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## G = D12⋊20D6order 288 = 25·32

### 4th semidirect product of D12 and D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12⋊20D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — C32⋊2D8 — D12⋊20D6
 Lower central C32 — C3×C6 — C3×C12 — D12⋊20D6
 Upper central C1 — C2 — C2×C4

Generators and relations for D1220D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=a-1, ac=ca, dad=a7, cbc-1=a6b, dbd=a3b, dcd=c-1 >

Subgroups: 530 in 146 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, Dic6, C4×S3, D12, D12, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C8⋊C22, C3×Dic3, C3×C12, S3×C6, C62, C4.Dic3, D4⋊S3, D4.S3, Q82S3, C2×D12, C4○D12, C6×D4, C3×C4○D4, C324C8, C3×Dic6, S3×C12, C3×D12, C3×D12, C3×D12, C3×C3⋊D4, C6×C12, S3×C2×C6, D126C22, D4⋊D6, C322D8, Dic6⋊S3, C12.58D6, C6×D12, C3×C4○D12, D1220D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C8⋊C22, S32, C2×C3⋊D4, D6⋊S3, C2×S32, D126C22, D4⋊D6, C2×D6⋊S3, D1220D6

Smallest permutation representation of D1220D6
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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 24)(11 23)(12 22)(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 10 6)(3 11 7)(4 12 8)(13 23 21 19 17 15)(14 24 22 20 18 16)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 39 41 43 45 47)(38 40 42 44 46 48)
(1 30)(2 25)(3 32)(4 27)(5 34)(6 29)(7 36)(8 31)(9 26)(10 33)(11 28)(12 35)(13 41)(14 48)(15 43)(16 38)(17 45)(18 40)(19 47)(20 42)(21 37)(22 44)(23 39)(24 46)```

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 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 C3⋊D4 C3⋊D4 C8⋊C22 S32 D6⋊S3 C2×S32 D6⋊S3 D12⋊6C22 D4⋊D6 D12⋊20D6 kernel D12⋊20D6 C32⋊2D8 Dic6⋊S3 C12.58D6 C6×D12 C3×C4○D12 C2×D12 C4○D12 C3×C12 C62 Dic6 D12 C2×C12 C12 C2×C6 C32 C2×C4 C4 C4 C22 C3 C3 C1 # reps 1 2 2 1 1 1 1 1 1 1 1 3 2 4 4 1 1 1 1 1 2 2 4

Matrix representation of D1220D6 in GL4(𝔽73) generated by

 49 0 0 0 0 3 0 0 0 0 24 0 0 0 0 70
,
 0 3 0 0 49 0 0 0 0 0 0 65 0 0 9 0
,
 8 0 0 0 0 65 0 0 0 0 64 0 0 0 0 9
,
 0 0 64 0 0 0 0 9 8 0 0 0 0 65 0 0
`G:=sub<GL(4,GF(73))| [49,0,0,0,0,3,0,0,0,0,24,0,0,0,0,70],[0,49,0,0,3,0,0,0,0,0,0,9,0,0,65,0],[8,0,0,0,0,65,0,0,0,0,64,0,0,0,0,9],[0,0,8,0,0,0,0,65,64,0,0,0,0,9,0,0] >;`

D1220D6 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_{20}D_6`
`% in TeX`

`G:=Group("D12:20D6");`
`// GroupNames label`

`G:=SmallGroup(288,471);`
`// by ID`

`G=gap.SmallGroup(288,471);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,219,675,346,80,1356,9414]);`
`// Polycyclic`

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

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