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## G = D12.28D6order 288 = 25·32

### 3rd non-split extension by D12 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12.28D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D12.S3 — D12.28D6
 Lower central C32 — C3×C6 — C3×C12 — D12.28D6
 Upper central C1 — C2 — C2×C4

Generators and relations for D12.28D6
G = < a,b,c,d | a12=b2=c6=1, d2=a3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=a6c-1 >

Subgroups: 698 in 156 conjugacy classes, 44 normal (32 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4 [×2], C4, C22, C22 [×5], S3 [×6], C6 [×2], C6 [×6], C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, C32, Dic3 [×4], C12 [×4], C12 [×2], D6 [×8], C2×C6 [×2], C2×C6 [×5], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3 [×2], C3⋊S3, C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], Dic6 [×3], C4×S3 [×4], D12 [×2], D12 [×4], C3⋊D4 [×4], C2×C12 [×2], C2×C12, C3×D4 [×3], C22×S3, C22×C6, C8⋊C22, C3⋊Dic3, C3×C12 [×2], S3×C6 [×4], C2×C3⋊S3, C62, C24⋊C2 [×2], D24 [×2], C4.Dic3, D4⋊S3 [×2], D4.S3 [×2], C3×M4(2), C2×D12, C4○D12 [×3], C6×D4, C3×C3⋊C8 [×2], C3×D12 [×2], C3×D12, C324Q8, C4×C3⋊S3, C12⋊S3, C327D4, C6×C12, S3×C2×C6, C8⋊D6, D126C22, C3⋊D24 [×2], D12.S3 [×2], C3×C4.Dic3, C6×D12, C12.59D6, D12.28D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C8⋊C22, S32, C2×D12, C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C8⋊D6, D126C22, C2×C3⋊D12, D12.28D6

Smallest permutation representation of D12.28D6
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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 48)(8 47)(9 46)(10 45)(11 44)(12 43)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)
(1 15 5 19 9 23)(2 16 6 20 10 24)(3 17 7 21 11 13)(4 18 8 22 12 14)(25 41 33 37 29 45)(26 42 34 38 30 46)(27 43 35 39 31 47)(28 44 36 40 32 48)
(1 42 4 45 7 48 10 39)(2 43 5 46 8 37 11 40)(3 44 6 47 9 38 12 41)(13 30 16 33 19 36 22 27)(14 31 17 34 20 25 23 28)(15 32 18 35 21 26 24 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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25), (1,15,5,19,9,23)(2,16,6,20,10,24)(3,17,7,21,11,13)(4,18,8,22,12,14)(25,41,33,37,29,45)(26,42,34,38,30,46)(27,43,35,39,31,47)(28,44,36,40,32,48), (1,42,4,45,7,48,10,39)(2,43,5,46,8,37,11,40)(3,44,6,47,9,38,12,41)(13,30,16,33,19,36,22,27)(14,31,17,34,20,25,23,28)(15,32,18,35,21,26,24,29)>;`

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 8A 8B 12A 12B 12C ··· 12I 24A 24B 24C 24D order 1 2 2 2 2 2 3 3 3 4 4 4 6 6 6 6 6 6 6 6 6 6 6 6 8 8 12 12 12 ··· 12 24 24 24 24 size 1 1 2 12 12 36 2 2 4 2 2 36 2 2 2 2 4 4 4 4 12 12 12 12 12 12 2 2 4 ··· 4 12 12 12 12

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D12 C3⋊D4 D12 C3⋊D4 C8⋊C22 S32 C3⋊D12 C2×S32 C3⋊D12 C8⋊D6 D12⋊6C22 D12.28D6 kernel D12.28D6 C3⋊D24 D12.S3 C3×C4.Dic3 C6×D12 C12.59D6 C4.Dic3 C2×D12 C3×C12 C62 C3⋊C8 D12 C2×C12 C12 C12 C2×C6 C2×C6 C32 C2×C4 C4 C4 C22 C3 C3 C1 # reps 1 2 2 1 1 1 1 1 1 1 2 2 2 2 2 2 2 1 1 1 1 1 2 2 4

Matrix representation of D12.28D6 in GL4(𝔽73) generated by

 7 66 0 0 7 14 0 0 0 0 7 66 0 0 7 14
,
 0 0 27 27 0 0 0 46 46 46 0 0 0 27 0 0
,
 60 30 0 0 43 30 0 0 0 0 30 43 0 0 30 60
,
 0 0 43 30 0 0 43 13 0 27 0 0 46 46 0 0
`G:=sub<GL(4,GF(73))| [7,7,0,0,66,14,0,0,0,0,7,7,0,0,66,14],[0,0,46,0,0,0,46,27,27,0,0,0,27,46,0,0],[60,43,0,0,30,30,0,0,0,0,30,30,0,0,43,60],[0,0,0,46,0,0,27,46,43,43,0,0,30,13,0,0] >;`

D12.28D6 in GAP, Magma, Sage, TeX

`D_{12}._{28}D_6`
`% in TeX`

`G:=Group("D12.28D6");`
`// GroupNames label`

`G:=SmallGroup(288,478);`
`// by ID`

`G=gap.SmallGroup(288,478);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,219,675,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^6=1,d^2=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=a^6*c^-1>;`
`// generators/relations`

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