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## G = C52⋊2C12order 300 = 22·3·52

### The semidirect product of C52 and C12 acting via C12/C2=C6

Aliases: C522C12, (C5×C10).C6, C526C4⋊C3, C52⋊C34C4, C2.(C52⋊C6), (C2×C52⋊C3).2C2, SmallGroup(300,14)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52⋊2C12
 Chief series C1 — C52 — C5×C10 — C2×C52⋊C3 — C52⋊2C12
 Lower central C52 — C52⋊2C12
 Upper central C1 — C2

Generators and relations for C522C12
G = < a,b,c | a5=b5=c12=1, ab=ba, cac-1=a-1b2, cbc-1=ab2 >

Character table of C522C12

 class 1 2 3A 3B 4A 4B 5A 5B 5C 5D 6A 6B 10A 10B 10C 10D 12A 12B 12C 12D size 1 1 25 25 25 25 6 6 6 6 25 25 6 6 6 6 25 25 25 25 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 ζ3 ζ32 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ4 1 1 ζ32 ζ3 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ5 1 1 ζ3 ζ32 -1 -1 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ6 1 1 ζ32 ζ3 -1 -1 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ7 1 -1 1 1 i -i 1 1 1 1 -1 -1 -1 -1 -1 -1 -i i -i i linear of order 4 ρ8 1 -1 1 1 -i i 1 1 1 1 -1 -1 -1 -1 -1 -1 i -i i -i linear of order 4 ρ9 1 -1 ζ32 ζ3 -i i 1 1 1 1 ζ65 ζ6 -1 -1 -1 -1 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 linear of order 12 ρ10 1 -1 ζ3 ζ32 -i i 1 1 1 1 ζ6 ζ65 -1 -1 -1 -1 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 linear of order 12 ρ11 1 -1 ζ32 ζ3 i -i 1 1 1 1 ζ65 ζ6 -1 -1 -1 -1 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 linear of order 12 ρ12 1 -1 ζ3 ζ32 i -i 1 1 1 1 ζ6 ζ65 -1 -1 -1 -1 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 linear of order 12 ρ13 6 6 0 0 0 0 -3-√5/2 -3+√5/2 1-√5 1+√5 0 0 1-√5 1+√5 -3+√5/2 -3-√5/2 0 0 0 0 orthogonal lifted from C52⋊C6 ρ14 6 6 0 0 0 0 1-√5 1+√5 -3+√5/2 -3-√5/2 0 0 -3+√5/2 -3-√5/2 1+√5 1-√5 0 0 0 0 orthogonal lifted from C52⋊C6 ρ15 6 6 0 0 0 0 1+√5 1-√5 -3-√5/2 -3+√5/2 0 0 -3-√5/2 -3+√5/2 1-√5 1+√5 0 0 0 0 orthogonal lifted from C52⋊C6 ρ16 6 6 0 0 0 0 -3+√5/2 -3-√5/2 1+√5 1-√5 0 0 1+√5 1-√5 -3-√5/2 -3+√5/2 0 0 0 0 orthogonal lifted from C52⋊C6 ρ17 6 -6 0 0 0 0 -3-√5/2 -3+√5/2 1-√5 1+√5 0 0 -1+√5 -1-√5 3-√5/2 3+√5/2 0 0 0 0 symplectic faithful, Schur index 2 ρ18 6 -6 0 0 0 0 1-√5 1+√5 -3+√5/2 -3-√5/2 0 0 3-√5/2 3+√5/2 -1-√5 -1+√5 0 0 0 0 symplectic faithful, Schur index 2 ρ19 6 -6 0 0 0 0 -3+√5/2 -3-√5/2 1+√5 1-√5 0 0 -1-√5 -1+√5 3+√5/2 3-√5/2 0 0 0 0 symplectic faithful, Schur index 2 ρ20 6 -6 0 0 0 0 1+√5 1-√5 -3-√5/2 -3+√5/2 0 0 3+√5/2 3-√5/2 -1+√5 -1-√5 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C522C12
On 60 points
Generators in S60
```(2 36 58 45 13)(3 25 59 46 14)(5 16 48 49 27)(6 17 37 50 28)(8 30 52 39 19)(9 31 53 40 20)(11 22 42 55 33)(12 23 43 56 34)
(1 44 35 24 57)(2 36 58 45 13)(3 46 25 14 59)(4 60 15 26 47)(5 16 48 49 27)(6 50 17 28 37)(7 38 29 18 51)(8 30 52 39 19)(9 40 31 20 53)(10 54 21 32 41)(11 22 42 55 33)(12 56 23 34 43)
(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)```

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

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

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

Matrix representation of C522C12 in GL7(𝔽61)

 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 60 17 0 0 0 0 0 44 44 0 0 0 0 0 0 0 44 44 0 0 0 0 0 17 60
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 60 17 0 0 0 0 0 0 0 0 1 0 0 0 0 0 60 17 0 0 0 0 0 0 0 44 44 0 0 0 0 0 17 60
,
 50 0 0 0 0 0 0 0 0 0 8 41 0 0 0 0 0 55 53 0 0 0 0 0 0 0 8 41 0 0 0 0 0 55 53 0 8 41 0 0 0 0 0 55 53 0 0 0 0

`G:=sub<GL(7,GF(61))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,44,0,0,0,0,0,17,44,0,0,0,0,0,0,0,44,17,0,0,0,0,0,44,60],[1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,1,17,0,0,0,0,0,0,0,0,60,0,0,0,0,0,1,17,0,0,0,0,0,0,0,44,17,0,0,0,0,0,44,60],[50,0,0,0,0,0,0,0,0,0,0,0,8,55,0,0,0,0,0,41,53,0,8,55,0,0,0,0,0,41,53,0,0,0,0,0,0,0,8,55,0,0,0,0,0,41,53,0,0] >;`

C522C12 in GAP, Magma, Sage, TeX

`C_5^2\rtimes_2C_{12}`
`% in TeX`

`G:=Group("C5^2:2C12");`
`// GroupNames label`

`G:=SmallGroup(300,14);`
`// by ID`

`G=gap.SmallGroup(300,14);`
`# by ID`

`G:=PCGroup([5,-2,-3,-2,-5,5,30,963,1568,6004,909]);`
`// Polycyclic`

`G:=Group<a,b,c|a^5=b^5=c^12=1,a*b=b*a,c*a*c^-1=a^-1*b^2,c*b*c^-1=a*b^2>;`
`// generators/relations`

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