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## G = C20.17D4order 160 = 25·5

### 17th non-split extension by C20 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C20.17D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — C20.17D4
 Lower central C5 — C2×C10 — C20.17D4
 Upper central C1 — C22 — C2×D4

Generators and relations for C20.17D4
G = < a,b,c | a20=b4=1, c2=a10, bab-1=a9, cac-1=a-1, cbc-1=a10b-1 >

Subgroups: 208 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C10, C10 [×2], C10 [×2], C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×4], C20 [×2], C2×C10, C2×C10 [×6], C4.4D4, Dic10 [×2], C2×Dic5 [×4], C2×C20, C5×D4 [×2], C22×C10 [×2], C4×Dic5, C23.D5 [×4], C2×Dic10, D4×C10, C20.17D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C5⋊D4 [×2], C22×D5, D42D5 [×2], C2×C5⋊D4, C20.17D4

Smallest permutation representation of C20.17D4
On 80 points
Generators in S80
```(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)
(1 67 36 52)(2 76 37 41)(3 65 38 50)(4 74 39 59)(5 63 40 48)(6 72 21 57)(7 61 22 46)(8 70 23 55)(9 79 24 44)(10 68 25 53)(11 77 26 42)(12 66 27 51)(13 75 28 60)(14 64 29 49)(15 73 30 58)(16 62 31 47)(17 71 32 56)(18 80 33 45)(19 69 34 54)(20 78 35 43)
(1 62 11 72)(2 61 12 71)(3 80 13 70)(4 79 14 69)(5 78 15 68)(6 77 16 67)(7 76 17 66)(8 75 18 65)(9 74 19 64)(10 73 20 63)(21 42 31 52)(22 41 32 51)(23 60 33 50)(24 59 34 49)(25 58 35 48)(26 57 36 47)(27 56 37 46)(28 55 38 45)(29 54 39 44)(30 53 40 43)```

`G:=sub<Sym(80)| (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), (1,67,36,52)(2,76,37,41)(3,65,38,50)(4,74,39,59)(5,63,40,48)(6,72,21,57)(7,61,22,46)(8,70,23,55)(9,79,24,44)(10,68,25,53)(11,77,26,42)(12,66,27,51)(13,75,28,60)(14,64,29,49)(15,73,30,58)(16,62,31,47)(17,71,32,56)(18,80,33,45)(19,69,34,54)(20,78,35,43), (1,62,11,72)(2,61,12,71)(3,80,13,70)(4,79,14,69)(5,78,15,68)(6,77,16,67)(7,76,17,66)(8,75,18,65)(9,74,19,64)(10,73,20,63)(21,42,31,52)(22,41,32,51)(23,60,33,50)(24,59,34,49)(25,58,35,48)(26,57,36,47)(27,56,37,46)(28,55,38,45)(29,54,39,44)(30,53,40,43)>;`

`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), (1,67,36,52)(2,76,37,41)(3,65,38,50)(4,74,39,59)(5,63,40,48)(6,72,21,57)(7,61,22,46)(8,70,23,55)(9,79,24,44)(10,68,25,53)(11,77,26,42)(12,66,27,51)(13,75,28,60)(14,64,29,49)(15,73,30,58)(16,62,31,47)(17,71,32,56)(18,80,33,45)(19,69,34,54)(20,78,35,43), (1,62,11,72)(2,61,12,71)(3,80,13,70)(4,79,14,69)(5,78,15,68)(6,77,16,67)(7,76,17,66)(8,75,18,65)(9,74,19,64)(10,73,20,63)(21,42,31,52)(22,41,32,51)(23,60,33,50)(24,59,34,49)(25,58,35,48)(26,57,36,47)(27,56,37,46)(28,55,38,45)(29,54,39,44)(30,53,40,43) );`

`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)], [(1,67,36,52),(2,76,37,41),(3,65,38,50),(4,74,39,59),(5,63,40,48),(6,72,21,57),(7,61,22,46),(8,70,23,55),(9,79,24,44),(10,68,25,53),(11,77,26,42),(12,66,27,51),(13,75,28,60),(14,64,29,49),(15,73,30,58),(16,62,31,47),(17,71,32,56),(18,80,33,45),(19,69,34,54),(20,78,35,43)], [(1,62,11,72),(2,61,12,71),(3,80,13,70),(4,79,14,69),(5,78,15,68),(6,77,16,67),(7,76,17,66),(8,75,18,65),(9,74,19,64),(10,73,20,63),(21,42,31,52),(22,41,32,51),(23,60,33,50),(24,59,34,49),(25,58,35,48),(26,57,36,47),(27,56,37,46),(28,55,38,45),(29,54,39,44),(30,53,40,43)])`

34 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 10A ··· 10F 10G ··· 10N 20A 20B 20C 20D order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 10 ··· 10 10 ··· 10 20 20 20 20 size 1 1 1 1 4 4 2 2 10 10 10 10 20 20 2 2 2 ··· 2 4 ··· 4 4 4 4 4

34 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 C5⋊D4 D4⋊2D5 kernel C20.17D4 C4×Dic5 C23.D5 C2×Dic10 D4×C10 C20 C2×D4 C10 C2×C4 C23 C4 C2 # reps 1 1 4 1 1 2 2 4 2 4 8 4

Matrix representation of C20.17D4 in GL4(𝔽41) generated by

 37 0 0 0 0 10 0 0 0 0 40 18 0 0 9 1
,
 0 1 0 0 40 0 0 0 0 0 9 2 0 0 1 32
,
 0 40 0 0 40 0 0 0 0 0 32 0 0 0 40 9
`G:=sub<GL(4,GF(41))| [37,0,0,0,0,10,0,0,0,0,40,9,0,0,18,1],[0,40,0,0,1,0,0,0,0,0,9,1,0,0,2,32],[0,40,0,0,40,0,0,0,0,0,32,40,0,0,0,9] >;`

C20.17D4 in GAP, Magma, Sage, TeX

`C_{20}._{17}D_4`
`% in TeX`

`G:=Group("C20.17D4");`
`// GroupNames label`

`G:=SmallGroup(160,157);`
`// by ID`

`G=gap.SmallGroup(160,157);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,55,506,116,4613]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^4=1,c^2=a^10,b*a*b^-1=a^9,c*a*c^-1=a^-1,c*b*c^-1=a^10*b^-1>;`
`// generators/relations`

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