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## G = C42⋊13D6order 192 = 26·3

### 11st semidirect product of C42 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C42⋊13D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×S3×D4 — C42⋊13D6
 Lower central C3 — C6 — C42⋊13D6
 Upper central C1 — C22 — C4×D4

Generators and relations for C4213D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=a-1, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 904 in 338 conjugacy classes, 151 normal (43 characteristic)
C1, C2 [×3], C2 [×10], C3, C4 [×2], C4 [×10], C22, C22 [×4], C22 [×24], S3 [×6], C6 [×3], C6 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×17], D4 [×4], D4 [×12], C23 [×2], C23 [×15], Dic3 [×2], Dic3 [×4], C12 [×2], C12 [×4], D6 [×6], D6 [×14], C2×C6, C2×C6 [×4], C2×C6 [×4], C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×7], C2×D4, C2×D4 [×11], C24 [×2], C4×S3 [×4], C4×S3 [×4], D12 [×4], C2×Dic3 [×3], C2×Dic3 [×2], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×3], C2×C12 [×2], C2×C12 [×2], C3×D4 [×4], C22×S3, C22×S3 [×10], C22×S3 [×4], C22×C6 [×2], C2×C22⋊C4 [×4], C42⋊C2 [×2], C4×D4, C4×D4 [×7], C22×D4, C4×Dic3, C4×Dic3 [×2], Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], D6⋊C4 [×6], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, S3×C2×C4 [×4], C2×D12, S3×D4 [×8], C22×Dic3 [×2], C2×C3⋊D4 [×2], C22×C12 [×2], C6×D4, S3×C23 [×2], C22.11C24, C422S3, C4×D12, S3×C22⋊C4 [×2], Dic34D4 [×2], C4⋊C47S3, Dic35D4, C2×D6⋊C4 [×2], C4×C3⋊D4 [×2], D4×Dic3, D4×C12, C2×S3×D4, C4213D6
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C24, C4×S3 [×4], C22×S3 [×7], C23×C4, 2+ 1+4 [×2], S3×C2×C4 [×6], S3×C23, C22.11C24, S3×C22×C4, D46D6, D4○D12, C4213D6

Smallest permutation representation of C4213D6
On 48 points
Generators in S48
```(1 41 17 31)(2 32 18 42)(3 37 13 33)(4 34 14 38)(5 39 15 35)(6 36 16 40)(7 23 44 27)(8 28 45 24)(9 19 46 29)(10 30 47 20)(11 21 48 25)(12 26 43 22)
(1 19 4 22)(2 20 5 23)(3 21 6 24)(7 42 47 35)(8 37 48 36)(9 38 43 31)(10 39 44 32)(11 40 45 33)(12 41 46 34)(13 25 16 28)(14 26 17 29)(15 27 18 30)
(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 6)(2 5)(3 4)(7 47)(8 46)(9 45)(10 44)(11 43)(12 48)(13 14)(15 18)(16 17)(19 28)(20 27)(21 26)(22 25)(23 30)(24 29)(31 36)(32 35)(33 34)(37 38)(39 42)(40 41)```

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C4 S3 D6 D6 D6 D6 D6 C4×S3 2+ 1+4 D4⋊6D6 D4○D12 kernel C42⋊13D6 C42⋊2S3 C4×D12 S3×C22⋊C4 Dic3⋊4D4 C4⋊C4⋊7S3 Dic3⋊5D4 C2×D6⋊C4 C4×C3⋊D4 D4×Dic3 D4×C12 C2×S3×D4 S3×D4 C4×D4 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D4 C6 C2 C2 # reps 1 1 1 2 2 1 1 2 2 1 1 1 16 1 1 2 1 2 1 8 2 2 2

Matrix representation of C4213D6 in GL6(𝔽13)

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

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

C4213D6 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_{13}D_6`
`% in TeX`

`G:=Group("C4^2:13D6");`
`// GroupNames label`

`G:=SmallGroup(192,1104);`
`// by ID`

`G=gap.SmallGroup(192,1104);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,1123,80,6278]);`
`// Polycyclic`

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

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