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C14×C3⋊D4
 Chief series C1 — C3 — C6 — C42 — S3×C14 — S3×C2×C14 — C14×C3⋊D4
 Lower central C3 — C6 — C14×C3⋊D4
 Upper central C1 — C2×C14 — C22×C14

Generators and relations for C14×C3⋊D4
G = < a,b,c,d | a14=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 216 in 108 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C7, C2×C4, D4, C23, C23, Dic3, D6, D6, C2×C6, C2×C6, C2×C6, C14, C14, C14, C2×D4, C21, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C28, C2×C14, C2×C14, C2×C14, S3×C7, C42, C42, C42, C2×C3⋊D4, C2×C28, C7×D4, C22×C14, C22×C14, C7×Dic3, S3×C14, S3×C14, C2×C42, C2×C42, C2×C42, D4×C14, Dic3×C14, C7×C3⋊D4, S3×C2×C14, C22×C42, C14×C3⋊D4
Quotients: C1, C2, C22, S3, C7, D4, C23, D6, C14, C2×D4, C3⋊D4, C22×S3, C2×C14, S3×C7, C2×C3⋊D4, C7×D4, C22×C14, S3×C14, D4×C14, C7×C3⋊D4, S3×C2×C14, C14×C3⋊D4

Smallest permutation representation of C14×C3⋊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 71)(2 108 72)(3 109 73)(4 110 74)(5 111 75)(6 112 76)(7 99 77)(8 100 78)(9 101 79)(10 102 80)(11 103 81)(12 104 82)(13 105 83)(14 106 84)(15 123 153)(16 124 154)(17 125 141)(18 126 142)(19 113 143)(20 114 144)(21 115 145)(22 116 146)(23 117 147)(24 118 148)(25 119 149)(26 120 150)(27 121 151)(28 122 152)(29 65 166)(30 66 167)(31 67 168)(32 68 155)(33 69 156)(34 70 157)(35 57 158)(36 58 159)(37 59 160)(38 60 161)(39 61 162)(40 62 163)(41 63 164)(42 64 165)(43 95 133)(44 96 134)(45 97 135)(46 98 136)(47 85 137)(48 86 138)(49 87 139)(50 88 140)(51 89 127)(52 90 128)(53 91 129)(54 92 130)(55 93 131)(56 94 132)
(1 150 66 98)(2 151 67 85)(3 152 68 86)(4 153 69 87)(5 154 70 88)(6 141 57 89)(7 142 58 90)(8 143 59 91)(9 144 60 92)(10 145 61 93)(11 146 62 94)(12 147 63 95)(13 148 64 96)(14 149 65 97)(15 33 139 74)(16 34 140 75)(17 35 127 76)(18 36 128 77)(19 37 129 78)(20 38 130 79)(21 39 131 80)(22 40 132 81)(23 41 133 82)(24 42 134 83)(25 29 135 84)(26 30 136 71)(27 31 137 72)(28 32 138 73)(43 104 117 164)(44 105 118 165)(45 106 119 166)(46 107 120 167)(47 108 121 168)(48 109 122 155)(49 110 123 156)(50 111 124 157)(51 112 125 158)(52 99 126 159)(53 100 113 160)(54 101 114 161)(55 102 115 162)(56 103 116 163)
(15 49)(16 50)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 166)(30 167)(31 168)(32 155)(33 156)(34 157)(35 158)(36 159)(37 160)(38 161)(39 162)(40 163)(41 164)(42 165)(71 107)(72 108)(73 109)(74 110)(75 111)(76 112)(77 99)(78 100)(79 101)(80 102)(81 103)(82 104)(83 105)(84 106)(85 151)(86 152)(87 153)(88 154)(89 141)(90 142)(91 143)(92 144)(93 145)(94 146)(95 147)(96 148)(97 149)(98 150)(113 129)(114 130)(115 131)(116 132)(117 133)(118 134)(119 135)(120 136)(121 137)(122 138)(123 139)(124 140)(125 127)(126 128)

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

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

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

126 conjugacy classes

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

126 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 C7 C14 C14 C14 C14 S3 D4 D6 C3⋊D4 S3×C7 C7×D4 S3×C14 C7×C3⋊D4 kernel C14×C3⋊D4 Dic3×C14 C7×C3⋊D4 S3×C2×C14 C22×C42 C2×C3⋊D4 C2×Dic3 C3⋊D4 C22×S3 C22×C6 C22×C14 C42 C2×C14 C14 C23 C6 C22 C2 # reps 1 1 4 1 1 6 6 24 6 6 1 2 3 4 6 12 18 24

Matrix representation of C14×C3⋊D4 in GL3(𝔽337) generated by

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

C14×C3⋊D4 in GAP, Magma, Sage, TeX

C_{14}\times C_3\rtimes D_4
% in TeX

G:=Group("C14xC3:D4");
// GroupNames label

G:=SmallGroup(336,193);
// by ID

G=gap.SmallGroup(336,193);
# by ID

G:=PCGroup([6,-2,-2,-2,-7,-2,-3,1082,8069]);
// Polycyclic

G:=Group<a,b,c,d|a^14=b^3=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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