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G = D4.D19order 304 = 24·19

The non-split extension by D4 of D19 acting via D19/C19=C2

Aliases: D4.D19, C4.2D38, C38.8D4, C192SD16, Dic382C2, C76.2C22, C19⋊C82C2, (D4×C19).1C2, C2.5(C19⋊D4), SmallGroup(304,15)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C76 — D4.D19
 Chief series C1 — C19 — C38 — C76 — Dic38 — D4.D19
 Lower central C19 — C38 — C76 — D4.D19
 Upper central C1 — C2 — C4 — D4

Generators and relations for D4.D19
G = < a,b,c,d | a4=b2=c19=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

Smallest permutation representation of D4.D19
On 152 points
Generators in S152
```(1 67 24 56)(2 68 25 57)(3 69 26 39)(4 70 27 40)(5 71 28 41)(6 72 29 42)(7 73 30 43)(8 74 31 44)(9 75 32 45)(10 76 33 46)(11 58 34 47)(12 59 35 48)(13 60 36 49)(14 61 37 50)(15 62 38 51)(16 63 20 52)(17 64 21 53)(18 65 22 54)(19 66 23 55)(77 115 96 134)(78 116 97 135)(79 117 98 136)(80 118 99 137)(81 119 100 138)(82 120 101 139)(83 121 102 140)(84 122 103 141)(85 123 104 142)(86 124 105 143)(87 125 106 144)(88 126 107 145)(89 127 108 146)(90 128 109 147)(91 129 110 148)(92 130 111 149)(93 131 112 150)(94 132 113 151)(95 133 114 152)
(1 56)(2 57)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 63)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 71)(29 72)(30 73)(31 74)(32 75)(33 76)(34 58)(35 59)(36 60)(37 61)(38 62)(77 96)(78 97)(79 98)(80 99)(81 100)(82 101)(83 102)(84 103)(85 104)(86 105)(87 106)(88 107)(89 108)(90 109)(91 110)(92 111)(93 112)(94 113)(95 114)
(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)
(1 114 24 95)(2 113 25 94)(3 112 26 93)(4 111 27 92)(5 110 28 91)(6 109 29 90)(7 108 30 89)(8 107 31 88)(9 106 32 87)(10 105 33 86)(11 104 34 85)(12 103 35 84)(13 102 36 83)(14 101 37 82)(15 100 38 81)(16 99 20 80)(17 98 21 79)(18 97 22 78)(19 96 23 77)(39 150 69 131)(40 149 70 130)(41 148 71 129)(42 147 72 128)(43 146 73 127)(44 145 74 126)(45 144 75 125)(46 143 76 124)(47 142 58 123)(48 141 59 122)(49 140 60 121)(50 139 61 120)(51 138 62 119)(52 137 63 118)(53 136 64 117)(54 135 65 116)(55 134 66 115)(56 152 67 133)(57 151 68 132)```

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

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

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

52 conjugacy classes

 class 1 2A 2B 4A 4B 8A 8B 19A ··· 19I 38A ··· 38I 38J ··· 38AA 76A ··· 76I order 1 2 2 4 4 8 8 19 ··· 19 38 ··· 38 38 ··· 38 76 ··· 76 size 1 1 4 2 76 38 38 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

52 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 4 type + + + + + + + - image C1 C2 C2 C2 D4 SD16 D19 D38 C19⋊D4 D4.D19 kernel D4.D19 C19⋊C8 Dic38 D4×C19 C38 C19 D4 C4 C2 C1 # reps 1 1 1 1 1 2 9 9 18 9

Matrix representation of D4.D19 in GL4(𝔽457) generated by

 1 0 0 0 0 1 0 0 0 0 203 327 0 0 317 254
,
 1 0 0 0 0 1 0 0 0 0 203 327 0 0 92 254
,
 381 1 0 0 452 433 0 0 0 0 1 0 0 0 0 1
,
 422 394 0 0 150 35 0 0 0 0 81 337 0 0 108 376
`G:=sub<GL(4,GF(457))| [1,0,0,0,0,1,0,0,0,0,203,317,0,0,327,254],[1,0,0,0,0,1,0,0,0,0,203,92,0,0,327,254],[381,452,0,0,1,433,0,0,0,0,1,0,0,0,0,1],[422,150,0,0,394,35,0,0,0,0,81,108,0,0,337,376] >;`

D4.D19 in GAP, Magma, Sage, TeX

`D_4.D_{19}`
`% in TeX`

`G:=Group("D4.D19");`
`// GroupNames label`

`G:=SmallGroup(304,15);`
`// by ID`

`G=gap.SmallGroup(304,15);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-19,40,61,182,97,42,7204]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=c^19=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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