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## G = D4.D17order 272 = 24·17

### The non-split extension by D4 of D17 acting via D17/C17=C2

Aliases: D4.D17, C34.8D4, C4.2D34, C172SD16, Dic342C2, C68.2C22, C173C82C2, (D4×C17).1C2, C2.5(C17⋊D4), SmallGroup(272,16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C68 — D4.D17
 Chief series C1 — C17 — C34 — C68 — Dic34 — D4.D17
 Lower central C17 — C34 — C68 — D4.D17
 Upper central C1 — C2 — C4 — D4

Generators and relations for D4.D17
G = < a,b,c,d | a4=b2=c17=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

Smallest permutation representation of D4.D17
On 136 points
Generators in S136
```(1 64 30 35)(2 65 31 36)(3 66 32 37)(4 67 33 38)(5 68 34 39)(6 52 18 40)(7 53 19 41)(8 54 20 42)(9 55 21 43)(10 56 22 44)(11 57 23 45)(12 58 24 46)(13 59 25 47)(14 60 26 48)(15 61 27 49)(16 62 28 50)(17 63 29 51)(69 103 86 120)(70 104 87 121)(71 105 88 122)(72 106 89 123)(73 107 90 124)(74 108 91 125)(75 109 92 126)(76 110 93 127)(77 111 94 128)(78 112 95 129)(79 113 96 130)(80 114 97 131)(81 115 98 132)(82 116 99 133)(83 117 100 134)(84 118 101 135)(85 119 102 136)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 65)(32 66)(33 67)(34 68)(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 97)(81 98)(82 99)(83 100)(84 101)(85 102)
(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)
(1 102 30 85)(2 101 31 84)(3 100 32 83)(4 99 33 82)(5 98 34 81)(6 97 18 80)(7 96 19 79)(8 95 20 78)(9 94 21 77)(10 93 22 76)(11 92 23 75)(12 91 24 74)(13 90 25 73)(14 89 26 72)(15 88 27 71)(16 87 28 70)(17 86 29 69)(35 136 64 119)(36 135 65 118)(37 134 66 117)(38 133 67 116)(39 132 68 115)(40 131 52 114)(41 130 53 113)(42 129 54 112)(43 128 55 111)(44 127 56 110)(45 126 57 109)(46 125 58 108)(47 124 59 107)(48 123 60 106)(49 122 61 105)(50 121 62 104)(51 120 63 103)```

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

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

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

47 conjugacy classes

 class 1 2A 2B 4A 4B 8A 8B 17A ··· 17H 34A ··· 34H 34I ··· 34X 68A ··· 68H order 1 2 2 4 4 8 8 17 ··· 17 34 ··· 34 34 ··· 34 68 ··· 68 size 1 1 4 2 68 34 34 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

47 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 4 type + + + + + + + - image C1 C2 C2 C2 D4 SD16 D17 D34 C17⋊D4 D4.D17 kernel D4.D17 C17⋊3C8 Dic34 D4×C17 C34 C17 D4 C4 C2 C1 # reps 1 1 1 1 1 2 8 8 16 8

Matrix representation of D4.D17 in GL4(𝔽137) generated by

 1 0 0 0 0 1 0 0 0 0 1 119 0 0 61 136
,
 1 0 0 0 0 1 0 0 0 0 1 119 0 0 0 136
,
 103 1 0 0 22 92 0 0 0 0 1 0 0 0 0 1
,
 71 76 0 0 13 66 0 0 0 0 0 89 0 0 20 0
`G:=sub<GL(4,GF(137))| [1,0,0,0,0,1,0,0,0,0,1,61,0,0,119,136],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,119,136],[103,22,0,0,1,92,0,0,0,0,1,0,0,0,0,1],[71,13,0,0,76,66,0,0,0,0,0,20,0,0,89,0] >;`

D4.D17 in GAP, Magma, Sage, TeX

`D_4.D_{17}`
`% in TeX`

`G:=Group("D4.D17");`
`// GroupNames label`

`G:=SmallGroup(272,16);`
`// by ID`

`G=gap.SmallGroup(272,16);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-17,40,61,182,97,42,6404]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=c^17=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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