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## G = C42⋊10D6order 192 = 26·3

### 8th semidirect product of C42 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C42⋊10D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22×C4 — C42⋊10D6
 Lower central C3 — C2×C6 — C42⋊10D6
 Upper central C1 — C2×C4 — C42⋊C2

Generators and relations for C4210D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=ab2, dad=a-1, bc=cb, bd=db, dcd=c-1 >

Subgroups: 904 in 330 conjugacy classes, 115 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×4], C4 [×8], C22, C22 [×2], C22 [×24], S3 [×6], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×18], D4 [×14], Q8 [×2], C23, C23 [×10], Dic3 [×4], C12 [×4], C12 [×4], D6 [×4], D6 [×18], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×4], C22×C4, C22×C4 [×11], C2×D4 [×7], C2×Q8, C4○D4 [×4], C24, Dic6 [×2], C4×S3 [×12], D12 [×10], C2×Dic3 [×4], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×8], C22×S3 [×4], C22×S3 [×6], C22×C6, C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, C4⋊Dic3 [×4], D6⋊C4 [×8], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6, S3×C2×C4 [×6], S3×C2×C4 [×4], C2×D12, C2×D12 [×4], C4○D12 [×4], C22×Dic3, C2×C3⋊D4 [×2], C22×C12, S3×C23, C22.19C24, C4×D12 [×4], D6⋊D4 [×2], C23.21D6 [×2], C12⋊D4 [×2], C4.D12 [×2], C3×C42⋊C2, S3×C22×C4, C2×C4○D12, C4210D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, D12 [×4], C22×S3 [×7], C22×D4, C2×C4○D4 [×2], C2×D12 [×6], S3×C23, C22.19C24, C22×D12, S3×C4○D4 [×2], C4210D6

Smallest permutation representation of C4210D6
On 48 points
Generators in S48
```(1 26 4 21)(2 19 5 30)(3 28 6 23)(7 32 46 35)(8 40 47 37)(9 34 48 31)(10 42 43 39)(11 36 44 33)(12 38 45 41)(13 22 16 27)(14 25 17 20)(15 24 18 29)
(1 34 15 41)(2 35 16 42)(3 36 17 37)(4 31 18 38)(5 32 13 39)(6 33 14 40)(7 27 43 19)(8 28 44 20)(9 29 45 21)(10 30 46 22)(11 25 47 23)(12 26 48 24)
(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 14)(2 13)(3 18)(4 17)(5 16)(6 15)(7 43)(8 48)(9 47)(10 46)(11 45)(12 44)(19 27)(20 26)(21 25)(22 30)(23 29)(24 28)(31 37)(32 42)(33 41)(34 40)(35 39)(36 38)```

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D6 C4○D4 D12 S3×C4○D4 kernel C42⋊10D6 C4×D12 D6⋊D4 C23.21D6 C12⋊D4 C4.D12 C3×C42⋊C2 S3×C22×C4 C2×C4○D12 C42⋊C2 C2×C12 C42 C22⋊C4 C4⋊C4 C22×C4 D6 C2×C4 C2 # reps 1 4 2 2 2 2 1 1 1 1 4 2 2 2 1 8 8 4

Matrix representation of C4210D6 in GL6(𝔽13)

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

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

C4210D6 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_{10}D_6`
`% in TeX`

`G:=Group("C4^2:10D6");`
`// GroupNames label`

`G:=SmallGroup(192,1083);`
`// by ID`

`G=gap.SmallGroup(192,1083);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,570,80,6278]);`
`// Polycyclic`

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

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