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## G = C42⋊25D6order 192 = 26·3

### 23rd semidirect product of C42 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C42⋊25D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22⋊C4 — C42⋊25D6
 Lower central C3 — C2×C6 — C42⋊25D6
 Upper central C1 — C22 — C42⋊2C2

Generators and relations for C4225D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=a-1b2, cbc-1=a2b, dbd=b-1, dcd=c-1 >

Subgroups: 784 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×10], C22, C22 [×20], S3 [×5], C6 [×3], C6, C2×C4 [×6], C2×C4 [×8], D4 [×9], Q8, C23, C23 [×8], Dic3 [×4], C12 [×6], D6 [×2], D6 [×15], C2×C6, C2×C6 [×3], C42, C42, C22⋊C4 [×3], C22⋊C4 [×11], C4⋊C4 [×3], C4⋊C4 [×3], C22×C4 [×4], C2×D4 [×7], C2×Q8, C24, Dic6, C4×S3 [×4], D12 [×7], C2×Dic3 [×4], C3⋊D4 [×2], C2×C12 [×6], C22×S3 [×4], C22×S3 [×4], C22×C6, C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C422C2, C422C2, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×10], C6.D4, C4×C12, C3×C22⋊C4 [×3], C3×C4⋊C4 [×3], C2×Dic6, S3×C2×C4 [×4], C2×D12 [×5], C2×C3⋊D4 [×2], S3×C23, C22.32C24, C4×D12, C427S3, S3×C22⋊C4, D6⋊D4 [×2], C23.9D6, Dic3⋊D4, C23.11D6, Dic35D4, D6.D4, C12⋊D4 [×2], D6⋊Q8, C4⋊C4⋊S3, C3×C422C2, C4225D6
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, S3×C4○D4, D4○D12 [×2], C4225D6

Smallest permutation representation of C4225D6
On 48 points
Generators in S48
```(1 31 4 28)(2 35 5 26)(3 33 6 30)(7 34 10 25)(8 32 11 29)(9 36 12 27)(13 41 16 43)(14 47 17 39)(15 37 18 45)(19 46 22 38)(20 42 23 44)(21 48 24 40)
(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 41 31 38)(26 44 32 47)(27 37 33 40)(28 46 34 43)(29 39 35 42)(30 48 36 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 6)(2 5)(3 4)(7 12)(8 11)(9 10)(13 21)(14 20)(15 19)(16 24)(17 23)(18 22)(25 33)(26 32)(27 31)(28 36)(29 35)(30 34)(37 41)(38 40)(43 45)(46 48)```

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

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

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

36 conjugacy classes

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

36 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⋊25D6 C4×D12 C42⋊7S3 S3×C22⋊C4 D6⋊D4 C23.9D6 Dic3⋊D4 C23.11D6 Dic3⋊5D4 D6.D4 C12⋊D4 D6⋊Q8 C4⋊C4⋊S3 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 4 2 2 4

Matrix representation of C4225D6 in GL6(𝔽13)

 8 0 0 0 0 0 0 8 0 0 0 0 0 0 12 0 7 1 0 0 0 12 12 6 0 0 10 7 1 0 0 0 6 3 0 1
,
 9 3 0 0 0 0 3 4 0 0 0 0 0 0 3 6 11 0 0 0 7 10 0 11 0 0 12 0 10 7 0 0 0 12 6 3
,
 1 0 0 0 0 0 7 12 0 0 0 0 0 0 1 1 0 0 0 0 12 0 0 0 0 0 10 3 12 12 0 0 10 7 1 0
,
 1 0 0 0 0 0 7 12 0 0 0 0 0 0 12 12 0 0 0 0 0 1 0 0 0 0 10 3 12 12 0 0 6 3 0 1

`G:=sub<GL(6,GF(13))| [8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,10,6,0,0,0,12,7,3,0,0,7,12,1,0,0,0,1,6,0,1],[9,3,0,0,0,0,3,4,0,0,0,0,0,0,3,7,12,0,0,0,6,10,0,12,0,0,11,0,10,6,0,0,0,11,7,3],[1,7,0,0,0,0,0,12,0,0,0,0,0,0,1,12,10,10,0,0,1,0,3,7,0,0,0,0,12,1,0,0,0,0,12,0],[1,7,0,0,0,0,0,12,0,0,0,0,0,0,12,0,10,6,0,0,12,1,3,3,0,0,0,0,12,0,0,0,0,0,12,1] >;`

C4225D6 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_{25}D_6`
`% in TeX`

`G:=Group("C4^2:25D6");`
`// GroupNames label`

`G:=SmallGroup(192,1263);`
`// by ID`

`G=gap.SmallGroup(192,1263);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,675,570,192,6278]);`
`// 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=b^-1,d*c*d=c^-1>;`
`// generators/relations`

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