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## G = D12.13D6order 288 = 25·32

### 13rd non-split extension by D12 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12.13D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — D6.6D6 — D12.13D6
 Lower central C32 — C3×C6 — C3×C12 — D12.13D6
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D12.13D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=dad=a-1, cac-1=a5, cbc-1=a10b, dbd=ab, dcd=a6c-1 >

Subgroups: 554 in 134 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12, D12, C3⋊D4, C2×C12, C3×D4, C3×Q8, C3×Q8, C4○D8, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, S3×C8, D24, C2×C3⋊C8, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C3×Q16, C4○D12, Q83S3, Q83S3, C3×C4○D4, C3×C3⋊C8, C324C8, C6.D6, C3⋊D12, C3×Dic6, S3×C12, S3×C12, C3×D12, C3×D12, C12⋊S3, Q8×C32, D24⋊C2, Q8.13D6, S3×C3⋊C8, C3⋊D24, Dic6⋊S3, C3×C3⋊Q16, C3211SD16, D6.6D6, C3×Q83S3, D12.13D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C4○D8, S32, S3×D4, C2×C3⋊D4, C2×S32, D24⋊C2, Q8.13D6, S3×C3⋊D4, D12.13D6

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

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 24A 24B order 1 2 2 2 2 3 3 3 4 4 4 4 4 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 12 12 12 12 12 24 24 size 1 1 6 12 36 2 2 4 2 3 3 4 12 2 2 4 12 12 12 6 6 18 18 4 4 4 4 6 6 8 8 8 8 24 12 12

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 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 C4○D8 S32 S3×D4 C2×S32 D24⋊C2 Q8.13D6 S3×C3⋊D4 D12.13D6 kernel D12.13D6 S3×C3⋊C8 C3⋊D24 Dic6⋊S3 C3×C3⋊Q16 C32⋊11SD16 D6.6D6 C3×Q8⋊3S3 C3⋊Q16 Q8⋊3S3 C3×Dic3 S3×C6 C3⋊C8 Dic6 C4×S3 D12 C3×Q8 Dic3 D6 C32 Q8 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 4 1 1 1 2 2 2 1

Matrix representation of D12.13D6 in GL6(𝔽73)

 72 3 0 0 0 0 48 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 32 25 0 0 0 0 35 41 0 0 0 0 0 0 72 0 0 0 0 0 72 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 46 8 0 0 0 0 55 27 0 0 0 0 0 0 0 72 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 1 1
,
 72 0 0 0 0 0 48 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(73))| [72,48,0,0,0,0,3,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,35,0,0,0,0,25,41,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[46,55,0,0,0,0,8,27,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,1],[72,48,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

D12.13D6 in GAP, Magma, Sage, TeX

`D_{12}._{13}D_6`
`% in TeX`

`G:=Group("D12.13D6");`
`// GroupNames label`

`G:=SmallGroup(288,597);`
`// by ID`

`G=gap.SmallGroup(288,597);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,422,135,100,346,185,80,1356,9414]);`
`// 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^10*b,d*b*d=a*b,d*c*d=a^6*c^-1>;`
`// generators/relations`

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