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## G = D12.D6order 288 = 25·32

### 5th 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.D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D6⋊D6 — D12.D6
 Lower central C32 — C3×C6 — C3×C12 — D12.D6
 Upper central C1 — C2 — C4 — D4

Generators and relations for D12.D6
G = < a,b,c,d | a12=b2=c6=1, d2=a3, bab=a-1, cac-1=a7, ad=da, cbc-1=dbd-1=a9b, dcd-1=a3c-1 >

Subgroups: 762 in 156 conjugacy classes, 38 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C3×D4, C22×S3, C8⋊C22, C3⋊Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, C62, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D42S3, C3×C3⋊C8, D6⋊S3, C3×D12, C324Q8, C4×C3⋊S3, C2×C3⋊Dic3, C327D4, D4×C32, C2×S32, D8⋊S3, C12.31D6, D12.S3, C3×D4⋊S3, D6⋊D6, C12.D6, D12.D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C8⋊C22, S32, S3×D4, C2×S32, D8⋊S3, Dic3⋊D6, D12.D6

Character table of D12.D6

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 6H 6I 8A 8B 12A 12B 12C 24A 24B 24C 24D size 1 1 4 12 12 18 2 2 4 2 18 36 2 2 4 8 8 8 8 24 24 12 12 4 4 8 12 12 12 12 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 1 -1 1 1 1 -1 1 1 -1 linear of order 2 ρ3 1 1 -1 -1 1 -1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 1 1 -1 -1 1 linear of order 2 ρ4 1 1 -1 1 1 1 1 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 -1 1 -1 -1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 1 1 1 -1 1 1 -1 linear of order 2 ρ6 1 1 -1 -1 -1 1 1 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ7 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ8 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 1 -1 1 1 1 1 1 -1 -1 1 linear of order 2 ρ9 2 2 -2 0 2 0 -1 2 -1 2 0 0 -1 2 -1 1 1 -2 1 -1 0 -2 0 -1 2 -1 0 1 1 0 orthogonal lifted from D6 ρ10 2 2 2 0 -2 0 -1 2 -1 2 0 0 -1 2 -1 -1 -1 2 -1 1 0 -2 0 -1 2 -1 0 1 1 0 orthogonal lifted from D6 ρ11 2 2 0 0 0 2 2 2 2 -2 -2 0 2 2 2 0 0 0 0 0 0 0 0 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 2 0 0 2 -1 -1 2 0 0 2 -1 -1 1 1 1 -2 0 -1 0 -2 2 -1 -1 1 0 0 1 orthogonal lifted from D6 ρ13 2 2 2 -2 0 0 2 -1 -1 2 0 0 2 -1 -1 -1 -1 -1 2 0 1 0 -2 2 -1 -1 1 0 0 1 orthogonal lifted from D6 ρ14 2 2 2 0 2 0 -1 2 -1 2 0 0 -1 2 -1 -1 -1 2 -1 -1 0 2 0 -1 2 -1 0 -1 -1 0 orthogonal lifted from S3 ρ15 2 2 0 0 0 -2 2 2 2 -2 2 0 2 2 2 0 0 0 0 0 0 0 0 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ16 2 2 2 2 0 0 2 -1 -1 2 0 0 2 -1 -1 -1 -1 -1 2 0 -1 0 2 2 -1 -1 -1 0 0 -1 orthogonal lifted from S3 ρ17 2 2 -2 0 -2 0 -1 2 -1 2 0 0 -1 2 -1 1 1 -2 1 1 0 2 0 -1 2 -1 0 -1 -1 0 orthogonal lifted from D6 ρ18 2 2 -2 -2 0 0 2 -1 -1 2 0 0 2 -1 -1 1 1 1 -2 0 1 0 2 2 -1 -1 -1 0 0 -1 orthogonal lifted from D6 ρ19 4 4 0 0 0 0 -2 -2 1 -4 0 0 -2 -2 1 3 -3 0 0 0 0 0 0 2 2 -1 0 0 0 0 orthogonal lifted from Dic3⋊D6 ρ20 4 4 0 0 0 0 -2 -2 1 -4 0 0 -2 -2 1 -3 3 0 0 0 0 0 0 2 2 -1 0 0 0 0 orthogonal lifted from Dic3⋊D6 ρ21 4 4 0 0 0 0 -2 4 -2 -4 0 0 -2 4 -2 0 0 0 0 0 0 0 0 2 -4 2 0 0 0 0 orthogonal lifted from S3×D4 ρ22 4 4 0 0 0 0 4 -2 -2 -4 0 0 4 -2 -2 0 0 0 0 0 0 0 0 -4 2 2 0 0 0 0 orthogonal lifted from S3×D4 ρ23 4 4 4 0 0 0 -2 -2 1 4 0 0 -2 -2 1 1 1 -2 -2 0 0 0 0 -2 -2 1 0 0 0 0 orthogonal lifted from S32 ρ24 4 -4 0 0 0 0 4 4 4 0 0 0 -4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ25 4 4 -4 0 0 0 -2 -2 1 4 0 0 -2 -2 1 -1 -1 2 2 0 0 0 0 -2 -2 1 0 0 0 0 orthogonal lifted from C2×S32 ρ26 4 -4 0 0 0 0 -2 4 -2 0 0 0 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 0 √-6 -√-6 0 complex lifted from D8⋊S3 ρ27 4 -4 0 0 0 0 -2 4 -2 0 0 0 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 0 -√-6 √-6 0 complex lifted from D8⋊S3 ρ28 4 -4 0 0 0 0 4 -2 -2 0 0 0 -4 2 2 0 0 0 0 0 0 0 0 0 0 0 √-6 0 0 -√-6 complex lifted from D8⋊S3 ρ29 4 -4 0 0 0 0 4 -2 -2 0 0 0 -4 2 2 0 0 0 0 0 0 0 0 0 0 0 -√-6 0 0 √-6 complex lifted from D8⋊S3 ρ30 8 -8 0 0 0 0 -4 -4 2 0 0 0 4 4 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of D12.D6 in GL8(𝔽73)

 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 1 1 0 0 0 0 71 0 72 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 1 1 2 0 0 0 0 0 72 72 72
,
 71 71 71 72 0 0 0 0 71 71 72 71 0 0 0 0 2 3 2 2 0 0 0 0 3 2 2 2 0 0 0 0 0 0 0 0 32 46 0 0 0 0 0 0 46 41 0 0 0 0 0 0 41 41 68 9 0 0 0 0 0 32 46 5
,
 0 1 0 0 0 0 0 0 72 1 0 0 0 0 0 0 48 46 72 72 0 0 0 0 50 48 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 72 72 72 71 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 71 71 72 71 0 0 0 0 71 71 71 72 0 0 0 0 51 52 2 2 0 0 0 0 52 51 2 2 0 0 0 0 0 0 0 0 27 27 59 54 0 0 0 0 32 32 5 64 0 0 0 0 46 41 0 0 0 0 0 0 0 46 41 14

`G:=sub<GL(8,GF(73))| [1,1,0,71,0,0,0,0,72,0,2,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,1,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,2,72],[71,71,2,3,0,0,0,0,71,71,3,2,0,0,0,0,71,72,2,2,0,0,0,0,72,71,2,2,0,0,0,0,0,0,0,0,32,46,41,0,0,0,0,0,46,41,41,32,0,0,0,0,0,0,68,46,0,0,0,0,0,0,9,5],[0,72,48,50,0,0,0,0,1,1,46,48,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,71,0,1],[71,71,51,52,0,0,0,0,71,71,52,51,0,0,0,0,72,71,2,2,0,0,0,0,71,72,2,2,0,0,0,0,0,0,0,0,27,32,46,0,0,0,0,0,27,32,41,46,0,0,0,0,59,5,0,41,0,0,0,0,54,64,0,14] >;`

D12.D6 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(288,575);`
`// by ID`

`G=gap.SmallGroup(288,575);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,303,675,346,185,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^6=1,d^2=a^3,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=d*b*d^-1=a^9*b,d*c*d^-1=a^3*c^-1>;`
`// generators/relations`

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