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## G = C62.60D4order 288 = 25·32

### 44th non-split extension by C62 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.60D4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic3⋊Dic3 — C62.60D4
 Lower central C32 — C62 — C62.60D4
 Upper central C1 — C22 — C23

Generators and relations for C62.60D4
G = < a,b,c,d | a6=b6=c4=1, d2=b3, ab=ba, cac-1=ab3, dad-1=a-1b3, cbc-1=dbd-1=b-1, dcd-1=c-1 >

Subgroups: 738 in 183 conjugacy classes, 52 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×3], C3 [×2], C3, C4 [×5], C22, C22 [×2], C22 [×5], S3 [×4], C6 [×2], C6 [×4], C6 [×9], C2×C4 [×7], D4 [×2], C23, C23, C32, Dic3 [×8], C12 [×4], D6 [×10], C2×C6 [×2], C2×C6 [×4], C2×C6 [×9], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, C3⋊S3, C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3 [×4], C2×Dic3 [×5], C3⋊D4 [×8], C2×C12 [×6], C22×S3 [×3], C22×C6 [×2], C22×C6, C22.D4, C3×Dic3 [×4], C3⋊Dic3, C2×C3⋊S3 [×3], C62, C62 [×2], C62 [×2], Dic3⋊C4 [×2], C4⋊Dic3 [×2], D6⋊C4 [×4], C6.D4, C3×C22⋊C4, C22×Dic3, C2×C3⋊D4 [×3], C22×C12, C6×Dic3 [×4], C6×Dic3 [×2], C2×C3⋊Dic3, C327D4 [×2], C22×C3⋊S3, C2×C62, C23.21D6, C23.28D6, C6.D12 [×2], Dic3⋊Dic3 [×2], C3×C6.D4, Dic3×C2×C6, C2×C327D4, C62.60D4
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, C4○D12 [×2], D42S3 [×2], C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C23.21D6, C23.28D6, D6.3D6 [×2], C2×C3⋊D12, C62.60D4

Smallest permutation representation of C62.60D4
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 11 3 7 5 9)(2 12 4 8 6 10)(13 43 17 47 15 45)(14 44 18 48 16 46)(19 28 21 30 23 26)(20 29 22 25 24 27)(31 37 35 41 33 39)(32 38 36 42 34 40)
(1 31 19 44)(2 42 20 17)(3 33 21 46)(4 38 22 13)(5 35 23 48)(6 40 24 15)(7 41 30 16)(8 32 25 45)(9 37 26 18)(10 34 27 47)(11 39 28 14)(12 36 29 43)
(1 4 7 10)(2 9 8 3)(5 6 11 12)(13 41 47 31)(14 36 48 40)(15 39 43 35)(16 34 44 38)(17 37 45 33)(18 32 46 42)(19 22 30 27)(20 26 25 21)(23 24 28 29)```

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

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

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

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 12A ··· 12H 12I 12J 12K 12L order 1 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 12 12 12 12 size 1 1 1 1 2 2 36 2 2 4 6 6 6 6 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 4 4 4 4 4 type + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 C4○D4 D12 C3⋊D4 C4○D12 S32 D4⋊2S3 C3⋊D12 C2×S32 D6.3D6 kernel C62.60D4 C6.D12 Dic3⋊Dic3 C3×C6.D4 Dic3×C2×C6 C2×C32⋊7D4 C6.D4 C22×Dic3 C62 C2×Dic3 C22×C6 C3×C6 C2×C6 C2×C6 C6 C23 C6 C22 C22 C2 # reps 1 2 2 1 1 1 1 1 2 4 2 4 4 4 8 1 2 2 1 4

Matrix representation of C62.60D4 in GL8(𝔽13)

 12 0 0 0 0 0 0 0 0 12 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 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 12 12
,
 1 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 0 1 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
,
 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 5 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 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 1 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 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 12 12

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

C62.60D4 in GAP, Magma, Sage, TeX

`C_6^2._{60}D_4`
`% in TeX`

`G:=Group("C6^2.60D4");`
`// GroupNames label`

`G:=SmallGroup(288,614);`
`// by ID`

`G=gap.SmallGroup(288,614);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,64,254,1356,9414]);`
`// 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*b^3,d*a*d^-1=a^-1*b^3,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^-1>;`
`// generators/relations`

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