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## G = C42⋊11D6order 192 = 26·3

### 9th semidirect product of C42 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C42⋊11D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C22×D12 — C42⋊11D6
 Lower central C3 — C2×C6 — C42⋊11D6
 Upper central C1 — C22 — C42⋊C2

Generators and relations for C4211D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=ab2, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 1128 in 334 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×4], C4 [×6], C22, C22 [×2], C22 [×28], S3 [×6], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×6], D4 [×22], Q8 [×2], C23, C23 [×14], Dic3 [×2], C12 [×4], C12 [×4], D6 [×26], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C2×D4 [×19], C2×Q8, C4○D4 [×4], C24 [×2], Dic6 [×2], C4×S3 [×4], D12 [×18], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×8], C22×S3 [×6], C22×S3 [×8], C22×C6, C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4 [×2], C22×D4, C2×C4○D4, D6⋊C4 [×8], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6, S3×C2×C4 [×2], C2×D12, C2×D12 [×12], C2×D12 [×4], C4○D12 [×4], C2×C3⋊D4 [×2], C22×C12, S3×C23 [×2], C22.29C24, C4⋊D12 [×2], C427S3 [×2], D6⋊D4 [×4], C12⋊D4 [×4], C3×C42⋊C2, C22×D12, C2×C4○D12, C4211D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C22×D4, 2+ 1+4 [×2], C2×D12 [×6], S3×C23, C22.29C24, C22×D12, D4○D12 [×2], C4211D6

Smallest permutation representation of C4211D6
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 40)(2 39)(3 38)(4 37)(5 42)(6 41)(7 27)(8 26)(9 25)(10 30)(11 29)(12 28)(13 35)(14 34)(15 33)(16 32)(17 31)(18 36)(19 43)(20 48)(21 47)(22 46)(23 45)(24 44)```

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D6 D12 2+ 1+4 D4○D12 kernel C42⋊11D6 C4⋊D12 C42⋊7S3 D6⋊D4 C12⋊D4 C3×C42⋊C2 C22×D12 C2×C4○D12 C42⋊C2 C2×C12 C42 C22⋊C4 C4⋊C4 C22×C4 C2×C4 C6 C2 # reps 1 2 2 4 4 1 1 1 1 4 2 2 2 1 8 2 4

Matrix representation of C4211D6 in GL6(𝔽13)

 3 6 0 0 0 0 7 10 0 0 0 0 0 0 1 0 11 0 0 0 0 1 0 11 0 0 1 0 12 0 0 0 0 1 0 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 10 6 0 0 0 0 7 3 0 0 0 0 0 0 10 6 0 0 0 0 7 3
,
 12 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 12 1 0 0 0 0 0 1 0 12 0 0 12 1 1 12
,
 1 1 0 0 0 0 0 12 0 0 0 0 0 0 3 3 0 0 0 0 6 10 0 0 0 0 3 3 10 10 0 0 6 10 7 3

`G:=sub<GL(6,GF(13))| [3,7,0,0,0,0,6,10,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,11,0,12,0,0,0,0,11,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,10,7,0,0,0,0,6,3,0,0,0,0,0,0,10,7,0,0,0,0,6,3],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,12,0,0,1,1,1,1,0,0,0,0,0,1,0,0,0,0,12,12],[1,0,0,0,0,0,1,12,0,0,0,0,0,0,3,6,3,6,0,0,3,10,3,10,0,0,0,0,10,7,0,0,0,0,10,3] >;`

C4211D6 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_{11}D_6`
`% in TeX`

`G:=Group("C4^2:11D6");`
`// GroupNames label`

`G:=SmallGroup(192,1084);`
`// by ID`

`G=gap.SmallGroup(192,1084);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,675,570,80,6278]);`
`// 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,d*b*d=b^-1,d*c*d=c^-1>;`
`// generators/relations`

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