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## G = C6.D30order 360 = 23·32·5

### 3rd non-split extension by C6 of D30 acting via D30/D15=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — C6.D30
 Chief series C1 — C5 — C15 — C3×C15 — C3×C30 — C3×Dic15 — C6.D30
 Lower central C3×C15 — C6.D30
 Upper central C1 — C2

Generators and relations for C6.D30
G = < a,b,c | a6=c2=1, b30=a3, bab-1=cac=a-1, cbc=b29 >

Subgroups: 580 in 74 conjugacy classes, 26 normal (24 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, C2×C4, C32, D5, C10, Dic3, Dic3, C12, D6, C15, C15, C3⋊S3, C3×C6, Dic5, C20, D10, C4×S3, D15, C30, C30, C3×Dic3, C3×Dic3, C2×C3⋊S3, C4×D5, C3×C15, C5×Dic3, C3×Dic5, Dic15, C60, D30, C6.D6, C3⋊D15, C3×C30, D30.C2, C4×D15, Dic3×C15, C3×Dic15, C2×C3⋊D15, C6.D30
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, D6, D10, C4×S3, D15, S32, C4×D5, S3×D5, D30, C6.D6, D30.C2, C4×D15, S3×D15, C6.D30

Smallest permutation representation of C6.D30
On 60 points
Generators in S60
```(1 11 21 31 41 51)(2 52 42 32 22 12)(3 13 23 33 43 53)(4 54 44 34 24 14)(5 15 25 35 45 55)(6 56 46 36 26 16)(7 17 27 37 47 57)(8 58 48 38 28 18)(9 19 29 39 49 59)(10 60 50 40 30 20)
(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 45)(2 14)(3 43)(4 12)(5 41)(6 10)(7 39)(9 37)(11 35)(13 33)(15 31)(16 60)(17 29)(18 58)(19 27)(20 56)(21 25)(22 54)(24 52)(26 50)(28 48)(30 46)(32 44)(34 42)(36 40)(47 59)(49 57)(51 55)```

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 5A 5B 6A 6B 6C 10A 10B 12A 12B 12C 12D 15A 15B 15C 15D 15E ··· 15J 20A 20B 20C 20D 30A 30B 30C 30D 30E ··· 30J 60A ··· 60H order 1 2 2 2 3 3 3 4 4 4 4 5 5 6 6 6 10 10 12 12 12 12 15 15 15 15 15 ··· 15 20 20 20 20 30 30 30 30 30 ··· 30 60 ··· 60 size 1 1 45 45 2 2 4 3 3 15 15 2 2 2 2 4 2 2 6 6 30 30 2 2 2 2 4 ··· 4 6 6 6 6 2 2 2 2 4 ··· 4 6 ··· 6

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C4 S3 S3 D5 D6 D10 C4×S3 D15 C4×D5 D30 C4×D15 S32 S3×D5 C6.D6 D30.C2 S3×D15 C6.D30 kernel C6.D30 Dic3×C15 C3×Dic15 C2×C3⋊D15 C3⋊D15 C5×Dic3 Dic15 C3×Dic3 C30 C3×C6 C15 Dic3 C32 C6 C3 C10 C6 C5 C3 C2 C1 # reps 1 1 1 1 4 1 1 2 2 2 4 4 4 4 8 1 2 1 2 4 4

Matrix representation of C6.D30 in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 1 0 0 0 0 60 0
,
 17 60 0 0 0 0 1 0 0 0 0 0 0 0 0 50 0 0 0 0 11 50 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 17 60 0 0 0 0 44 44 0 0 0 0 0 0 1 60 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[17,1,0,0,0,0,60,0,0,0,0,0,0,0,0,11,0,0,0,0,50,50,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[17,44,0,0,0,0,60,44,0,0,0,0,0,0,1,0,0,0,0,0,60,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

C6.D30 in GAP, Magma, Sage, TeX

`C_6.D_{30}`
`% in TeX`

`G:=Group("C6.D30");`
`// GroupNames label`

`G:=SmallGroup(360,79);`
`// by ID`

`G=gap.SmallGroup(360,79);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,31,201,1444,10373]);`
`// Polycyclic`

`G:=Group<a,b,c|a^6=c^2=1,b^30=a^3,b*a*b^-1=c*a*c=a^-1,c*b*c=b^29>;`
`// generators/relations`

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