Copied to
clipboard

G = D12.22D6order 288 = 25·32

7th non-split extension by D12 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12.22D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — D12⋊5S3 — D12.22D6
 Lower central C32 — C3×C6 — C3×C12 — D12.22D6
 Upper central C1 — C2 — C4 — D4

Generators and relations for D12.22D6
G = < a,b,c,d | a12=b2=c6=1, d2=a6, bab=a-1, cac-1=dad-1=a7, cbc-1=a3b, dbd-1=a9b, dcd-1=a6c-1 >

Subgroups: 490 in 134 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×2], C6 [×2], C6 [×6], C8 [×2], C2×C4 [×3], D4, D4 [×3], Q8 [×2], C32, Dic3, Dic3 [×4], C12 [×2], C12 [×3], D6, D6, C2×C6 [×5], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×S3 [×2], C3×C6, C3×C6, C3⋊C8, C3⋊C8 [×3], C24, Dic6, Dic6 [×3], C4×S3, C4×S3, D12, C2×Dic3 [×2], C3⋊D4 [×3], C2×C12 [×2], C3×D4 [×2], C3×D4 [×3], C3×Q8, C4○D8, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C62, S3×C8, Dic12, C2×C3⋊C8, D4⋊S3, D4.S3 [×4], Q82S3, C3⋊Q16, C3×D8, C4○D12, D42S3, D42S3, C3×C4○D4, C3×C3⋊C8, C324C8, S3×Dic3, D6⋊S3, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C324Q8, D4×C32, D83S3, Q8.13D6, S3×C3⋊C8, Dic6⋊S3, C323Q16, C3×D4⋊S3, C329SD16, D125S3, C3×D42S3, D12.22D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C3⋊D4 [×2], C22×S3 [×2], C4○D8, S32, S3×D4, C2×C3⋊D4, C2×S32, D83S3, Q8.13D6, S3×C3⋊D4, D12.22D6

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

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 24A 24B order 1 2 2 2 2 3 3 3 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 12 24 24 size 1 1 4 6 12 2 2 4 2 3 3 12 36 2 2 4 4 4 8 8 8 12 24 6 6 18 18 4 4 6 6 8 12 12 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 D8⋊3S3 Q8.13D6 S3×C3⋊D4 D12.22D6 kernel D12.22D6 S3×C3⋊C8 Dic6⋊S3 C32⋊3Q16 C3×D4⋊S3 C32⋊9SD16 D12⋊5S3 C3×D4⋊2S3 D4⋊S3 D4⋊2S3 C3×Dic3 S3×C6 C3⋊C8 Dic6 C4×S3 D12 C3×D4 Dic3 D6 C32 D4 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 D12.22D6 in GL6(𝔽73)

 0 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 57 57 0 0 0 0 57 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 72 1
,
 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 46 0 0 0 0 46 0 0 0 0 0 0 0 1 0 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[57,57,0,0,0,0,57,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,1],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,46,0,0,0,0,46,0,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

D12.22D6 in GAP, Magma, Sage, TeX

`D_{12}._{22}D_6`
`% in TeX`

`G:=Group("D12.22D6");`
`// GroupNames label`

`G:=SmallGroup(288,581);`
`// by ID`

`G=gap.SmallGroup(288,581);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,422,135,346,185,80,1356,9414]);`
`// Polycyclic`

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

׿
×
𝔽