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## G = C17⋊D4order 136 = 23·17

### The semidirect product of C17 and D4 acting via D4/C22=C2

Aliases: C172D4, C22⋊D17, D342C2, Dic17⋊C2, C2.5D34, C34.5C22, (C2×C34)⋊2C2, SmallGroup(136,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C34 — C17⋊D4
 Chief series C1 — C17 — C34 — D34 — C17⋊D4
 Lower central C17 — C34 — C17⋊D4
 Upper central C1 — C2 — C22

Generators and relations for C17⋊D4
G = < a,b,c | a17=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

Smallest permutation representation of C17⋊D4
On 68 points
Generators in S68
```(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)
(1 36 28 67)(2 35 29 66)(3 51 30 65)(4 50 31 64)(5 49 32 63)(6 48 33 62)(7 47 34 61)(8 46 18 60)(9 45 19 59)(10 44 20 58)(11 43 21 57)(12 42 22 56)(13 41 23 55)(14 40 24 54)(15 39 25 53)(16 38 26 52)(17 37 27 68)
(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)(18 21)(19 20)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(35 68)(36 67)(37 66)(38 65)(39 64)(40 63)(41 62)(42 61)(43 60)(44 59)(45 58)(46 57)(47 56)(48 55)(49 54)(50 53)(51 52)```

`G:=sub<Sym(68)| (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), (1,36,28,67)(2,35,29,66)(3,51,30,65)(4,50,31,64)(5,49,32,63)(6,48,33,62)(7,47,34,61)(8,46,18,60)(9,45,19,59)(10,44,20,58)(11,43,21,57)(12,42,22,56)(13,41,23,55)(14,40,24,54)(15,39,25,53)(16,38,26,52)(17,37,27,68), (2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(18,21)(19,20)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)>;`

`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), (1,36,28,67)(2,35,29,66)(3,51,30,65)(4,50,31,64)(5,49,32,63)(6,48,33,62)(7,47,34,61)(8,46,18,60)(9,45,19,59)(10,44,20,58)(11,43,21,57)(12,42,22,56)(13,41,23,55)(14,40,24,54)(15,39,25,53)(16,38,26,52)(17,37,27,68), (2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(18,21)(19,20)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52) );`

`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)], [(1,36,28,67),(2,35,29,66),(3,51,30,65),(4,50,31,64),(5,49,32,63),(6,48,33,62),(7,47,34,61),(8,46,18,60),(9,45,19,59),(10,44,20,58),(11,43,21,57),(12,42,22,56),(13,41,23,55),(14,40,24,54),(15,39,25,53),(16,38,26,52),(17,37,27,68)], [(2,17),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10),(18,21),(19,20),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(35,68),(36,67),(37,66),(38,65),(39,64),(40,63),(41,62),(42,61),(43,60),(44,59),(45,58),(46,57),(47,56),(48,55),(49,54),(50,53),(51,52)])`

C17⋊D4 is a maximal subgroup of
D685C2  D4×D17  D42D17  C51⋊D4  C17⋊D12  C517D4  C17⋊S4
C17⋊D4 is a maximal quotient of
C34.D4  D34⋊C4  D4⋊D17  D4.D17  Q8⋊D17  C17⋊Q16  C23.D17  C51⋊D4  C17⋊D12  C517D4

37 conjugacy classes

 class 1 2A 2B 2C 4 17A ··· 17H 34A ··· 34X order 1 2 2 2 4 17 ··· 17 34 ··· 34 size 1 1 2 34 34 2 ··· 2 2 ··· 2

37 irreducible representations

 dim 1 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 D4 D17 D34 C17⋊D4 kernel C17⋊D4 Dic17 D34 C2×C34 C17 C22 C2 C1 # reps 1 1 1 1 1 8 8 16

Matrix representation of C17⋊D4 in GL2(𝔽137) generated by

 0 1 136 58
,
 10 28 60 127
,
 1 0 58 136
`G:=sub<GL(2,GF(137))| [0,136,1,58],[10,60,28,127],[1,58,0,136] >;`

C17⋊D4 in GAP, Magma, Sage, TeX

`C_{17}\rtimes D_4`
`% in TeX`

`G:=Group("C17:D4");`
`// GroupNames label`

`G:=SmallGroup(136,8);`
`// by ID`

`G=gap.SmallGroup(136,8);`
`# by ID`

`G:=PCGroup([4,-2,-2,-2,-17,49,2051]);`
`// Polycyclic`

`G:=Group<a,b,c|a^17=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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