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## G = C62.D4order 288 = 25·32

### 1st non-split extension by C62 of D4 acting faithfully

Series: Derived Chief Lower central Upper central

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

Generators and relations for C62.D4
G = < a,b,c,d | a6=b6=c4=1, d2=b3, ab=ba, cac-1=a3b4, dad-1=a-1, cbc-1=a2b3, bd=db, dcd-1=a3b3c-1 >

Subgroups: 648 in 130 conjugacy classes, 31 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C4 [×6], C22, C22 [×6], S3 [×8], C6 [×6], C2×C4 [×12], C23, C32, Dic3 [×4], C12 [×4], D6 [×12], C2×C6 [×2], C22×C4 [×3], C3⋊S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C4×S3 [×8], C2×Dic3 [×2], C2×C12 [×2], C22×S3 [×2], C2.C42, C3×Dic3 [×4], C32⋊C4 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×4], C62, S3×C2×C4 [×2], C6.D6 [×4], C6.D6 [×2], C6×Dic3 [×2], C2×C32⋊C4 [×2], C2×C32⋊C4 [×2], C22×C3⋊S3, C2×C6.D6 [×2], C22×C32⋊C4, C62.D4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, S3≀C2, S32⋊C4 [×2], C3⋊S3.Q8, C62.D4

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

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A ··· 4H 4I 4J 4K 4L 6A ··· 6F 12A ··· 12H order 1 2 2 2 2 2 2 2 3 3 4 ··· 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 1 1 9 9 9 9 4 4 6 ··· 6 18 18 18 18 4 ··· 4 12 ··· 12

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 4 4 4 4 type + + + + - + + + image C1 C2 C2 C4 C4 D4 Q8 D4 S3≀C2 S32⋊C4 S32⋊C4 C3⋊S3.Q8 kernel C62.D4 C2×C6.D6 C22×C32⋊C4 C6.D6 C2×C32⋊C4 C2×C3⋊S3 C2×C3⋊S3 C62 C22 C2 C2 C2 # reps 1 2 1 8 4 2 1 1 4 4 4 4

Matrix representation of C62.D4 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 1 0 0 0 0 0 0 1 12 0 0 0 0 1 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 1 0 0 0 0 0 6 12 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 8 5 0 0 0 0 0 5 0 0
,
 4 3 0 0 0 0 3 9 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

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[1,6,0,0,0,0,0,12,0,0,0,0,0,0,0,0,8,0,0,0,0,0,5,5,0,0,5,0,0,0,0,0,0,5,0,0],[4,3,0,0,0,0,3,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C62.D4 in GAP, Magma, Sage, TeX

`C_6^2.D_4`
`% in TeX`

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

`G:=SmallGroup(288,385);`
`// by ID`

`G=gap.SmallGroup(288,385);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,112,85,64,2693,2028,691,797,2372]);`
`// 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^3*b^4,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^3,b*d=d*b,d*c*d^-1=a^3*b^3*c^-1>;`
`// generators/relations`

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