Copied to
clipboard

## G = D12.33D6order 288 = 25·32

### 8th non-split extension by D12 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D12.33D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — S3×Dic3 — S3×Dic6 — D12.33D6
 Lower central C32 — C3×C6 — D12.33D6
 Upper central C1 — C2 — C2×C4

Generators and relations for D12.33D6
G = < a,b,c,d | a12=b2=1, c6=d2=a6, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a6b, dcd-1=a6c5 >

Subgroups: 1050 in 312 conjugacy classes, 108 normal (36 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×Q8, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, 2- 1+4, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C62, C2×Dic6, C2×Dic6, C4○D12, C4○D12, D42S3, S3×Q8, Q83S3, C6×Q8, C3×C4○D4, S3×Dic3, C6.D6, C3⋊D12, C322Q8, C3×Dic6, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C324Q8, C4×C3⋊S3, C12⋊S3, C327D4, C6×C12, Q8.15D6, Q8○D12, S3×Dic6, D12⋊S3, Dic3.D6, D6.6D6, D6.3D6, C6×Dic6, C3×C4○D12, C12.59D6, D12.33D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2- 1+4, S32, S3×C23, C2×S32, Q8.15D6, Q8○D12, C22×S32, D12.33D6

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 3C 4A 4B 4C ··· 4H 4I 4J 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 12A 12B 12C ··· 12I 12J ··· 12O order 1 2 2 2 2 2 2 3 3 3 4 4 4 ··· 4 4 4 6 6 6 6 6 6 6 6 6 6 12 12 12 ··· 12 12 ··· 12 size 1 1 2 6 6 18 18 2 2 4 2 2 6 ··· 6 18 18 2 2 2 2 4 4 4 4 12 12 2 2 4 ··· 4 12 ··· 12

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 D6 D6 2- 1+4 S32 C2×S32 C2×S32 Q8.15D6 Q8○D12 D12.33D6 kernel D12.33D6 S3×Dic6 D12⋊S3 Dic3.D6 D6.6D6 D6.3D6 C6×Dic6 C3×C4○D12 C12.59D6 C2×Dic6 C4○D12 Dic6 C4×S3 D12 C2×Dic3 C3⋊D4 C2×C12 C32 C2×C4 C4 C22 C3 C3 C1 # reps 1 2 2 2 2 4 1 1 1 1 1 5 2 1 2 2 2 1 1 2 1 2 2 4

Matrix representation of D12.33D6 in GL4(𝔽13) generated by

 8 8 0 0 5 0 0 0 0 0 5 5 0 0 8 0
,
 0 0 12 12 0 0 0 1 12 12 0 0 0 1 0 0
,
 7 10 0 0 3 10 0 0 0 0 10 3 0 0 10 7
,
 0 0 3 10 0 0 3 6 7 10 0 0 3 10 0 0
`G:=sub<GL(4,GF(13))| [8,5,0,0,8,0,0,0,0,0,5,8,0,0,5,0],[0,0,12,0,0,0,12,1,12,0,0,0,12,1,0,0],[7,3,0,0,10,10,0,0,0,0,10,10,0,0,3,7],[0,0,7,3,0,0,10,10,3,3,0,0,10,6,0,0] >;`

D12.33D6 in GAP, Magma, Sage, TeX

`D_{12}._{33}D_6`
`% in TeX`

`G:=Group("D12.33D6");`
`// GroupNames label`

`G:=SmallGroup(288,945);`
`// by ID`

`G=gap.SmallGroup(288,945);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,100,675,1356,9414]);`
`// Polycyclic`

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

׿
×
𝔽