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## G = C42⋊20D6order 192 = 26·3

### 18th semidirect product of C42 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C42⋊20D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×S3×D4 — C42⋊20D6
 Lower central C3 — C2×C6 — C42⋊20D6
 Upper central C1 — C22 — C4.4D4

Generators and relations for C4220D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=a2b-1, dbd=b-1, dcd=c-1 >

Subgroups: 1104 in 334 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×2], C4 [×8], C22, C22 [×30], S3 [×6], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], C2×C4 [×11], D4 [×22], Q8 [×2], C23 [×2], C23 [×13], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×4], D6 [×2], D6 [×22], C2×C6, C2×C6 [×6], C42, C42, C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×18], C2×Q8, C4○D4 [×4], C24 [×2], C4×S3 [×4], C4×S3 [×4], D12 [×12], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×8], C2×C12, C2×C12 [×4], C3×D4 [×2], C3×Q8 [×2], C22×S3, C22×S3 [×4], C22×S3 [×8], C22×C6 [×2], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4, C4.4D4, C41D4 [×2], C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4 [×2], D6⋊C4 [×6], C4×C12, C3×C22⋊C4 [×4], S3×C2×C4, S3×C2×C4 [×2], C2×D12 [×2], C2×D12 [×6], S3×D4 [×4], Q83S3 [×4], C2×C3⋊D4 [×6], C6×D4, C6×Q8, S3×C23 [×2], C22.29C24, C422S3, C4⋊D12, D6⋊D4 [×4], Dic3⋊D4 [×4], C123D4, C12.23D4, C3×C4.4D4, C2×S3×D4, C2×Q83S3, C4220D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C22×D4, 2+ 1+4 [×2], S3×D4 [×2], S3×C23, C22.29C24, C2×S3×D4, D4○D12 [×2], C4220D6

Smallest permutation representation of C4220D6
On 48 points
Generators in S48
```(1 23 6 20)(2 21 4 24)(3 19 5 22)(7 26 10 29)(8 30 11 27)(9 28 12 25)(13 40 31 44)(14 45 32 41)(15 42 33 46)(16 47 34 37)(17 38 35 48)(18 43 36 39)
(1 47 7 40)(2 45 8 38)(3 43 9 42)(4 41 11 48)(5 39 12 46)(6 37 10 44)(13 20 16 29)(14 27 17 24)(15 22 18 25)(19 36 28 33)(21 32 30 35)(23 34 26 31)
(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 4)(2 6)(3 5)(7 11)(8 10)(9 12)(13 14)(15 18)(16 17)(20 24)(21 23)(26 30)(27 29)(31 32)(33 36)(34 35)(37 38)(39 42)(40 41)(43 46)(44 45)(47 48)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 6D 6E 12A ··· 12F 12G 12H order 1 2 2 2 2 2 2 2 2 2 2 2 3 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 12 12 2 2 2 4 4 4 4 6 6 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 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D6 2+ 1+4 S3×D4 D4○D12 kernel C42⋊20D6 C42⋊2S3 C4⋊D12 D6⋊D4 Dic3⋊D4 C12⋊3D4 C12.23D4 C3×C4.4D4 C2×S3×D4 C2×Q8⋊3S3 C4.4D4 C4×S3 C42 C22⋊C4 C2×D4 C2×Q8 C6 C4 C2 # reps 1 1 1 4 4 1 1 1 1 1 1 4 1 4 1 1 2 2 4

Matrix representation of C4220D6 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 3 7 0 0 0 0 6 10 0 0 3 7 0 0 0 0 6 10 0 0
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 1 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 0 1

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

C4220D6 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_{20}D_6`
`% in TeX`

`G:=Group("C4^2:20D6");`
`// GroupNames label`

`G:=SmallGroup(192,1233);`
`// by ID`

`G=gap.SmallGroup(192,1233);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,675,570,297,192,6278]);`
`// 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,c*b*c^-1=a^2*b^-1,d*b*d=b^-1,d*c*d=c^-1>;`
`// generators/relations`

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