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## G = C42⋊23D6order 192 = 26·3

### 21st semidirect product of C42 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C42⋊23D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22⋊C4 — C42⋊23D6
 Lower central C3 — C2×C6 — C42⋊23D6
 Upper central C1 — C22 — C4.4D4

Generators and relations for C4223D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 672 in 248 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×11], C22, C22 [×18], S3 [×4], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23 [×7], Dic3 [×6], C12 [×5], D6 [×4], D6 [×8], C2×C6, C2×C6 [×6], C42, C42 [×2], C22⋊C4 [×4], C22⋊C4 [×10], C4⋊C4 [×8], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×Q8, C24, C4×S3 [×6], C2×Dic3 [×6], C2×Dic3, C3⋊D4 [×4], C2×C12, C2×C12 [×4], C3×D4, C3×Q8, C22×S3 [×2], C22×S3 [×5], C22×C6 [×2], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C422C2 [×2], C4×Dic3 [×2], Dic3⋊C4 [×6], C4⋊Dic3 [×2], D6⋊C4 [×6], C6.D4 [×2], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×4], S3×C2×C4 [×4], C22×Dic3, C2×C3⋊D4 [×2], C6×D4, C6×Q8, S3×C23, C22.45C24, C422S3 [×2], C23.8D6 [×2], S3×C22⋊C4 [×2], Dic34D4 [×2], C23.9D6 [×2], C23.23D6, C232D6, D63Q8 [×2], C3×C4.4D4, C4223D6
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, S3×C23, C22.45C24, D46D6, S3×C4○D4 [×2], C4223D6

Smallest permutation representation of C4223D6
On 48 points
Generators in S48
```(1 38 7 31)(2 42 8 35)(3 40 9 33)(4 41 10 34)(5 39 11 32)(6 37 12 36)(13 28 46 25)(14 23 47 20)(15 30 48 27)(16 19 43 22)(17 26 44 29)(18 21 45 24)
(1 25 4 19)(2 29 5 23)(3 27 6 21)(7 28 10 22)(8 26 11 20)(9 30 12 24)(13 41 43 38)(14 35 44 32)(15 37 45 40)(16 31 46 34)(17 39 47 42)(18 33 48 36)
(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 9)(2 8)(3 7)(4 12)(5 11)(6 10)(13 15)(16 18)(19 21)(22 24)(25 27)(28 30)(31 40)(32 39)(33 38)(34 37)(35 42)(36 41)(43 45)(46 48)```

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 6A 6B 6C 6D 6E 12A ··· 12F 12G 12H order 1 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 6 6 6 6 6 12 ··· 12 12 12 size 1 1 1 1 4 4 6 6 6 6 2 2 2 2 2 4 4 4 6 6 6 6 12 12 12 12 2 2 2 8 8 4 ··· 4 8 8

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 C4○D4 2+ 1+4 D4⋊6D6 S3×C4○D4 kernel C42⋊23D6 C42⋊2S3 C23.8D6 S3×C22⋊C4 Dic3⋊4D4 C23.9D6 C23.23D6 C23⋊2D6 D6⋊3Q8 C3×C4.4D4 C4.4D4 C42 C22⋊C4 C2×D4 C2×Q8 D6 C6 C2 C2 # reps 1 2 2 2 2 2 1 1 2 1 1 1 4 1 1 8 1 2 4

Matrix representation of C4223D6 in GL6(𝔽13)

 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 5 0 0 0 0 0 0 5
,
 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 1 11 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 1 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 12 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 12 1

`G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,11,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,1,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,12,12,0,0,0,0,0,1] >;`

C4223D6 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_{23}D_6`
`% in TeX`

`G:=Group("C4^2:23D6");`
`// GroupNames label`

`G:=SmallGroup(192,1238);`
`// by ID`

`G=gap.SmallGroup(192,1238);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,387,100,346,136,6278]);`
`// Polycyclic`

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

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