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## G = C20⋊2D4order 160 = 25·5

### 2nd semidirect product of C20 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C20⋊2D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C20⋊2D4
 Lower central C5 — C2×C10 — C20⋊2D4
 Upper central C1 — C22 — C2×D4

Generators and relations for C202D4
G = < a,b,c | a20=b4=c2=1, bab-1=a-1, cac=a9, cbc=b-1 >

Subgroups: 304 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×3], C22, C22 [×10], C5, C2×C4, C2×C4 [×5], D4 [×6], C23 [×2], C23, D5 [×2], C10 [×3], C10 [×2], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], Dic5 [×3], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×6], C4⋊D4, C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×C10 [×2], C4⋊Dic5, C23.D5 [×2], C2×C4×D5, C2×C5⋊D4 [×2], D4×C10, C202D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C5⋊D4 [×2], C22×D5, D4×D5, D42D5, C2×C5⋊D4, C202D4

Smallest permutation representation of C202D4
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 76 56 23)(2 75 57 22)(3 74 58 21)(4 73 59 40)(5 72 60 39)(6 71 41 38)(7 70 42 37)(8 69 43 36)(9 68 44 35)(10 67 45 34)(11 66 46 33)(12 65 47 32)(13 64 48 31)(14 63 49 30)(15 62 50 29)(16 61 51 28)(17 80 52 27)(18 79 53 26)(19 78 54 25)(20 77 55 24)
(1 11)(2 20)(3 9)(4 18)(5 7)(6 16)(8 14)(10 12)(13 19)(15 17)(21 68)(22 77)(23 66)(24 75)(25 64)(26 73)(27 62)(28 71)(29 80)(30 69)(31 78)(32 67)(33 76)(34 65)(35 74)(36 63)(37 72)(38 61)(39 70)(40 79)(41 51)(42 60)(43 49)(44 58)(45 47)(46 56)(48 54)(50 52)(53 59)(55 57)```

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

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

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

34 conjugacy classes

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

34 irreducible representations

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

Matrix representation of C202D4 in GL4(𝔽41) generated by

 1 1 0 0 5 6 0 0 0 0 8 14 0 0 10 33
,
 20 3 0 0 3 21 0 0 0 0 1 23 0 0 0 40
,
 6 7 0 0 36 35 0 0 0 0 40 0 0 0 0 40
`G:=sub<GL(4,GF(41))| [1,5,0,0,1,6,0,0,0,0,8,10,0,0,14,33],[20,3,0,0,3,21,0,0,0,0,1,0,0,0,23,40],[6,36,0,0,7,35,0,0,0,0,40,0,0,0,0,40] >;`

C202D4 in GAP, Magma, Sage, TeX

`C_{20}\rtimes_2D_4`
`% in TeX`

`G:=Group("C20:2D4");`
`// GroupNames label`

`G:=SmallGroup(160,159);`
`// by ID`

`G=gap.SmallGroup(160,159);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,218,188,4613]);`
`// Polycyclic`

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

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