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## G = D12⋊D7order 336 = 24·3·7

### 3rd semidirect product of D12 and D7 acting via D7/C7=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — D12⋊D7
 Chief series C1 — C7 — C21 — C42 — C3×Dic7 — S3×Dic7 — D12⋊D7
 Lower central C21 — C42 — D12⋊D7
 Upper central C1 — C2 — C4

Generators and relations for D12⋊D7
G = < a,b,c,d | a12=b2=c7=d2=1, bab=a-1, ac=ca, dad=a5, bc=cb, dbd=a10b, dcd=c-1 >

Subgroups: 436 in 80 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C7, C2×C4, D4, Q8, Dic3, C12, C12, D6, D6, D7, C14, C14, C4○D4, C21, C4×S3, D12, D12, C3×Q8, Dic7, Dic7, C28, D14, C2×C14, S3×C7, D21, C42, Q83S3, Dic14, C4×D7, C2×Dic7, C7⋊D4, C7×D4, C3×Dic7, Dic21, C84, S3×C14, D42, D42D7, S3×Dic7, C7⋊D12, C3×Dic14, C7×D12, C4×D21, D12⋊D7
Quotients: C1, C2, C22, S3, C23, D6, D7, C4○D4, C22×S3, D14, Q83S3, C22×D7, S3×D7, D42D7, C2×S3×D7, D12⋊D7

Smallest permutation representation of D12⋊D7
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)
(1 121)(2 132)(3 131)(4 130)(5 129)(6 128)(7 127)(8 126)(9 125)(10 124)(11 123)(12 122)(13 112)(14 111)(15 110)(16 109)(17 120)(18 119)(19 118)(20 117)(21 116)(22 115)(23 114)(24 113)(25 142)(26 141)(27 140)(28 139)(29 138)(30 137)(31 136)(32 135)(33 134)(34 133)(35 144)(36 143)(37 59)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(48 60)(61 85)(62 96)(63 95)(64 94)(65 93)(66 92)(67 91)(68 90)(69 89)(70 88)(71 87)(72 86)(73 100)(74 99)(75 98)(76 97)(77 108)(78 107)(79 106)(80 105)(81 104)(82 103)(83 102)(84 101)(145 166)(146 165)(147 164)(148 163)(149 162)(150 161)(151 160)(152 159)(153 158)(154 157)(155 168)(156 167)
(1 138 115 99 52 167 72)(2 139 116 100 53 168 61)(3 140 117 101 54 157 62)(4 141 118 102 55 158 63)(5 142 119 103 56 159 64)(6 143 120 104 57 160 65)(7 144 109 105 58 161 66)(8 133 110 106 59 162 67)(9 134 111 107 60 163 68)(10 135 112 108 49 164 69)(11 136 113 97 50 165 70)(12 137 114 98 51 166 71)(13 77 47 147 89 124 32)(14 78 48 148 90 125 33)(15 79 37 149 91 126 34)(16 80 38 150 92 127 35)(17 81 39 151 93 128 36)(18 82 40 152 94 129 25)(19 83 41 153 95 130 26)(20 84 42 154 96 131 27)(21 73 43 155 85 132 28)(22 74 44 156 86 121 29)(23 75 45 145 87 122 30)(24 76 46 146 88 123 31)
(1 72)(2 65)(3 70)(4 63)(5 68)(6 61)(7 66)(8 71)(9 64)(10 69)(11 62)(12 67)(13 37)(14 42)(15 47)(16 40)(17 45)(18 38)(19 43)(20 48)(21 41)(22 46)(23 39)(24 44)(25 150)(26 155)(27 148)(28 153)(29 146)(30 151)(31 156)(32 149)(33 154)(34 147)(35 152)(36 145)(49 112)(50 117)(51 110)(52 115)(53 120)(54 113)(55 118)(56 111)(57 116)(58 109)(59 114)(60 119)(73 83)(74 76)(75 81)(77 79)(78 84)(80 82)(85 130)(86 123)(87 128)(88 121)(89 126)(90 131)(91 124)(92 129)(93 122)(94 127)(95 132)(96 125)(97 101)(98 106)(100 104)(103 107)(133 166)(134 159)(135 164)(136 157)(137 162)(138 167)(139 160)(140 165)(141 158)(142 163)(143 168)(144 161)```

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 S3 D6 D6 D7 C4○D4 D14 D14 Q8⋊3S3 S3×D7 D4⋊2D7 C2×S3×D7 D12⋊D7 kernel D12⋊D7 S3×Dic7 C7⋊D12 C3×Dic14 C7×D12 C4×D21 Dic14 Dic7 C28 D12 C21 C12 D6 C7 C4 C3 C2 C1 # reps 1 2 2 1 1 1 1 2 1 3 2 3 6 1 3 3 3 6

Matrix representation of D12⋊D7 in GL6(𝔽337)

 189 0 0 0 0 0 266 148 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 141 0 0 0 0 43 335
,
 68 63 0 0 0 0 55 269 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 141 0 0 0 0 43 335
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 336 1 0 0 0 0 32 304 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 276 336 0 0 0 0 0 0 336 0 0 0 0 0 32 1 0 0 0 0 0 0 336 196 0 0 0 0 0 1

`G:=sub<GL(6,GF(337))| [189,266,0,0,0,0,0,148,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,43,0,0,0,0,141,335],[68,55,0,0,0,0,63,269,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,43,0,0,0,0,141,335],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,336,32,0,0,0,0,1,304,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,276,0,0,0,0,0,336,0,0,0,0,0,0,336,32,0,0,0,0,0,1,0,0,0,0,0,0,336,0,0,0,0,0,196,1] >;`

D12⋊D7 in GAP, Magma, Sage, TeX

`D_{12}\rtimes D_7`
`% in TeX`

`G:=Group("D12:D7");`
`// GroupNames label`

`G:=SmallGroup(336,141);`
`// by ID`

`G=gap.SmallGroup(336,141);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-7,48,218,116,50,490,10373]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^7=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;`
`// generators/relations`

׿
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