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## G = C62⋊5D4order 288 = 25·32

### 2nd semidirect product of C62 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62⋊5D4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×C3⋊D12 — C62⋊5D4
 Lower central C32 — C62 — C62⋊5D4
 Upper central C1 — C22 — C23

Generators and relations for C625D4
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=dad=a-1b3, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 1186 in 287 conjugacy classes, 60 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×7], C3 [×2], C3, C4 [×3], C22, C22 [×2], C22 [×21], S3 [×8], C6 [×2], C6 [×4], C6 [×13], C2×C4 [×3], D4 [×6], C23, C23 [×9], C32, Dic3 [×6], C12 [×2], D6 [×4], D6 [×22], C2×C6 [×2], C2×C6 [×4], C2×C6 [×25], C22⋊C4 [×3], C2×D4 [×3], C24, C3×S3 [×4], C3⋊S3, C3×C6, C3×C6 [×2], C3×C6 [×2], D12 [×4], C2×Dic3 [×2], C2×Dic3 [×3], C3⋊D4 [×12], C2×C12 [×2], C22×S3 [×2], C22×S3 [×9], C22×C6 [×2], C22×C6 [×9], C22≀C2, C3×Dic3 [×2], C3⋊Dic3, S3×C6 [×4], S3×C6 [×12], C2×C3⋊S3 [×3], C62, C62 [×2], C62 [×2], D6⋊C4 [×2], C6.D4, C6.D4 [×2], C3×C22⋊C4, C2×D12 [×2], C2×C3⋊D4 [×5], S3×C23, C23×C6, C3⋊D12 [×4], C6×Dic3 [×2], C2×C3⋊Dic3, C327D4 [×2], S3×C2×C6 [×2], S3×C2×C6 [×6], C22×C3⋊S3, C2×C62, D6⋊D4, C244S3, D6⋊Dic3 [×2], C3×C6.D4, C2×C3⋊D12 [×2], C2×C327D4, S3×C22×C6, C625D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×6], C23, D6 [×6], C2×D4 [×3], D12 [×2], C3⋊D4 [×6], C22×S3 [×2], C22≀C2, S32, C2×D12, S3×D4 [×2], C2×C3⋊D4 [×3], C3⋊D12 [×2], C2×S32, D6⋊D4, C244S3, C2×C3⋊D12, S3×C3⋊D4 [×2], C625D4

Smallest permutation representation of C625D4
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 31 3 33 5 35)(2 32 4 34 6 36)(7 14 9 16 11 18)(8 15 10 17 12 13)(19 40 23 38 21 42)(20 41 24 39 22 37)(25 46 29 44 27 48)(26 47 30 45 28 43)
(1 47 10 19)(2 27 11 37)(3 45 12 23)(4 25 7 41)(5 43 8 21)(6 29 9 39)(13 38 33 28)(14 24 34 46)(15 42 35 26)(16 22 36 44)(17 40 31 30)(18 20 32 48)
(1 10)(2 18)(3 8)(4 16)(5 12)(6 14)(7 36)(9 34)(11 32)(13 33)(15 31)(17 35)(20 37)(21 23)(22 41)(24 39)(25 44)(26 30)(27 48)(29 46)(40 42)(43 45)```

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 2 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 D4 D6 D6 D6 C3⋊D4 D12 C3⋊D4 S32 S3×D4 C3⋊D12 C2×S32 S3×C3⋊D4 kernel C62⋊5D4 D6⋊Dic3 C3×C6.D4 C2×C3⋊D12 C2×C32⋊7D4 S3×C22×C6 C6.D4 S3×C23 S3×C6 C62 C2×Dic3 C22×S3 C22×C6 D6 C2×C6 C2×C6 C23 C6 C22 C22 C2 # reps 1 2 1 2 1 1 1 1 4 2 2 2 2 8 4 4 1 2 2 1 4

Matrix representation of C625D4 in GL8(ℤ)

 0 -1 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 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 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 -1 1 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 -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 1 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 0 1 0 0 0 0 0 0 1 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 1 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 0 1 0 0 0 0 0 0 1 0

`G:=sub<GL(8,Integers())| [0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0],[-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,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;`

C625D4 in GAP, Magma, Sage, TeX

`C_6^2\rtimes_5D_4`
`% in TeX`

`G:=Group("C6^2:5D4");`
`// GroupNames label`

`G:=SmallGroup(288,625);`
`// by ID`

`G=gap.SmallGroup(288,625);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,219,1356,9414]);`
`// Polycyclic`

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

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