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## G = D12.29D6order 288 = 25·32

### 4th non-split extension by D12 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12.29D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D12.S3 — D12.29D6
 Lower central C32 — C3×C6 — C3×C12 — D12.29D6
 Upper central C1 — C2 — C2×C4

Generators and relations for D12.29D6
G = < a,b,c,d | a12=b2=1, c6=a6, d2=a9, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=a6c5 >

Subgroups: 514 in 139 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×3], C22, C22, S3, C6 [×2], C6 [×6], C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×4], C32, Dic3 [×9], C12 [×4], C12 [×3], D6, C2×C6 [×2], C2×C6 [×2], M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], Dic6, Dic6 [×10], C4×S3, D12, C2×Dic3 [×4], C3⋊D4, C2×C12 [×2], C2×C12 [×2], C3×D4 [×2], C3×Q8, C8.C22, C3×Dic3, C3⋊Dic3 [×2], C3×C12 [×2], S3×C6, C62, C24⋊C2 [×2], Dic12 [×2], C4.Dic3, D4.S3 [×2], C3⋊Q16 [×2], C3×M4(2), C2×Dic6 [×3], C4○D12, C3×C4○D4, C3×C3⋊C8 [×2], C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C324Q8 [×2], C324Q8, C2×C3⋊Dic3, C6×C12, C8.D6, Q8.14D6, D12.S3 [×2], C323Q16 [×2], C3×C4.Dic3, C3×C4○D12, C2×C324Q8, D12.29D6
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, Q8.14D6, C2×C3⋊D12, D12.29D6

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

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C ··· 6G 6H 6I 8A 8B 12A 12B 12C 12D 12E ··· 12J 12K 12L 24A 24B 24C 24D order 1 2 2 2 3 3 3 4 4 4 4 4 6 6 6 ··· 6 6 6 8 8 12 12 12 12 12 ··· 12 12 12 24 24 24 24 size 1 1 2 12 2 2 4 2 2 12 36 36 2 2 4 ··· 4 12 12 12 12 2 2 2 2 4 ··· 4 12 12 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 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + - + + + + - - - image C1 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D6 D12 C3⋊D4 D12 C3⋊D4 C8.C22 S32 C3⋊D12 C2×S32 C3⋊D12 C8.D6 Q8.14D6 D12.29D6 kernel D12.29D6 D12.S3 C32⋊3Q16 C3×C4.Dic3 C3×C4○D12 C2×C32⋊4Q8 C4.Dic3 C4○D12 C3×C12 C62 C3⋊C8 Dic6 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 1 1 2 2 2 2 2 1 1 1 1 1 2 2 4

Matrix representation of D12.29D6 in GL4(𝔽73) generated by

 14 66 0 0 7 7 0 0 7 0 7 66 0 66 7 14
,
 22 22 39 41 10 10 41 2 60 70 51 63 70 48 51 63
,
 59 7 0 0 66 66 0 0 0 7 59 66 66 0 7 66
,
 7 7 52 0 14 14 52 52 56 55 59 66 56 56 59 66
`G:=sub<GL(4,GF(73))| [14,7,7,0,66,7,0,66,0,0,7,7,0,0,66,14],[22,10,60,70,22,10,70,48,39,41,51,51,41,2,63,63],[59,66,0,66,7,66,7,0,0,0,59,7,0,0,66,66],[7,14,56,56,7,14,55,56,52,52,59,59,0,52,66,66] >;`

D12.29D6 in GAP, Magma, Sage, TeX

`D_{12}._{29}D_6`
`% in TeX`

`G:=Group("D12.29D6");`
`// GroupNames label`

`G:=SmallGroup(288,479);`
`// by ID`

`G=gap.SmallGroup(288,479);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,219,100,675,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=1,c^6=a^6,d^2=a^9,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^5>;`
`// generators/relations`

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