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## G = C42⋊7D6order 192 = 26·3

### 5th semidirect product of C42 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C42⋊7D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — D4⋊6D6 — C42⋊7D6
 Lower central C3 — C6 — C2×C12 — C42⋊7D6
 Upper central C1 — C2 — C2×C4 — C4.4D4

Generators and relations for C427D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=a-1b2, dad=a-1b, cbc-1=b-1, bd=db, dcd=c-1 >

Subgroups: 496 in 152 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2 [×5], C3, C4 [×2], C4 [×4], C22, C22 [×8], S3 [×2], C6, C6 [×3], C8 [×2], C2×C4, C2×C4 [×6], D4 [×10], Q8 [×4], C23 [×2], C23 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6 [×4], C42, C22⋊C4 [×2], M4(2) [×2], SD16 [×2], Q16 [×2], C2×D4, C2×D4 [×4], C2×Q8, C4○D4 [×4], C3⋊C8 [×2], Dic6 [×2], C4×S3 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×6], C2×C12, C2×C12 [×2], C3×D4 [×2], C3×Q8 [×2], C22×S3 [×2], C22×C6 [×2], C4.D4, C4≀C2 [×2], C4.4D4, C8.C22 [×2], 2+ 1+4, C4.Dic3 [×2], Q82S3 [×2], C3⋊Q16 [×2], C4×C12, C3×C22⋊C4 [×2], C4○D12 [×2], S3×D4 [×2], D42S3 [×2], C2×C3⋊D4 [×2], C6×D4, C6×Q8, D4.9D4, C424S3 [×2], C12.D4, Q8.11D6 [×2], C3×C4.4D4, D46D6, C427D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C22≀C2, S3×D4 [×2], C2×C3⋊D4, D4.9D4, C232D6, C427D6

Character table of C427D6

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 6A 6B 6C 6D 6E 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 2 4 4 12 12 2 2 2 4 4 8 12 12 2 2 2 8 8 24 24 4 4 4 4 4 4 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 1 -1 1 1 1 1 -1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 linear of order 2 ρ4 1 1 1 -1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 -1 1 1 1 1 1 1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ6 1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ7 1 1 1 -1 -1 1 -1 1 1 1 1 1 -1 1 -1 1 1 1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 linear of order 2 ρ8 1 1 1 -1 -1 1 -1 1 1 1 -1 -1 1 1 -1 1 1 1 -1 -1 1 -1 -1 -1 1 1 -1 -1 1 1 linear of order 2 ρ9 2 2 2 -2 -2 0 0 -1 2 2 2 2 -2 0 0 -1 -1 -1 1 1 0 0 -1 -1 -1 -1 -1 -1 1 1 orthogonal lifted from D6 ρ10 2 2 2 -2 -2 0 0 -1 2 2 -2 -2 2 0 0 -1 -1 -1 1 1 0 0 1 1 -1 -1 1 1 -1 -1 orthogonal lifted from D6 ρ11 2 2 -2 0 0 2 0 2 2 -2 0 0 0 -2 0 2 -2 -2 0 0 0 0 0 0 -2 2 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 0 0 0 -2 2 -2 2 0 0 0 0 2 2 -2 -2 0 0 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 -2 0 0 2 -2 -2 0 0 0 0 0 2 2 2 2 -2 0 0 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 -2 0 0 -2 0 2 2 -2 0 0 0 2 0 2 -2 -2 0 0 0 0 0 0 -2 2 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 2 2 0 0 -1 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ16 2 2 -2 0 0 0 2 2 -2 2 0 0 0 0 -2 2 -2 -2 0 0 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from D4 ρ17 2 2 2 2 2 0 0 -1 2 2 -2 -2 -2 0 0 -1 -1 -1 -1 -1 0 0 1 1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ18 2 2 2 -2 2 0 0 2 -2 -2 0 0 0 0 0 2 2 2 -2 2 0 0 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 -2 0 0 -1 -2 -2 0 0 0 0 0 -1 -1 -1 -1 1 0 0 √-3 √-3 1 1 -√-3 -√-3 -√-3 √-3 complex lifted from C3⋊D4 ρ20 2 2 2 -2 2 0 0 -1 -2 -2 0 0 0 0 0 -1 -1 -1 1 -1 0 0 -√-3 -√-3 1 1 √-3 √-3 -√-3 √-3 complex lifted from C3⋊D4 ρ21 2 2 2 -2 2 0 0 -1 -2 -2 0 0 0 0 0 -1 -1 -1 1 -1 0 0 √-3 √-3 1 1 -√-3 -√-3 √-3 -√-3 complex lifted from C3⋊D4 ρ22 2 2 2 2 -2 0 0 -1 -2 -2 0 0 0 0 0 -1 -1 -1 -1 1 0 0 -√-3 -√-3 1 1 √-3 √-3 √-3 -√-3 complex lifted from C3⋊D4 ρ23 4 4 -4 0 0 0 0 -2 -4 4 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 -2 2 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 4 -4 0 0 0 0 -2 4 -4 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from S3×D4 ρ25 4 -4 0 0 0 0 0 4 0 0 -2i 2i 0 0 0 -4 0 0 0 0 0 0 2i -2i 0 0 2i -2i 0 0 complex lifted from D4.9D4 ρ26 4 -4 0 0 0 0 0 4 0 0 2i -2i 0 0 0 -4 0 0 0 0 0 0 -2i 2i 0 0 -2i 2i 0 0 complex lifted from D4.9D4 ρ27 4 -4 0 0 0 0 0 -2 0 0 -2i 2i 0 0 0 2 -2√-3 2√-3 0 0 0 0 2ζ4ζ32 2ζ43ζ32 0 0 2ζ4ζ3 2ζ43ζ3 0 0 complex faithful ρ28 4 -4 0 0 0 0 0 -2 0 0 2i -2i 0 0 0 2 2√-3 -2√-3 0 0 0 0 2ζ43ζ3 2ζ4ζ3 0 0 2ζ43ζ32 2ζ4ζ32 0 0 complex faithful ρ29 4 -4 0 0 0 0 0 -2 0 0 2i -2i 0 0 0 2 -2√-3 2√-3 0 0 0 0 2ζ43ζ32 2ζ4ζ32 0 0 2ζ43ζ3 2ζ4ζ3 0 0 complex faithful ρ30 4 -4 0 0 0 0 0 -2 0 0 -2i 2i 0 0 0 2 2√-3 -2√-3 0 0 0 0 2ζ4ζ3 2ζ43ζ3 0 0 2ζ4ζ32 2ζ43ζ32 0 0 complex faithful

Smallest permutation representation of C427D6
On 48 points
Generators in S48
```(1 24 13 32)(2 47 14 38)(3 20 15 34)(4 43 16 40)(5 22 17 36)(6 45 18 42)(7 23 28 31)(8 46 29 37)(9 19 30 33)(10 48 25 39)(11 21 26 35)(12 44 27 41)
(1 16 29 11)(2 12 30 17)(3 18 25 7)(4 8 26 13)(5 14 27 9)(6 10 28 15)(19 22 38 41)(20 42 39 23)(21 24 40 37)(31 34 45 48)(32 43 46 35)(33 36 47 44)
(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 13)(8 18)(9 17)(10 16)(11 15)(12 14)(19 38)(20 37)(21 42)(22 41)(23 40)(24 39)(25 26)(27 30)(28 29)(31 35)(32 34)(43 45)(46 48)```

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

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

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

Matrix representation of C427D6 in GL4(𝔽73) generated by

 17 66 63 66 7 10 7 56 10 7 17 66 66 17 7 10
,
 0 0 1 0 0 0 0 1 72 0 0 0 0 72 0 0
,
 0 0 13 30 0 0 43 43 13 30 0 0 43 43 0 0
,
 0 0 43 60 0 0 30 30 30 13 0 0 43 43 0 0
`G:=sub<GL(4,GF(73))| [17,7,10,66,66,10,7,17,63,7,17,7,66,56,66,10],[0,0,72,0,0,0,0,72,1,0,0,0,0,1,0,0],[0,0,13,43,0,0,30,43,13,43,0,0,30,43,0,0],[0,0,30,43,0,0,13,43,43,30,0,0,60,30,0,0] >;`

C427D6 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_7D_6`
`% in TeX`

`G:=Group("C4^2:7D6");`
`// GroupNames label`

`G:=SmallGroup(192,620);`
`// by ID`

`G=gap.SmallGroup(192,620);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,184,1123,570,297,136,1684,6278]);`
`// 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^-1*b^2,d*a*d=a^-1*b,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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