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## G = C42⋊6D6order 192 = 26·3

### 4th semidirect product of C42 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C42⋊6D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4○D12 — C2×C4○D12 — C42⋊6D6
 Lower central C3 — C6 — C12 — C42⋊6D6
 Upper central C1 — C4 — C22×C4 — C42⋊C2

Generators and relations for C426D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=ab2, dad=ab-1, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 424 in 154 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2 [×5], C3, C4 [×4], C4 [×4], C22 [×3], C22 [×5], S3 [×2], C6, C6 [×3], C8 [×2], C2×C4 [×6], C2×C4 [×7], D4 [×7], Q8 [×3], C23, C23, Dic3 [×2], C12 [×4], C12 [×2], D6 [×4], C2×C6 [×3], C2×C6, C42 [×2], C22⋊C4, C4⋊C4, C2×C8, M4(2) [×3], C22×C4, C22×C4, C2×D4 [×2], C2×Q8, C4○D4 [×6], C3⋊C8 [×2], Dic6 [×2], Dic6, C4×S3 [×4], D12 [×2], D12, C2×Dic3, C3⋊D4 [×4], C2×C12 [×6], C2×C12 [×2], C22×S3, C22×C6, C4≀C2 [×4], C42⋊C2, C2×M4(2), C2×C4○D4, C2×C3⋊C8, C4.Dic3 [×2], C4.Dic3, C4×C12 [×2], C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12 [×4], C4○D12 [×2], C2×C3⋊D4, C22×C12, C42⋊C22, C424S3 [×4], C2×C4.Dic3, C3×C42⋊C2, C2×C4○D12, C426D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, D6⋊C4 [×4], S3×C2×C4, C2×D12, C2×C3⋊D4, C42⋊C22, C2×D6⋊C4, C426D6

Smallest permutation representation of C426D6
On 48 points
Generators in S48
```(1 37 16 28)(2 41 17 26)(3 39 18 30)(4 40 13 25)(5 38 14 29)(6 42 15 27)(7 43 22 34)(8 47 23 32)(9 45 24 36)(10 46 19 31)(11 44 20 35)(12 48 21 33)
(1 10 4 7)(2 11 5 8)(3 12 6 9)(13 22 16 19)(14 23 17 20)(15 24 18 21)(25 34 28 31)(26 35 29 32)(27 36 30 33)(37 46 40 43)(38 47 41 44)(39 48 42 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 3)(4 6)(7 12)(8 11)(9 10)(13 18)(14 17)(15 16)(19 21)(22 24)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(37 45)(38 44)(39 43)(40 48)(41 47)(42 46)```

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 S3 D4 D4 D6 D6 C4×S3 D12 C3⋊D4 C3⋊D4 C42⋊C22 C42⋊6D6 kernel C42⋊6D6 C42⋊4S3 C2×C4.Dic3 C3×C42⋊C2 C2×C4○D12 C2×Dic6 C2×D12 C4○D12 C42⋊C2 C2×C12 C22×C6 C42 C22×C4 C2×C4 C2×C4 C2×C4 C23 C3 C1 # reps 1 4 1 1 1 2 2 4 1 3 1 2 1 4 4 2 2 2 4

Matrix representation of C426D6 in GL4(𝔽73) generated by

 0 0 72 0 0 0 0 72 43 13 0 0 60 30 0 0
,
 66 59 0 0 14 7 0 0 0 0 66 59 0 0 14 7
,
 72 72 0 0 1 0 0 0 0 0 1 1 0 0 72 0
,
 72 0 0 0 1 1 0 0 0 0 66 59 0 0 66 7
`G:=sub<GL(4,GF(73))| [0,0,43,60,0,0,13,30,72,0,0,0,0,72,0,0],[66,14,0,0,59,7,0,0,0,0,66,14,0,0,59,7],[72,1,0,0,72,0,0,0,0,0,1,72,0,0,1,0],[72,1,0,0,0,1,0,0,0,0,66,66,0,0,59,7] >;`

C426D6 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_6D_6`
`% in TeX`

`G:=Group("C4^2:6D6");`
`// GroupNames label`

`G:=SmallGroup(192,564);`
`// by ID`

`G=gap.SmallGroup(192,564);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,387,58,1123,1684,102,6278]);`
`// 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,d*a*d=a*b^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;`
`// generators/relations`

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