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## G = D12⋊5D7order 336 = 24·3·7

### The semidirect product of D12 and D7 acting through Inn(D12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — D12⋊5D7
 Chief series C1 — C7 — C21 — C42 — C6×D7 — C21⋊D4 — D12⋊5D7
 Lower central C21 — C42 — D12⋊5D7
 Upper central C1 — C2 — C4

Generators and relations for D125D7
G = < a,b,c,d | a12=b2=c7=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a6b, dcd=c-1 >

Subgroups: 404 in 80 conjugacy classes, 32 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C7, C2×C4, D4, Q8, Dic3, C12, C12, D6, C2×C6, D7, C14, C14, C4○D4, C21, Dic6, C4×S3, D12, C3⋊D4, C2×C12, Dic7, Dic7, C28, D14, C2×C14, S3×C7, C3×D7, C42, C4○D12, Dic14, C4×D7, C2×Dic7, C7⋊D4, C7×D4, C3×Dic7, Dic21, C84, C6×D7, S3×C14, D42D7, S3×Dic7, C21⋊D4, C12×D7, C7×D12, Dic42, D125D7
Quotients: C1, C2, C22, S3, C23, D6, D7, C4○D4, C22×S3, D14, C4○D12, C22×D7, S3×D7, D42D7, C2×S3×D7, D125D7

Smallest permutation representation of D125D7
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 39)(2 38)(3 37)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 124)(14 123)(15 122)(16 121)(17 132)(18 131)(19 130)(20 129)(21 128)(22 127)(23 126)(24 125)(25 92)(26 91)(27 90)(28 89)(29 88)(30 87)(31 86)(32 85)(33 96)(34 95)(35 94)(36 93)(49 136)(50 135)(51 134)(52 133)(53 144)(54 143)(55 142)(56 141)(57 140)(58 139)(59 138)(60 137)(61 105)(62 104)(63 103)(64 102)(65 101)(66 100)(67 99)(68 98)(69 97)(70 108)(71 107)(72 106)(73 166)(74 165)(75 164)(76 163)(77 162)(78 161)(79 160)(80 159)(81 158)(82 157)(83 168)(84 167)(109 150)(110 149)(111 148)(112 147)(113 146)(114 145)(115 156)(116 155)(117 154)(118 153)(119 152)(120 151)
(1 128 166 106 117 35 59)(2 129 167 107 118 36 60)(3 130 168 108 119 25 49)(4 131 157 97 120 26 50)(5 132 158 98 109 27 51)(6 121 159 99 110 28 52)(7 122 160 100 111 29 53)(8 123 161 101 112 30 54)(9 124 162 102 113 31 55)(10 125 163 103 114 32 56)(11 126 164 104 115 33 57)(12 127 165 105 116 34 58)(13 77 64 146 86 142 43)(14 78 65 147 87 143 44)(15 79 66 148 88 144 45)(16 80 67 149 89 133 46)(17 81 68 150 90 134 47)(18 82 69 151 91 135 48)(19 83 70 152 92 136 37)(20 84 71 153 93 137 38)(21 73 72 154 94 138 39)(22 74 61 155 95 139 40)(23 75 62 156 96 140 41)(24 76 63 145 85 141 42)
(1 59)(2 60)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 92)(14 93)(15 94)(16 95)(17 96)(18 85)(19 86)(20 87)(21 88)(22 89)(23 90)(24 91)(25 130)(26 131)(27 132)(28 121)(29 122)(30 123)(31 124)(32 125)(33 126)(34 127)(35 128)(36 129)(37 142)(38 143)(39 144)(40 133)(41 134)(42 135)(43 136)(44 137)(45 138)(46 139)(47 140)(48 141)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 148)(74 149)(75 150)(76 151)(77 152)(78 153)(79 154)(80 155)(81 156)(82 145)(83 146)(84 147)(109 158)(110 159)(111 160)(112 161)(113 162)(114 163)(115 164)(116 165)(117 166)(118 167)(119 168)(120 157)```

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

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

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

45 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 28B 28C 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 42 42 42 84 ··· 84 size 1 1 6 6 14 2 2 7 7 42 42 2 14 14 2 2 2 2 2 14 14 2 2 2 12 ··· 12 4 4 4 4 4 4 4 4 4 4 ··· 4

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 S3 D6 D6 D6 D7 C4○D4 D14 D14 C4○D12 S3×D7 D4⋊2D7 C2×S3×D7 D12⋊5D7 kernel D12⋊5D7 S3×Dic7 C21⋊D4 C12×D7 C7×D12 Dic42 C4×D7 Dic7 C28 D14 D12 C21 C12 D6 C7 C4 C3 C2 C1 # reps 1 2 2 1 1 1 1 1 1 1 3 2 3 6 4 3 3 3 6

Matrix representation of D125D7 in GL6(𝔽337)

 336 0 0 0 0 0 0 336 0 0 0 0 0 0 2 196 0 0 0 0 294 336 0 0 0 0 0 0 189 0 0 0 0 0 189 148
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 93 0 0 0 0 29 0 0 0 0 0 0 0 148 41 0 0 0 0 148 189
,
 34 1 0 0 0 0 335 109 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 109 336 0 0 0 0 85 228 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 336

`G:=sub<GL(6,GF(337))| [336,0,0,0,0,0,0,336,0,0,0,0,0,0,2,294,0,0,0,0,196,336,0,0,0,0,0,0,189,189,0,0,0,0,0,148],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,29,0,0,0,0,93,0,0,0,0,0,0,0,148,148,0,0,0,0,41,189],[34,335,0,0,0,0,1,109,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[109,85,0,0,0,0,336,228,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,336] >;`

D125D7 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_5D_7`
`% in TeX`

`G:=Group("D12:5D7");`
`// GroupNames label`

`G:=SmallGroup(336,145);`
`// by ID`

`G=gap.SmallGroup(336,145);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-7,121,116,50,490,10373]);`
`// 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,a*d=d*a,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;`
`// generators/relations`

׿
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