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## G = C42⋊26D6order 192 = 26·3

### 24th semidirect product of C42 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C42⋊26D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22⋊C4 — C42⋊26D6
 Lower central C3 — C2×C6 — C42⋊26D6
 Upper central C1 — C22 — C42⋊2C2

Generators and relations for C4226D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=a-1b2, cbc-1=a2b, dbd=a2b-1, dcd=c-1 >

Subgroups: 704 in 248 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×11], C22, C22 [×18], S3 [×5], C6 [×3], C6, C2×C4 [×6], C2×C4 [×12], D4 [×5], Q8, C23, C23 [×8], Dic3 [×5], C12 [×6], D6 [×4], D6 [×11], C2×C6, C2×C6 [×3], C42, C42 [×2], C22⋊C4 [×3], C22⋊C4 [×11], C4⋊C4 [×3], C4⋊C4 [×5], C22×C4 [×5], C2×D4 [×3], C2×Q8, C24, Dic6, C4×S3 [×7], D12 [×4], C2×Dic3 [×5], C3⋊D4, C2×C12 [×6], C22×S3 [×3], C22×S3 [×5], C22×C6, C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C422C2, C422C2, C4×Dic3 [×2], Dic3⋊C4 [×3], C4⋊Dic3 [×2], D6⋊C4 [×9], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×3], C3×C4⋊C4 [×3], C2×Dic6, S3×C2×C4 [×5], C2×D12 [×2], C2×C3⋊D4, S3×C23, C22.45C24, C422S3, C4×D12, C23.8D6, S3×C22⋊C4 [×2], D6⋊D4, C23.9D6, C23.11D6, C4⋊C47S3, Dic35D4, D6.D4 [×2], D6⋊Q8, C4.D12, C3×C422C2, C4226D6
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, S3×C4○D4 [×2], D4○D12, C4226D6

Smallest permutation representation of C4226D6
On 48 points
Generators in S48
```(1 38 4 28)(2 42 5 26)(3 40 6 30)(7 41 10 25)(8 39 11 29)(9 37 12 27)(13 46 16 34)(14 32 17 44)(15 48 18 36)(19 31 22 43)(20 47 23 35)(21 33 24 45)
(1 22 10 13)(2 20 11 17)(3 24 12 15)(4 19 7 16)(5 23 8 14)(6 21 9 18)(25 46 38 43)(26 35 39 32)(27 48 40 45)(28 31 41 34)(29 44 42 47)(30 33 37 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 3)(4 6)(7 9)(10 12)(13 21)(14 20)(15 19)(16 24)(17 23)(18 22)(25 30)(26 29)(27 28)(31 33)(34 36)(37 38)(39 42)(40 41)(43 45)(46 48)```

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

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

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

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 C4○D4 2+ 1+4 S3×C4○D4 D4○D12 kernel C42⋊26D6 C42⋊2S3 C4×D12 C23.8D6 S3×C22⋊C4 D6⋊D4 C23.9D6 C23.11D6 C4⋊C4⋊7S3 Dic3⋊5D4 D6.D4 D6⋊Q8 C4.D12 C3×C42⋊2C2 C42⋊2C2 C42 C22⋊C4 C4⋊C4 D6 C6 C2 C2 # reps 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 3 3 8 1 4 2

Matrix representation of C4226D6 in GL6(𝔽13)

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

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

C4226D6 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_{26}D_6`
`% in TeX`

`G:=Group("C4^2:26D6");`
`// GroupNames label`

`G:=SmallGroup(192,1264);`
`// by ID`

`G=gap.SmallGroup(192,1264);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,1571,570,192,6278]);`
`// 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,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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