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## G = C42⋊3D6order 192 = 26·3

### 1st semidirect product of C42 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C42⋊3D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — S3×C4○D4 — C42⋊3D6
 Lower central C3 — C6 — C12 — C42⋊3D6
 Upper central C1 — C4 — C2×C4 — C4≀C2

Generators and relations for C423D6
G = < a,b,c,d | a4=b4=c6=d2=1, cac-1=ab=ba, dad=ab-1, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 448 in 154 conjugacy classes, 51 normal (all characteristic)
C1, C2, C2 [×5], C3, C4 [×2], C4 [×6], C22, C22 [×7], S3 [×3], C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×12], D4, D4 [×6], Q8, Q8 [×2], C23 [×2], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×4], C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2) [×2], C22×C4 [×2], C2×D4 [×2], C2×Q8, C4○D4, C4○D4 [×5], C3⋊C8, C24, Dic6, Dic6, C4×S3 [×4], C4×S3 [×3], D12, D12, C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×3], C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C22×S3, C22×S3, C4≀C2, C4≀C2 [×3], C42⋊C2, C2×M4(2), C2×C4○D4, S3×C8, C8⋊S3, C4.Dic3, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, C3×M4(2), S3×C2×C4, S3×C2×C4, C4○D12, C4○D12, S3×D4, S3×D4, D42S3, D42S3, S3×Q8, Q83S3, C3×C4○D4, C42⋊C22, C424S3, D12⋊C4, Q83Dic3, C3×C4≀C2, C422S3, S3×M4(2), S3×C4○D4, C423D6
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], C22×S3, C2×C22⋊C4, S3×C2×C4, S3×D4 [×2], C42⋊C22, S3×C22⋊C4, C423D6

Smallest permutation representation of C423D6
On 48 points
Generators in S48
```(1 15 22 25)(2 13 20 29)(3 17 24 27)(4 31 38 18)(5 35 42 16)(6 33 40 14)(7 30 37 46)(8 28 41 44)(9 26 39 48)(10 45 23 32)(11 43 21 36)(12 47 19 34)
(1 38 12 41)(2 42 10 39)(3 40 11 37)(4 19 8 22)(5 23 9 20)(6 21 7 24)(13 16 45 48)(14 43 46 17)(15 18 47 44)(25 31 34 28)(26 29 35 32)(27 33 36 30)
(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 8)(11 12)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 24)(20 23)(21 22)(25 33)(26 32)(27 31)(28 36)(29 35)(30 34)(37 38)(39 42)(40 41)```

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

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

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

36 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 8A 8B 8C 8D 12A 12B 12C ··· 12G 12H 24A 24B order 1 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 6 6 6 8 8 8 8 12 12 12 ··· 12 12 24 24 size 1 1 2 4 6 6 12 2 1 1 2 4 4 4 6 6 12 12 12 2 4 8 4 4 12 12 2 2 4 ··· 4 8 8 8

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 S3 D4 D4 D4 D6 D6 D6 C4×S3 C4×S3 S3×D4 S3×D4 C42⋊C22 C42⋊3D6 kernel C42⋊3D6 C42⋊4S3 D12⋊C4 Q8⋊3Dic3 C3×C4≀C2 C42⋊2S3 S3×M4(2) S3×C4○D4 S3×D4 D4⋊2S3 S3×Q8 Q8⋊3S3 C4≀C2 C4×S3 C2×Dic3 C22×S3 C42 M4(2) C4○D4 D4 Q8 C4 C22 C3 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 1 2 1 1 1 1 1 2 2 1 1 2 4

Matrix representation of C423D6 in GL4(𝔽73) generated by

 55 37 55 37 36 18 36 18 18 36 55 37 37 55 36 18
,
 0 0 72 0 0 0 0 72 1 0 0 0 0 1 0 0
,
 0 0 72 72 0 0 1 0 72 72 0 0 1 0 0 0
,
 0 0 1 1 0 0 0 72 1 1 0 0 0 72 0 0
`G:=sub<GL(4,GF(73))| [55,36,18,37,37,18,36,55,55,36,55,36,37,18,37,18],[0,0,1,0,0,0,0,1,72,0,0,0,0,72,0,0],[0,0,72,1,0,0,72,0,72,1,0,0,72,0,0,0],[0,0,1,0,0,0,1,72,1,0,0,0,1,72,0,0] >;`

C423D6 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_3D_6`
`% in TeX`

`G:=Group("C4^2:3D6");`
`// GroupNames label`

`G:=SmallGroup(192,380);`
`// by ID`

`G=gap.SmallGroup(192,380);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,58,136,851,438,102,6278]);`
`// Polycyclic`

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

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