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## G = C42⋊12D6order 192 = 26·3

### 10th semidirect product of C42 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C42⋊12D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22⋊C4 — C42⋊12D6
 Lower central C3 — C2×C6 — C42⋊12D6
 Upper central C1 — C22 — C42⋊C2

Generators and relations for C4212D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 696 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×11], C22, C22 [×2], C22 [×16], S3 [×4], C6 [×3], C6 [×2], C2×C4 [×6], C2×C4 [×12], D4 [×5], Q8, C23, C23 [×8], Dic3 [×5], C12 [×6], D6 [×2], D6 [×12], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4, C22×C4 [×4], C2×D4 [×3], C2×Q8, C24, Dic6, C4×S3 [×4], D12 [×3], C2×Dic3 [×5], C2×Dic3, C3⋊D4 [×2], C2×C12 [×6], C2×C12 [×2], C22×S3 [×3], C22×S3 [×5], C22×C6, C2×C22⋊C4 [×2], C42⋊C2, C42⋊C2, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C422C2 [×2], C4×Dic3, Dic3⋊C4 [×5], C4⋊Dic3, D6⋊C4 [×11], C6.D4, C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6, S3×C2×C4 [×3], C2×D12 [×2], C22×Dic3, C2×C3⋊D4, C22×C12, S3×C23, C22.45C24, C422S3, C4×D12, C427S3, C423S3, Dic3.D4, S3×C22⋊C4, Dic34D4, D6⋊D4, D6.D4 [×2], D6⋊Q8, C4⋊C4⋊S3, C2×D6⋊C4, C23.28D6, C3×C42⋊C2, C4212D6
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×4], C24, C22×S3 [×7], C2×C4○D4 [×2], 2+ 1+4, C4○D12 [×2], S3×C23, C22.45C24, C2×C4○D12, S3×C4○D4, D4○D12, C4212D6

Smallest permutation representation of C4212D6
On 48 points
Generators in S48
```(1 31 7 25)(2 35 8 29)(3 33 9 27)(4 34 10 28)(5 32 11 26)(6 36 12 30)(13 43 19 37)(14 47 20 41)(15 45 21 39)(16 46 22 40)(17 44 23 38)(18 48 24 42)
(1 16 4 13)(2 17 5 14)(3 18 6 15)(7 22 10 19)(8 23 11 20)(9 24 12 21)(25 40 28 37)(26 41 29 38)(27 42 30 39)(31 46 34 43)(32 47 35 44)(33 48 36 45)
(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 12)(8 11)(9 10)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(25 27)(28 30)(31 33)(34 36)(37 45)(38 44)(39 43)(40 48)(41 47)(42 46)```

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 C4○D4 C4○D4 C4○D12 2+ 1+4 S3×C4○D4 D4○D12 kernel C42⋊12D6 C42⋊2S3 C4×D12 C42⋊7S3 C42⋊3S3 Dic3.D4 S3×C22⋊C4 Dic3⋊4D4 D6⋊D4 D6.D4 D6⋊Q8 C4⋊C4⋊S3 C2×D6⋊C4 C23.28D6 C3×C42⋊C2 C42⋊C2 C42 C22⋊C4 C4⋊C4 C22×C4 D6 C2×C6 C22 C6 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 2 2 2 1 4 4 8 1 2 2

Matrix representation of C4212D6 in GL6(𝔽13)

 8 0 0 0 0 0 0 8 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 1 0
,
 0 1 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 8 0 0 0 0 0 0 8
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 12 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))| [8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1] >;`

C4212D6 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_{12}D_6`
`% in TeX`

`G:=Group("C4^2:12D6");`
`// GroupNames label`

`G:=SmallGroup(192,1086);`
`// by ID`

`G=gap.SmallGroup(192,1086);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,100,675,136,6278]);`
`// 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*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;`
`// generators/relations`

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