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## G = C20.48D4order 160 = 25·5

### 5th non-split extension by C20 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C20.48D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C2×Dic10 — C20.48D4
 Lower central C5 — C2×C10 — C20.48D4
 Upper central C1 — C22 — C22×C4

Generators and relations for C20.48D4
G = < a,b,c | a20=b4=1, c2=a10, bab-1=cac-1=a-1, cbc-1=a10b-1 >

Subgroups: 192 in 74 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, Q8, C23, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊Q8, Dic10, C2×Dic5, C2×C20, C2×C20, C22×C10, C10.D4, C4⋊Dic5, C23.D5, C2×Dic10, C22×C20, C20.48D4
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, Dic10, C5⋊D4, C22×D5, C2×Dic10, C4○D20, C2×C5⋊D4, C20.48D4

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

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

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

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

46 conjugacy classes

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

46 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 D4 Q8 D5 C4○D4 D10 D10 C5⋊D4 Dic10 C4○D20 kernel C20.48D4 C10.D4 C4⋊Dic5 C23.D5 C2×Dic10 C22×C20 C20 C2×C10 C22×C4 C10 C2×C4 C23 C4 C22 C2 # reps 1 2 1 2 1 1 2 2 2 2 4 2 8 8 8

Matrix representation of C20.48D4 in GL4(𝔽41) generated by

 39 0 0 0 0 20 0 0 0 0 18 17 0 0 0 16
,
 0 1 0 0 1 0 0 0 0 0 1 17 0 0 24 40
,
 0 1 0 0 40 0 0 0 0 0 40 0 0 0 17 1
`G:=sub<GL(4,GF(41))| [39,0,0,0,0,20,0,0,0,0,18,0,0,0,17,16],[0,1,0,0,1,0,0,0,0,0,1,24,0,0,17,40],[0,40,0,0,1,0,0,0,0,0,40,17,0,0,0,1] >;`

C20.48D4 in GAP, Magma, Sage, TeX

`C_{20}._{48}D_4`
`% in TeX`

`G:=Group("C20.48D4");`
`// GroupNames label`

`G:=SmallGroup(160,145);`
`// by ID`

`G=gap.SmallGroup(160,145);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,217,103,218,4613]);`
`// Polycyclic`

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

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