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## G = C62⋊4D4order 288 = 25·32

### 1st semidirect product of C62 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62⋊4D4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×D6⋊S3 — C62⋊4D4
 Lower central C32 — C62 — C62⋊4D4
 Upper central C1 — C22 — C23

Generators and relations for C624D4
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a-1b3, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 1018 in 277 conjugacy classes, 60 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×7], C3 [×2], C3, C4 [×3], C22, C22 [×2], C22 [×21], S3 [×5], C6 [×2], C6 [×4], C6 [×14], C2×C4 [×3], D4 [×6], C23, C23 [×9], C32, Dic3 [×7], C12, D6 [×4], D6 [×15], C2×C6 [×2], C2×C6 [×4], C2×C6 [×28], C22⋊C4 [×3], C2×D4 [×3], C24, C3×S3 [×5], C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3, C2×Dic3 [×6], C3⋊D4 [×10], C2×C12, C3×D4 [×2], C22×S3, C22×S3 [×2], C22×S3 [×6], C22×C6 [×2], C22×C6 [×10], C22≀C2, C3×Dic3, C3⋊Dic3 [×2], S3×C6 [×4], S3×C6 [×15], C62, C62 [×2], C62 [×2], D6⋊C4 [×2], C6.D4 [×5], C2×C3⋊D4, C2×C3⋊D4 [×4], C6×D4, S3×C23, C23×C6, D6⋊S3 [×4], C6×Dic3, C3×C3⋊D4 [×2], C2×C3⋊Dic3 [×2], S3×C2×C6, S3×C2×C6 [×2], S3×C2×C6 [×6], C2×C62, C232D6, C244S3, D6⋊Dic3 [×2], C625C4, C2×D6⋊S3 [×2], C6×C3⋊D4, S3×C22×C6, C624D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×6], C23, D6 [×6], C2×D4 [×3], C3⋊D4 [×8], C22×S3 [×2], C22≀C2, S32, S3×D4 [×2], C2×C3⋊D4 [×4], D6⋊S3 [×2], C2×S32, C232D6, C244S3, C2×D6⋊S3, S3×C3⋊D4 [×2], C624D4

Smallest permutation representation of C624D4
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 15 3 14 2 13)(4 20 5 21 6 19)(7 12 8 10 9 11)(16 24 18 23 17 22)(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 30 24 33)(2 26 22 35)(3 28 23 31)(4 44 10 41)(5 46 11 37)(6 48 12 39)(7 38 21 47)(8 40 19 43)(9 42 20 45)(13 25 16 34)(14 27 17 36)(15 29 18 32)
(1 20)(2 21)(3 19)(4 15)(5 13)(6 14)(7 22)(8 23)(9 24)(10 18)(11 16)(12 17)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 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), (1,15,3,14,2,13)(4,20,5,21,6,19)(7,12,8,10,9,11)(16,24,18,23,17,22)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,42,41,40,39,38)(43,48,47,46,45,44), (1,30,24,33)(2,26,22,35)(3,28,23,31)(4,44,10,41)(5,46,11,37)(6,48,12,39)(7,38,21,47)(8,40,19,43)(9,42,20,45)(13,25,16,34)(14,27,17,36)(15,29,18,32), (1,20)(2,21)(3,19)(4,15)(5,13)(6,14)(7,22)(8,23)(9,24)(10,18)(11,16)(12,17)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)>;`

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

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

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 6AB 6AC 12A 12B order 1 2 2 2 2 2 2 2 2 2 2 3 3 3 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 6 12 12 size 1 1 1 1 2 2 6 6 6 6 12 2 2 4 12 36 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 4 4 4 4 4 type + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 C3⋊D4 C3⋊D4 S32 S3×D4 D6⋊S3 C2×S32 S3×C3⋊D4 kernel C62⋊4D4 D6⋊Dic3 C62⋊5C4 C2×D6⋊S3 C6×C3⋊D4 S3×C22×C6 C2×C3⋊D4 S3×C23 S3×C6 C62 C2×Dic3 C22×S3 C22×C6 D6 C2×C6 C23 C6 C22 C22 C2 # reps 1 2 1 2 1 1 1 1 4 2 1 3 2 8 8 1 2 2 1 4

Matrix representation of C624D4 in GL8(𝔽13)

 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 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 3 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 1
,
 0 1 0 0 0 0 0 0 12 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 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
,
 1 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 1
,
 1 0 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 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

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

C624D4 in GAP, Magma, Sage, TeX

`C_6^2\rtimes_4D_4`
`% in TeX`

`G:=Group("C6^2:4D4");`
`// GroupNames label`

`G:=SmallGroup(288,624);`
`// by ID`

`G=gap.SmallGroup(288,624);`
`# by ID`

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

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

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