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## G = C20⋊D6order 240 = 24·3·5

### 2nd semidirect product of C20 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C20⋊D6
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — C20⋊D6
 Lower central C15 — C30 — C20⋊D6
 Upper central C1 — C2 — C4

Generators and relations for C20⋊D6
G = < a,b,c | a20=b6=c2=1, bab-1=a-1, cac=a9, cbc=b-1 >

Subgroups: 584 in 108 conjugacy classes, 34 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C2×D4, Dic5, C20, D10, D10, C2×C10, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C5×S3, C3×D5, D15, C30, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, S3×D4, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, D4×D5, C15⋊D4, C3×D20, C5×D12, C4×D15, C2×S3×D5, C20⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C22×D5, S3×D4, S3×D5, D4×D5, C2×S3×D5, C20⋊D6

Smallest permutation representation of C20⋊D6
On 60 points
Generators in S60
```(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)
(1 50 37 6 45 22)(2 49 38 5 46 21)(3 48 39 4 47 40)(7 44 23 20 51 36)(8 43 24 19 52 35)(9 42 25 18 53 34)(10 41 26 17 54 33)(11 60 27 16 55 32)(12 59 28 15 56 31)(13 58 29 14 57 30)
(1 27)(2 36)(3 25)(4 34)(5 23)(6 32)(7 21)(8 30)(9 39)(10 28)(11 37)(12 26)(13 35)(14 24)(15 33)(16 22)(17 31)(18 40)(19 29)(20 38)(41 59)(42 48)(43 57)(44 46)(45 55)(47 53)(49 51)(50 60)(52 58)(54 56)```

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 5A 5B 6A 6B 6C 10A 10B 10C 10D 10E 10F 12 15A 15B 20A 20B 30A 30B 60A 60B 60C 60D order 1 2 2 2 2 2 2 2 3 4 4 5 5 6 6 6 10 10 10 10 10 10 12 15 15 20 20 30 30 60 60 60 60 size 1 1 6 6 10 10 15 15 2 2 30 2 2 2 20 20 2 2 12 12 12 12 4 4 4 4 4 4 4 4 4 4 4

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D5 D6 D6 D10 D10 S3×D4 S3×D5 D4×D5 C2×S3×D5 C20⋊D6 kernel C20⋊D6 C15⋊D4 C3×D20 C5×D12 C4×D15 C2×S3×D5 D20 D15 D12 C20 D10 C12 D6 C5 C4 C3 C2 C1 # reps 1 2 1 1 1 2 1 2 2 1 2 2 4 1 2 2 2 4

Matrix representation of C20⋊D6 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 60 44 0 0 0 0 0 0 60 3 0 0 0 0 40 1
,
 1 1 0 0 0 0 60 0 0 0 0 0 0 0 44 17 0 0 0 0 1 17 0 0 0 0 0 0 1 58 0 0 0 0 0 60
,
 60 60 0 0 0 0 0 1 0 0 0 0 0 0 17 44 0 0 0 0 60 44 0 0 0 0 0 0 60 0 0 0 0 0 0 60

`G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,1,44,0,0,0,0,0,0,60,40,0,0,0,0,3,1],[1,60,0,0,0,0,1,0,0,0,0,0,0,0,44,1,0,0,0,0,17,17,0,0,0,0,0,0,1,0,0,0,0,0,58,60],[60,0,0,0,0,0,60,1,0,0,0,0,0,0,17,60,0,0,0,0,44,44,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;`

C20⋊D6 in GAP, Magma, Sage, TeX

`C_{20}\rtimes D_6`
`% in TeX`

`G:=Group("C20:D6");`
`// GroupNames label`

`G:=SmallGroup(240,138);`
`// by ID`

`G=gap.SmallGroup(240,138);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,218,116,50,490,6917]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^6=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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