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## G = C22.D20order 160 = 25·5

### 3rd non-split extension by C22 of D20 acting via D20/D10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C22.D20
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C5⋊D4 — C22.D20
 Lower central C5 — C2×C10 — C22.D20
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C22.D20
G = < a,b,c,d | a2=b2=c20=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=bc-1 >

Subgroups: 256 in 78 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C22.D4, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C4⋊Dic5, D10⋊C4, C5×C22⋊C4, C22×Dic5, C2×C5⋊D4, C22.D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22.D4, D20, C22×D5, C2×D20, D42D5, C22.D20

Smallest permutation representation of C22.D20
On 80 points
Generators in S80
```(1 31)(2 69)(3 33)(4 71)(5 35)(6 73)(7 37)(8 75)(9 39)(10 77)(11 21)(12 79)(13 23)(14 61)(15 25)(16 63)(17 27)(18 65)(19 29)(20 67)(22 57)(24 59)(26 41)(28 43)(30 45)(32 47)(34 49)(36 51)(38 53)(40 55)(42 64)(44 66)(46 68)(48 70)(50 72)(52 74)(54 76)(56 78)(58 80)(60 62)
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 41)(17 42)(18 43)(19 44)(20 45)(21 78)(22 79)(23 80)(24 61)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 73)(37 74)(38 75)(39 76)(40 77)
(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 45 46 20)(2 19 47 44)(3 43 48 18)(4 17 49 42)(5 41 50 16)(6 15 51 60)(7 59 52 14)(8 13 53 58)(9 57 54 12)(10 11 55 56)(21 40 78 77)(22 76 79 39)(23 38 80 75)(24 74 61 37)(25 36 62 73)(26 72 63 35)(27 34 64 71)(28 70 65 33)(29 32 66 69)(30 68 67 31)```

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 size 1 1 1 1 2 2 20 4 4 10 10 10 10 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 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D20 D4⋊2D5 kernel C22.D20 C4⋊Dic5 D10⋊C4 C5×C22⋊C4 C22×Dic5 C2×C5⋊D4 C2×C10 C22⋊C4 C10 C2×C4 C23 C22 C2 # reps 1 2 2 1 1 1 2 2 4 4 2 8 4

Matrix representation of C22.D20 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 21 1 0 0 11 20
,
 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 16 30 0 0 27 2 0 0 0 0 25 9 0 0 40 16
,
 39 30 0 0 4 2 0 0 0 0 25 9 0 0 17 16
`G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,21,11,0,0,1,20],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[16,27,0,0,30,2,0,0,0,0,25,40,0,0,9,16],[39,4,0,0,30,2,0,0,0,0,25,17,0,0,9,16] >;`

C22.D20 in GAP, Magma, Sage, TeX

`C_2^2.D_{20}`
`% in TeX`

`G:=Group("C2^2.D20");`
`// GroupNames label`

`G:=SmallGroup(160,107);`
`// by ID`

`G=gap.SmallGroup(160,107);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,218,188,122,4613]);`
`// Polycyclic`

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

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