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## G = D6.D14order 336 = 24·3·7

### 3rd non-split extension by D6 of D14 acting via D14/C14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — D6.D14
 Chief series C1 — C7 — C21 — C42 — C6×D7 — C21⋊D4 — D6.D14
 Lower central C21 — C42 — D6.D14
 Upper central C1 — C4

Generators and relations for D6.D14
G = < a,b,c,d | a14=b2=1, c6=d2=a7, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a7b, dcd-1=c5 >

Subgroups: 452 in 80 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C7, C2×C4, D4, Q8, Dic3, Dic3, C12, C12, D6, D6, C2×C6, D7, C14, C14, C4○D4, C21, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12, Dic7, Dic7, C28, C28, D14, D14, C2×C14, S3×C7, C3×D7, D21, C42, C4○D12, Dic14, C4×D7, C4×D7, D28, C7⋊D4, C2×C28, C7×Dic3, C3×Dic7, Dic21, C84, C6×D7, S3×C14, D42, C4○D28, C21⋊D4, C3⋊D28, C7⋊D12, C21⋊Q8, C12×D7, S3×C28, C4×D21, D6.D14
Quotients: C1, C2, C22, S3, C23, D6, D7, C4○D4, C22×S3, D14, C4○D12, C22×D7, S3×D7, C4○D28, C2×S3×D7, D6.D14

Smallest permutation representation of D6.D14
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)
(1 102)(2 101)(3 100)(4 99)(5 112)(6 111)(7 110)(8 109)(9 108)(10 107)(11 106)(12 105)(13 104)(14 103)(15 79)(16 78)(17 77)(18 76)(19 75)(20 74)(21 73)(22 72)(23 71)(24 84)(25 83)(26 82)(27 81)(28 80)(29 146)(30 145)(31 144)(32 143)(33 142)(34 141)(35 154)(36 153)(37 152)(38 151)(39 150)(40 149)(41 148)(42 147)(43 119)(44 118)(45 117)(46 116)(47 115)(48 114)(49 113)(50 126)(51 125)(52 124)(53 123)(54 122)(55 121)(56 120)(57 132)(58 131)(59 130)(60 129)(61 128)(62 127)(63 140)(64 139)(65 138)(66 137)(67 136)(68 135)(69 134)(70 133)(85 156)(86 155)(87 168)(88 167)(89 166)(90 165)(91 164)(92 163)(93 162)(94 161)(95 160)(96 159)(97 158)(98 157)
(1 63 54 83 29 167 8 70 47 76 36 160)(2 64 55 84 30 168 9 57 48 77 37 161)(3 65 56 71 31 155 10 58 49 78 38 162)(4 66 43 72 32 156 11 59 50 79 39 163)(5 67 44 73 33 157 12 60 51 80 40 164)(6 68 45 74 34 158 13 61 52 81 41 165)(7 69 46 75 35 159 14 62 53 82 42 166)(15 150 92 99 137 119 22 143 85 106 130 126)(16 151 93 100 138 120 23 144 86 107 131 113)(17 152 94 101 139 121 24 145 87 108 132 114)(18 153 95 102 140 122 25 146 88 109 133 115)(19 154 96 103 127 123 26 147 89 110 134 116)(20 141 97 104 128 124 27 148 90 111 135 117)(21 142 98 105 129 125 28 149 91 112 136 118)
(1 123 8 116)(2 124 9 117)(3 125 10 118)(4 126 11 119)(5 113 12 120)(6 114 13 121)(7 115 14 122)(15 163 22 156)(16 164 23 157)(17 165 24 158)(18 166 25 159)(19 167 26 160)(20 168 27 161)(21 155 28 162)(29 154 36 147)(30 141 37 148)(31 142 38 149)(32 143 39 150)(33 144 40 151)(34 145 41 152)(35 146 42 153)(43 106 50 99)(44 107 51 100)(45 108 52 101)(46 109 53 102)(47 110 54 103)(48 111 55 104)(49 112 56 105)(57 128 64 135)(58 129 65 136)(59 130 66 137)(60 131 67 138)(61 132 68 139)(62 133 69 140)(63 134 70 127)(71 91 78 98)(72 92 79 85)(73 93 80 86)(74 94 81 87)(75 95 82 88)(76 96 83 89)(77 97 84 90)```

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D7 C4○D4 D14 D14 D14 C4○D12 C4○D28 S3×D7 C2×S3×D7 D6.D14 kernel D6.D14 C21⋊D4 C3⋊D28 C7⋊D12 C21⋊Q8 C12×D7 S3×C28 C4×D21 C4×D7 Dic7 C28 D14 C4×S3 C21 Dic3 C12 D6 C7 C3 C4 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 3 2 3 3 3 4 12 3 3 6

Matrix representation of D6.D14 in GL4(𝔽337) generated by

 304 304 0 0 33 227 0 0 0 0 1 0 0 0 0 1
,
 80 257 0 0 219 257 0 0 0 0 336 0 0 0 0 336
,
 189 0 0 0 0 189 0 0 0 0 2 325 0 0 253 336
,
 243 225 0 0 112 94 0 0 0 0 2 325 0 0 253 335
`G:=sub<GL(4,GF(337))| [304,33,0,0,304,227,0,0,0,0,1,0,0,0,0,1],[80,219,0,0,257,257,0,0,0,0,336,0,0,0,0,336],[189,0,0,0,0,189,0,0,0,0,2,253,0,0,325,336],[243,112,0,0,225,94,0,0,0,0,2,253,0,0,325,335] >;`

D6.D14 in GAP, Magma, Sage, TeX

`D_6.D_{14}`
`% in TeX`

`G:=Group("D6.D14");`
`// GroupNames label`

`G:=SmallGroup(336,144);`
`// by ID`

`G=gap.SmallGroup(336,144);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-7,121,50,490,10373]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^14=b^2=1,c^6=d^2=a^7,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^7*b,d*c*d^-1=c^5>;`
`// generators/relations`

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