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## G = C62.57D4order 288 = 25·32

### 41st non-split extension by C62 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.57D4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — D6⋊Dic3 — C62.57D4
 Lower central C32 — C62 — C62.57D4
 Upper central C1 — C22 — C23

Generators and relations for C62.57D4
G = < a,b,c,d | a6=b6=c4=1, d2=b3, ab=ba, cac-1=a-1b3, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b3c-1 >

Subgroups: 626 in 183 conjugacy classes, 52 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×3], C3 [×2], C3, C4 [×5], C22, C22 [×2], C22 [×5], S3, C6 [×2], C6 [×4], C6 [×12], C2×C4 [×7], D4 [×2], C23, C23, C32, Dic3 [×11], C12 [×3], D6 [×3], C2×C6 [×2], C2×C6 [×4], C2×C6 [×14], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, C3×S3, C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3, C2×Dic3 [×2], C2×Dic3 [×14], C3⋊D4 [×2], C2×C12 [×3], C3×D4 [×2], C22×S3, C22×C6 [×2], C22×C6 [×2], C22.D4, C3×Dic3 [×3], C3⋊Dic3 [×2], S3×C6 [×3], C62, C62 [×2], C62 [×2], Dic3⋊C4 [×2], C4⋊Dic3 [×2], D6⋊C4 [×2], C6.D4, C6.D4 [×2], C3×C22⋊C4, C22×Dic3 [×3], C2×C3⋊D4, C6×D4, C6×Dic3, C6×Dic3 [×2], C3×C3⋊D4 [×2], C2×C3⋊Dic3 [×2], C2×C3⋊Dic3 [×2], S3×C2×C6, C2×C62, C23.21D6, C23.23D6, D6⋊Dic3 [×2], Dic3⋊Dic3 [×2], C3×C6.D4, C6×C3⋊D4, C22×C3⋊Dic3, C62.57D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C4○D4 [×2], D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C22.D4, S32, C2×D12, D42S3 [×4], C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C23.21D6, C23.23D6, D6.4D6 [×2], C2×C3⋊D12, C62.57D4

Smallest permutation representation of C62.57D4
On 48 points
Generators in S48
```(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)
(1 13 3 15 5 17)(2 14 4 16 6 18)(7 48 9 44 11 46)(8 43 10 45 12 47)(19 25 23 29 21 27)(20 26 24 30 22 28)(31 41 35 39 33 37)(32 42 36 40 34 38)
(1 27 18 30)(2 22 13 19)(3 25 14 28)(4 20 15 23)(5 29 16 26)(6 24 17 21)(7 39 47 42)(8 36 48 33)(9 37 43 40)(10 34 44 31)(11 41 45 38)(12 32 46 35)
(1 10 15 47)(2 9 16 46)(3 8 17 45)(4 7 18 44)(5 12 13 43)(6 11 14 48)(19 35 29 37)(20 34 30 42)(21 33 25 41)(22 32 26 40)(23 31 27 39)(24 36 28 38)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 6A ··· 6F 6G ··· 6Q 6R 6S 12A ··· 12F order 1 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 6 12 ··· 12 size 1 1 1 1 2 2 12 2 2 4 12 12 12 18 18 18 18 2 ··· 2 4 ··· 4 12 12 12 ··· 12

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 C4○D4 D12 C3⋊D4 S32 D4⋊2S3 C3⋊D12 C2×S32 D6.4D6 kernel C62.57D4 D6⋊Dic3 Dic3⋊Dic3 C3×C6.D4 C6×C3⋊D4 C22×C3⋊Dic3 C6.D4 C2×C3⋊D4 C62 C2×Dic3 C22×S3 C22×C6 C3×C6 C2×C6 C2×C6 C23 C6 C22 C22 C2 # reps 1 2 2 1 1 1 1 1 2 3 1 2 4 4 4 1 4 2 1 4

Matrix representation of C62.57D4 in GL8(𝔽13)

 1 0 0 0 0 0 0 0 10 12 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0
,
 12 8 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

`G:=sub<GL(8,GF(13))| [1,10,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[12,3,0,0,0,0,0,0,8,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,2,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;`

C62.57D4 in GAP, Magma, Sage, TeX

`C_6^2._{57}D_4`
`% in TeX`

`G:=Group("C6^2.57D4");`
`// GroupNames label`

`G:=SmallGroup(288,610);`
`// by ID`

`G=gap.SmallGroup(288,610);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,176,422,219,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=c^4=1,d^2=b^3,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^3*c^-1>;`
`// generators/relations`

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