Copied to
clipboard

## G = D12.10D6order 288 = 25·32

### 10th non-split extension by D12 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12.10D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D6⋊D6 — D12.10D6
 Lower central C32 — C3×C6 — C3×C12 — D12.10D6
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D12.10D6
G = < a,b,c,d | a12=b2=1, c6=a6, d2=a3, bab=a-1, cac-1=a7, ad=da, cbc-1=dbd-1=a3b, dcd-1=a9c5 >

Subgroups: 826 in 156 conjugacy classes, 38 normal (16 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4, C4 [×2], C22 [×6], S3 [×9], C6 [×2], C6 [×3], C8 [×2], C2×C4 [×2], D4 [×5], Q8, C23, C32, Dic3 [×3], C12 [×2], C12 [×5], D6 [×13], C2×C6 [×2], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3⋊C8 [×2], C24 [×2], C4×S3 [×7], D12 [×2], D12 [×7], C3⋊D4 [×2], C3×D4 [×2], C3×Q8 [×2], C3×Q8, C22×S3 [×2], C8⋊C22, C3⋊Dic3, C3×C12, C3×C12, S32 [×2], S3×C6 [×2], C2×C3⋊S3, C2×C3⋊S3, C8⋊S3 [×2], D24 [×2], D4⋊S3 [×2], Q82S3 [×2], C3×SD16 [×2], S3×D4 [×2], Q83S3 [×3], C3×C3⋊C8 [×2], D6⋊S3, C3×D12 [×2], C4×C3⋊S3, C4×C3⋊S3, C12⋊S3, C12⋊S3, Q8×C32, C2×S32, Q83D6 [×2], C12.31D6, C3⋊D24 [×2], C3×Q82S3 [×2], D6⋊D6, C12.26D6, D12.10D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C22×S3 [×2], C8⋊C22, S32, S3×D4 [×2], C2×S32, Q83D6 [×2], Dic3⋊D6, D12.10D6

Character table of D12.10D6

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 6A 6B 6C 6D 6E 8A 8B 12A 12B 12C 12D 12E 12F 12G 24A 24B 24C 24D size 1 1 12 12 18 36 2 2 4 2 4 18 2 2 4 24 24 12 12 4 4 8 8 8 8 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 0 0 -1 2 -1 2 2 0 2 -1 -1 -1 0 0 2 -1 2 2 -1 -1 -1 -1 0 -1 -1 0 orthogonal lifted from S3 ρ10 2 2 -2 0 0 0 -1 2 -1 2 -2 0 2 -1 -1 1 0 0 2 -1 2 -2 1 -1 1 1 0 -1 -1 0 orthogonal lifted from D6 ρ11 2 2 0 -2 0 0 2 -1 -1 2 -2 0 -1 2 -1 0 1 2 0 2 -1 1 1 -1 1 -2 -1 0 0 -1 orthogonal lifted from D6 ρ12 2 2 -2 0 0 0 -1 2 -1 2 2 0 2 -1 -1 1 0 0 -2 -1 2 2 -1 -1 -1 -1 0 1 1 0 orthogonal lifted from D6 ρ13 2 2 2 0 0 0 -1 2 -1 2 -2 0 2 -1 -1 -1 0 0 -2 -1 2 -2 1 -1 1 1 0 1 1 0 orthogonal lifted from D6 ρ14 2 2 0 2 0 0 2 -1 -1 2 2 0 -1 2 -1 0 -1 2 0 2 -1 -1 -1 -1 -1 2 -1 0 0 -1 orthogonal lifted from S3 ρ15 2 2 0 -2 0 0 2 -1 -1 2 2 0 -1 2 -1 0 1 -2 0 2 -1 -1 -1 -1 -1 2 1 0 0 1 orthogonal lifted from D6 ρ16 2 2 0 2 0 0 2 -1 -1 2 -2 0 -1 2 -1 0 -1 -2 0 2 -1 1 1 -1 1 -2 1 0 0 1 orthogonal lifted from D6 ρ17 2 2 0 0 2 0 2 2 2 -2 0 -2 2 2 2 0 0 0 0 -2 -2 0 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 0 0 -2 0 2 2 2 -2 0 2 2 2 2 0 0 0 0 -2 -2 0 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 4 4 0 0 0 0 -2 -2 1 4 -4 0 -2 -2 1 0 0 0 0 -2 -2 2 -1 1 -1 2 0 0 0 0 orthogonal lifted from C2×S32 ρ20 4 4 0 0 0 0 -2 4 -2 -4 0 0 4 -2 -2 0 0 0 0 2 -4 0 0 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ21 4 4 0 0 0 0 -2 -2 1 -4 0 0 -2 -2 1 0 0 0 0 2 2 0 -3 -1 3 0 0 0 0 0 orthogonal lifted from Dic3⋊D6 ρ22 4 4 0 0 0 0 4 -2 -2 -4 0 0 -2 4 -2 0 0 0 0 -4 2 0 0 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ23 4 4 0 0 0 0 -2 -2 1 4 4 0 -2 -2 1 0 0 0 0 -2 -2 -2 1 1 1 -2 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 0 0 0 0 -2 -2 1 -4 0 0 -2 -2 1 0 0 0 0 2 2 0 3 -1 -3 0 0 0 0 0 orthogonal lifted from Dic3⋊D6 ρ26 4 -4 0 0 0 0 -2 4 -2 0 0 0 -4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 √6 -√6 0 orthogonal lifted from Q8⋊3D6 ρ27 4 -4 0 0 0 0 -2 4 -2 0 0 0 -4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 -√6 √6 0 orthogonal lifted from Q8⋊3D6 ρ28 4 -4 0 0 0 0 4 -2 -2 0 0 0 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 √6 0 0 -√6 orthogonal lifted from Q8⋊3D6 ρ29 4 -4 0 0 0 0 4 -2 -2 0 0 0 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 -√6 0 0 √6 orthogonal lifted from Q8⋊3D6 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 orthogonal faithful, Schur index 2

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

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

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

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

Matrix representation of D12.10D6 in GL8(𝔽73)

 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 72 1 2 71 0 0 0 0 72 0 2 0 0 0 0 0 72 1 1 72 0 0 0 0 72 0 1 0
,
 0 0 1 72 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 9 0 24 0 0 0 0 0 9 64 24 49 0 0 0 0 21 0 64 0 0 0 0 0 21 52 64 9
,
 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 0 0 0 0 41 64 45 56 0 0 0 0 9 32 17 28 0 0 0 0 27 19 32 9 0 0 0 0 54 46 64 41
,
 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 68 10 5 63 0 0 0 0 63 5 10 68 0 0 0 0 34 5 0 0 0 0 0 0 68 39 0 0

`G:=sub<GL(8,GF(73))| [0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,72,72,72,72,0,0,0,0,1,0,1,0,0,0,0,0,2,2,1,1,0,0,0,0,71,0,72,0],[0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,9,9,21,21,0,0,0,0,0,64,0,52,0,0,0,0,24,24,64,64,0,0,0,0,0,49,0,9],[1,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,41,9,27,54,0,0,0,0,64,32,19,46,0,0,0,0,45,17,32,64,0,0,0,0,56,28,9,41],[0,0,1,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,68,63,34,68,0,0,0,0,10,5,5,39,0,0,0,0,5,10,0,0,0,0,0,0,63,68,0,0] >;`

D12.10D6 in GAP, Magma, Sage, TeX

`D_{12}._{10}D_6`
`% in TeX`

`G:=Group("D12.10D6");`
`// GroupNames label`

`G:=SmallGroup(288,589);`
`// by ID`

`G=gap.SmallGroup(288,589);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,303,100,675,346,185,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=1,c^6=a^6,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^3*b,d*c*d^-1=a^9*c^5>;`
`// generators/relations`

Export

׿
×
𝔽