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## G = C62.56D4order 288 = 25·32

### 40th non-split extension by C62 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.56D4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — D6⋊Dic3 — C62.56D4
 Lower central C32 — C62 — C62.56D4
 Upper central C1 — C22 — C23

Generators and relations for C62.56D4
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a-1b3, dad=ab3, cbc-1=dbd=b-1, dcd=b3c-1 >

Subgroups: 570 in 173 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 [×10], C2×C4 [×7], D4 [×2], C23, C23, C32, Dic3 [×9], C12 [×3], D6 [×3], C2×C6 [×2], C2×C6 [×4], C2×C6 [×12], 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 [×8], C3⋊D4 [×2], C2×C12 [×5], 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 [×4], D6⋊C4 [×2], C6.D4 [×5], C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C6×Dic3, C6×Dic3 [×2], C6×Dic3 [×2], C3×C3⋊D4 [×2], C2×C3⋊Dic3 [×2], S3×C2×C6, C2×C62, C23.28D6, C23.23D6, D6⋊Dic3 [×2], C62.C22 [×2], C625C4, Dic3×C2×C6, C6×C3⋊D4, C62.56D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C4○D4 [×2], C3⋊D4 [×4], C22×S3 [×2], C22.D4, S32, C4○D12 [×2], D42S3 [×2], C2×C3⋊D4 [×2], D6⋊S3 [×2], C2×S32, C23.28D6, C23.23D6, D6.3D6 [×2], C2×D6⋊S3, C62.56D4

Smallest permutation representation of C62.56D4
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 4 2 5 3 6)(7 12 8 10 9 11)(13 17 15 16 14 18)(19 23 21 22 20 24)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 42 41 40 39 38)(43 48 47 46 45 44)
(1 46 11 37)(2 48 12 39)(3 44 10 41)(4 47 7 38)(5 43 8 40)(6 45 9 42)(13 35 22 26)(14 31 23 28)(15 33 24 30)(16 32 19 29)(17 34 20 25)(18 36 21 27)
(1 29)(2 27)(3 25)(4 28)(5 26)(6 30)(7 31)(8 35)(9 33)(10 34)(11 32)(12 36)(13 37)(14 41)(15 39)(16 40)(17 38)(18 42)(19 43)(20 47)(21 45)(22 46)(23 44)(24 48)```

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 6A ··· 6J 6K ··· 6S 6T 6U 12A ··· 12H 12I 12J 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 12 12 size 1 1 1 1 2 2 12 2 2 4 6 6 6 6 12 36 36 2 ··· 2 4 ··· 4 12 12 6 ··· 6 12 12

48 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 C3⋊D4 C4○D12 S32 D4⋊2S3 D6⋊S3 C2×S32 D6.3D6 kernel C62.56D4 D6⋊Dic3 C62.C22 C62⋊5C4 Dic3×C2×C6 C6×C3⋊D4 C22×Dic3 C2×C3⋊D4 C62 C2×Dic3 C22×S3 C22×C6 C3×C6 C2×C6 C6 C23 C6 C22 C22 C2 # reps 1 2 2 1 1 1 1 1 2 3 1 2 4 8 8 1 2 2 1 4

Matrix representation of C62.56D4 in GL8(𝔽13)

 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 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 12 0 0 0 0 0 0 1 0
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 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
,
 0 5 0 0 0 0 0 0 8 0 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 3 1 0 0 0 0 0 0 3 10 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 0 1 0 0 0 0 0 0 1 0 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 10 7 0 0 0 0 0 0 10 3 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12

`G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,1,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,10,10,0,0,0,0,0,0,7,3,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12] >;`

C62.56D4 in GAP, Magma, Sage, TeX

`C_6^2._{56}D_4`
`% in TeX`

`G:=Group("C6^2.56D4");`
`// GroupNames label`

`G:=SmallGroup(288,609);`
`// by ID`

`G=gap.SmallGroup(288,609);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,120,422,1356,9414]);`
`// Polycyclic`

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

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