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## G = C42⋊22D6order 192 = 26·3

### 20th semidirect product of C42 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C42⋊22D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22⋊C4 — C42⋊22D6
 Lower central C3 — C2×C6 — C42⋊22D6
 Upper central C1 — C22 — C4.4D4

Generators and relations for C4222D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=ab2, cbc-1=a2b, dbd=a2b-1, dcd=c-1 >

Subgroups: 752 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×10], C22, C22 [×20], S3 [×4], C6 [×3], C6 [×2], C2×C4 [×5], C2×C4 [×9], D4 [×9], Q8, C23 [×2], C23 [×7], Dic3 [×5], C12 [×5], D6 [×2], D6 [×12], C2×C6, C2×C6 [×6], C42, C42, C22⋊C4 [×4], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×4], C2×D4, C2×D4 [×6], C2×Q8, C24, C4×S3 [×3], D12 [×3], C2×Dic3 [×5], C2×Dic3, C3⋊D4 [×5], C2×C12 [×5], C3×D4, C3×Q8, C22×S3 [×3], C22×S3 [×4], C22×C6 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C4.4D4, C422C2 [×2], C4×Dic3, Dic3⋊C4 [×4], C4⋊Dic3 [×2], D6⋊C4 [×8], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×4], S3×C2×C4 [×3], C2×D12 [×2], C22×Dic3, C2×C3⋊D4 [×4], C6×D4, C6×Q8, S3×C23, C22.32C24, C4×D12, C423S3, C23.8D6, S3×C22⋊C4, Dic34D4, D6⋊D4, C23.9D6, Dic3⋊D4 [×2], C23.21D6, C232D6, C23.14D6, D63Q8, C12.23D4, C3×C4.4D4, C4222D6
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, 2+ 1+4 [×2], S3×C23, C22.32C24, D46D6, S3×C4○D4, D4○D12, C4222D6

Smallest permutation representation of C4222D6
On 48 points
Generators in S48
```(1 19 12 45)(2 23 10 43)(3 21 11 47)(4 46 9 20)(5 44 7 24)(6 48 8 22)(13 29 16 37)(14 41 17 27)(15 25 18 39)(26 33 40 36)(28 35 42 32)(30 31 38 34)
(1 31 8 17)(2 35 9 15)(3 33 7 13)(4 18 10 32)(5 16 11 36)(6 14 12 34)(19 38 22 27)(20 25 23 42)(21 40 24 29)(26 44 37 47)(28 46 39 43)(30 48 41 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 7)(2 9)(3 8)(4 10)(5 12)(6 11)(13 14)(15 18)(16 17)(19 21)(22 24)(25 28)(26 27)(29 30)(31 36)(32 35)(33 34)(37 38)(39 42)(40 41)(44 48)(45 47)```

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

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

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

36 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 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 6 6 6 6 6 12 ··· 12 12 12 size 1 1 1 1 4 4 6 6 12 12 2 2 2 4 4 4 4 6 6 12 12 12 12 2 2 2 8 8 4 ··· 4 8 8

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 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 2+ 1+4 D4⋊6D6 S3×C4○D4 D4○D12 kernel C42⋊22D6 C4×D12 C42⋊3S3 C23.8D6 S3×C22⋊C4 Dic3⋊4D4 D6⋊D4 C23.9D6 Dic3⋊D4 C23.21D6 C23⋊2D6 C23.14D6 D6⋊3Q8 C12.23D4 C3×C4.4D4 C4.4D4 C42 C22⋊C4 C2×D4 C2×Q8 D6 C6 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 4 1 1 4 2 2 2 2

Matrix representation of C4222D6 in GL8(𝔽13)

 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 12 0 11 0 0 0 0 0 0 12 5 8 0 0 0 0 1 0 1 0 0 0 0 0 1 3 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 8 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0
,
 1 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 5 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 5 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 12 10 0 12
,
 1 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 8 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 3 0 1

`G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,1,1,0,0,0,0,0,12,0,3,0,0,0,0,11,5,1,0,0,0,0,0,0,8,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,12,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,3,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0],[1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,5,1,12,0,0,0,0,0,1,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,1,1,0,0,0,0,0,12,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;`

C4222D6 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_{22}D_6`
`% in TeX`

`G:=Group("C4^2:22D6");`
`// GroupNames label`

`G:=SmallGroup(192,1237);`
`// by ID`

`G=gap.SmallGroup(192,1237);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,675,570,80,6278]);`
`// Polycyclic`

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

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