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## G = D12.7D6order 288 = 25·32

### 7th non-split extension by D12 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12.7D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — D6.6D6 — D12.7D6
 Lower central C32 — C3×C6 — C3×C12 — D12.7D6
 Upper central C1 — C2 — C4 — D4

Generators and relations for D12.7D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=dad=a-1, cac-1=a5, cbc-1=a4b, dbd=a7b, dcd=a6c-1 >

Subgroups: 642 in 147 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4, C4 [×2], C22 [×6], S3 [×5], C6 [×2], C6 [×6], C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, C32, Dic3, Dic3, C12 [×2], C12 [×3], D6, D6 [×6], C2×C6 [×7], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3 [×2], C3⋊S3, C3×C6, C3×C6, C3⋊C8, C3⋊C8 [×3], C24, Dic6, C4×S3, C4×S3 [×2], D12, D12 [×4], C3⋊D4 [×2], C2×C12, C3×D4 [×2], C3×D4 [×3], C3×Q8, C22×S3, C22×C6, C8⋊C22, C3×Dic3, C3×Dic3, C3×C12, S3×C6, S3×C6 [×3], C2×C3⋊S3, C62, C8⋊S3, D24, C4.Dic3, D4⋊S3 [×4], D4.S3, D4.S3, Q82S3, C3×SD16, C4○D12, S3×D4, Q83S3, C6×D4, C3×C3⋊C8, C324C8, C6.D6, C3⋊D12, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C12⋊S3, D4×C32, S3×C2×C6, Q83D6, D126C22, D6.Dic3, C3⋊D24, Dic6⋊S3, C3×D4.S3, C327D8, D6.6D6, C3×S3×D4, D12.7D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C3⋊D4 [×2], C22×S3 [×2], C8⋊C22, S32, S3×D4, C2×C3⋊D4, C2×S32, Q83D6, D126C22, S3×C3⋊D4, D12.7D6

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

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

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

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

33 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 12D 12E 24A 24B 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 12 24 24 size 1 1 4 6 12 36 2 2 4 2 6 12 2 2 4 4 4 6 6 8 8 8 12 12 12 36 4 4 8 12 24 12 12

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D6 D6 C3⋊D4 C3⋊D4 C8⋊C22 S32 S3×D4 C2×S32 Q8⋊3D6 D12⋊6C22 S3×C3⋊D4 D12.7D6 kernel D12.7D6 D6.Dic3 C3⋊D24 Dic6⋊S3 C3×D4.S3 C32⋊7D8 D6.6D6 C3×S3×D4 D4.S3 S3×D4 C3×Dic3 S3×C6 C3⋊C8 Dic6 C4×S3 D12 C3×D4 Dic3 D6 C32 D4 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 2 2 2 1

Matrix representation of D12.7D6 in GL8(ℤ)

 0 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 -1 0 -1 -1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 -1 1 2 0 0 0 0 0 1 -1 -1
,
 1 1 1 2 0 0 0 0 1 1 2 1 0 0 0 0 -1 0 -1 -1 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 -2 0 1 2 0 0 0 0 2 1 0 -2 0 0 0 0 -2 -1 -1 1
,
 1 1 1 2 0 0 0 0 1 1 2 1 0 0 0 0 0 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 1 0 -2 -2 0 0 0 0 2 1 0 -2 0 0 0 0 -2 0 1 2 0 0 0 0 2 0 -2 -3
,
 -1 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 1 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 -1 -1 1 2 0 0 0 0 0 1 0 -1

`G:=sub<GL(8,Integers())| [0,1,-1,0,0,0,0,0,-1,-1,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1,0,-1,1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,2,-1],[1,1,-1,0,0,0,0,0,1,1,0,-1,0,0,0,0,1,2,-1,-1,0,0,0,0,2,1,-1,-1,0,0,0,0,0,0,0,0,-1,-2,2,-2,0,0,0,0,-1,0,1,-1,0,0,0,0,-1,1,0,-1,0,0,0,0,0,2,-2,1],[1,1,0,-1,0,0,0,0,1,1,-1,-1,0,0,0,0,1,2,-1,-1,0,0,0,0,2,1,-1,-1,0,0,0,0,0,0,0,0,1,2,-2,2,0,0,0,0,0,1,0,0,0,0,0,0,-2,0,1,-2,0,0,0,0,-2,-2,2,-3],[-1,-1,0,1,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,-1,0,-1,0,0,0,0,0,0,1,-1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,-1] >;`

D12.7D6 in GAP, Magma, Sage, TeX

`D_{12}._7D_6`
`% in TeX`

`G:=Group("D12.7D6");`
`// GroupNames label`

`G:=SmallGroup(288,582);`
`// by ID`

`G=gap.SmallGroup(288,582);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,135,346,185,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^4*b,d*b*d=a^7*b,d*c*d=a^6*c^-1>;`
`// generators/relations`

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