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## G = C42⋊18D6order 192 = 26·3

### 16th semidirect product of C42 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C42⋊18D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22⋊C4 — C42⋊18D6
 Lower central C3 — C2×C6 — C42⋊18D6
 Upper central C1 — C22 — C4×D4

Generators and relations for C4218D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=a-1b2, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 664 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×11], C22, C22 [×2], C22 [×16], S3 [×3], C6 [×3], C6 [×3], C2×C4 [×5], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23 [×7], Dic3 [×6], C12 [×5], D6 [×2], D6 [×9], C2×C6, C2×C6 [×2], C2×C6 [×5], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×7], C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C24, Dic6, C4×S3 [×3], C2×Dic3 [×6], C2×Dic3 [×2], C3⋊D4 [×3], C2×C12 [×5], C2×C12 [×2], C3×D4 [×2], C22×S3 [×2], C22×S3 [×5], C22×C6 [×2], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4, C4×D4, C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C422C2 [×2], C4×Dic3 [×2], Dic3⋊C4 [×6], C4⋊Dic3, D6⋊C4 [×8], C6.D4 [×4], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4 [×2], C22×Dic3, C2×C3⋊D4 [×2], C22×C12 [×2], C6×D4, S3×C23, C22.45C24, C422S3, C423S3, C23.16D6, S3×C22⋊C4, C23.9D6, C23.11D6, D6⋊Q8, C4⋊C4⋊S3, C12.48D4, C2×D6⋊C4, C4×C3⋊D4, C23.28D6, C23.23D6, C232D6, D4×C12, C4218D6
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, D46D6, S3×C4○D4, C4218D6

Smallest permutation representation of C4218D6
On 48 points
Generators in S48
```(1 41 10 34)(2 39 11 32)(3 37 12 36)(4 31 7 38)(5 35 8 42)(6 33 9 40)(13 28 22 43)(14 26 23 47)(15 30 24 45)(16 46 19 25)(17 44 20 29)(18 48 21 27)
(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 38 43 34)(26 39 44 35)(27 40 45 36)(28 41 46 31)(29 42 47 32)(30 37 48 33)
(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 3)(4 6)(7 9)(10 12)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(25 45)(26 44)(27 43)(28 48)(29 47)(30 46)(31 36)(32 35)(33 34)(37 38)(39 42)(40 41)```

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A ··· 4F 4G 4H 4I 4J 4K ··· 4O 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 4 4 4 ··· 4 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 size 1 1 1 1 2 2 4 6 6 12 2 2 ··· 2 4 4 6 6 12 ··· 12 2 2 2 4 4 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 1 2 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 C2 S3 D6 D6 D6 D6 D6 C4○D4 C4○D4 C4○D12 2+ 1+4 D4⋊6D6 S3×C4○D4 kernel C42⋊18D6 C42⋊2S3 C42⋊3S3 C23.16D6 S3×C22⋊C4 C23.9D6 C23.11D6 D6⋊Q8 C4⋊C4⋊S3 C12.48D4 C2×D6⋊C4 C4×C3⋊D4 C23.28D6 C23.23D6 C23⋊2D6 D4×C12 C4×D4 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D6 C2×C6 C22 C6 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 1 4 4 8 1 2 2

Matrix representation of C4218D6 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 12 0 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 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5 2 0 0 0 0 0 8
,
 0 12 0 0 0 0 1 12 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 0 1
,
 1 12 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 0 0 0 0 0 8 12

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,2,8],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,8,0,0,0,0,0,12] >;`

C4218D6 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_{18}D_6`
`% in TeX`

`G:=Group("C4^2:18D6");`
`// GroupNames label`

`G:=SmallGroup(192,1115);`
`// by ID`

`G=gap.SmallGroup(192,1115);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,1571,136,6278]);`
`// 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^-1*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;`
`// generators/relations`

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