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## G = D12⋊20D4order 192 = 26·3

### 8th semidirect product of D12 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — D12⋊20D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×S3×D4 — D12⋊20D4
 Lower central C3 — C2×C6 — D12⋊20D4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for D1220D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, cac-1=dad=a7, bc=cb, dbd=a6b, dcd=c-1 >

Subgroups: 992 in 334 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C3, C4 [×2], C4 [×8], C22, C22 [×29], S3 [×6], C6 [×3], C6 [×3], C2×C4 [×2], C2×C4 [×2], C2×C4 [×15], D4 [×18], Q8 [×2], C23, C23 [×2], C23 [×13], Dic3 [×5], C12 [×2], C12 [×3], D6 [×6], D6 [×14], C2×C6, C2×C6 [×9], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×10], C2×Q8, C4○D4 [×4], C24 [×2], Dic6 [×2], C4×S3 [×8], D12 [×4], C2×Dic3 [×3], C2×Dic3 [×2], C3⋊D4 [×10], C2×C12 [×2], C2×C12 [×2], C2×C12 [×2], C3×D4 [×4], C22×S3, C22×S3 [×2], C22×S3 [×10], C22×C6, C22×C6 [×2], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4, C4⋊D4 [×2], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3 [×2], D6⋊C4, D6⋊C4 [×4], C6.D4, C6.D4 [×4], C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4 [×4], C2×D12, C4○D12 [×4], S3×D4 [×4], C2×C3⋊D4, C2×C3⋊D4 [×4], C22×C12, C6×D4, C6×D4 [×2], S3×C23 [×2], D45D4, S3×C22⋊C4 [×2], C23.9D6 [×2], Dic35D4, C4.D12, C4×C3⋊D4, C23.12D6, C232D6 [×2], D63D4 [×2], C3×C4⋊D4, C2×C4○D12, C2×S3×D4, D1220D4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×D4, C2×C4○D4, 2+ 1+4, S3×D4 [×2], S3×C23, D45D4, C2×S3×D4, D46D6, S3×C4○D4, D1220D4

Smallest permutation representation of D1220D4
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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 22)(14 21)(15 20)(16 19)(17 18)(23 24)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 40)(38 39)(41 48)(42 47)(43 46)(44 45)
(1 21 39 32)(2 16 40 27)(3 23 41 34)(4 18 42 29)(5 13 43 36)(6 20 44 31)(7 15 45 26)(8 22 46 33)(9 17 47 28)(10 24 48 35)(11 19 37 30)(12 14 38 25)
(1 39)(2 46)(3 41)(4 48)(5 43)(6 38)(7 45)(8 40)(9 47)(10 42)(11 37)(12 44)(14 20)(16 22)(18 24)(25 31)(27 33)(29 35)```

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D6 C4○D4 2+ 1+4 S3×D4 D4⋊6D6 S3×C4○D4 kernel D12⋊20D4 S3×C22⋊C4 C23.9D6 Dic3⋊5D4 C4.D12 C4×C3⋊D4 C23.12D6 C23⋊2D6 D6⋊3D4 C3×C4⋊D4 C2×C4○D12 C2×S3×D4 C4⋊D4 D12 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D6 C6 C4 C2 C2 # reps 1 2 2 1 1 1 1 2 2 1 1 1 1 4 2 1 1 3 4 1 2 2 2

Matrix representation of D1220D4 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 12 0 0 0 0 0 0 0 0 8 0 0 0 0 8 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 12 0 0 0 0 0 0 0 8 0 0 0 0 5 0
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 0
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1

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

D1220D4 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_{20}D_4`
`% in TeX`

`G:=Group("D12:20D4");`
`// GroupNames label`

`G:=SmallGroup(192,1171);`
`// by ID`

`G=gap.SmallGroup(192,1171);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,1571,570,297,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^7,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;`
`// generators/relations`

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