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## G = D12⋊10D4order 192 = 26·3

### 3rd semidirect product of D12 and D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — D12⋊10D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×S3×D4 — D12⋊10D4
 Lower central C3 — C2×C6 — D12⋊10D4
 Upper central C1 — C22 — C4.4D4

Generators and relations for D1210D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, ac=ca, dad=a5, cbc-1=a6b, dbd=a10b, dcd=c-1 >

Subgroups: 1024 in 334 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C3, C4 [×2], C4 [×8], C22, C22 [×29], S3 [×7], C6 [×3], C6 [×2], C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×18], Q8 [×2], C23 [×2], C23 [×14], Dic3 [×4], C12 [×2], C12 [×4], D6 [×6], D6 [×17], C2×C6, C2×C6 [×6], C42, C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×6], C2×D4, C2×D4 [×12], C2×Q8, C4○D4 [×4], C24 [×2], C4×S3 [×10], D12 [×4], D12 [×6], C2×Dic3 [×2], C2×Dic3 [×2], C3⋊D4 [×6], C2×C12 [×3], C2×C12 [×2], C3×D4 [×2], C3×Q8 [×2], C22×S3 [×2], C22×S3 [×2], C22×S3 [×10], C22×C6 [×2], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, Dic3⋊C4 [×2], C4⋊Dic3 [×2], D6⋊C4 [×2], D6⋊C4 [×4], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×4], S3×C2×C4 [×2], S3×C2×C4 [×4], C2×D12 [×2], C2×D12 [×2], S3×D4 [×4], Q83S3 [×4], C2×C3⋊D4 [×4], C6×D4, C6×Q8, S3×C23 [×2], D45D4, C4×D12 [×2], S3×C22⋊C4 [×2], D6⋊D4 [×2], C23.9D6 [×2], Dic3⋊D4 [×2], D63D4, D63Q8, C3×C4.4D4, C2×S3×D4, C2×Q83S3, D1210D4
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, S3×C4○D4, D4○D12, D1210D4

Smallest permutation representation of D1210D4
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 20)(14 19)(15 18)(16 17)(21 24)(22 23)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 42)(38 41)(39 40)(43 48)(44 47)(45 46)
(1 20 35 43)(2 21 36 44)(3 22 25 45)(4 23 26 46)(5 24 27 47)(6 13 28 48)(7 14 29 37)(8 15 30 38)(9 16 31 39)(10 17 32 40)(11 18 33 41)(12 19 34 42)
(1 14)(2 19)(3 24)(4 17)(5 22)(6 15)(7 20)(8 13)(9 18)(10 23)(11 16)(12 21)(25 47)(26 40)(27 45)(28 38)(29 43)(30 48)(31 41)(32 46)(33 39)(34 44)(35 37)(36 42)```

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 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 S3 D4 D6 D6 D6 D6 C4○D4 2+ 1+4 S3×D4 S3×C4○D4 D4○D12 kernel D12⋊10D4 C4×D12 S3×C22⋊C4 D6⋊D4 C23.9D6 Dic3⋊D4 D6⋊3D4 D6⋊3Q8 C3×C4.4D4 C2×S3×D4 C2×Q8⋊3S3 C4.4D4 D12 C42 C22⋊C4 C2×D4 C2×Q8 D6 C6 C4 C2 C2 # reps 1 2 2 2 2 2 1 1 1 1 1 1 4 1 4 1 1 4 1 2 2 2

Matrix representation of D1210D4 in GL6(𝔽13)

 0 5 0 0 0 0 5 0 0 0 0 0 0 0 1 1 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 5 0 0 0 0 8 0 0 0 0 0 0 0 1 1 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 1 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 1 3 0 0 0 0 8 12
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 12 10 0 0 0 0 0 1

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

D1210D4 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_{10}D_4`
`% in TeX`

`G:=Group("D12:10D4");`
`// GroupNames label`

`G:=SmallGroup(192,1235);`
`// by ID`

`G=gap.SmallGroup(192,1235);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,1571,570,297,192,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^5,c*b*c^-1=a^6*b,d*b*d=a^10*b,d*c*d=c^-1>;`
`// generators/relations`

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