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## G = C42⋊8D6order 192 = 26·3

### 6th semidirect product of C42 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C42⋊8D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — D4⋊6D6 — C42⋊8D6
 Lower central C3 — C6 — C2×C12 — C42⋊8D6
 Upper central C1 — C2 — C2×C4 — C4⋊1D4

Generators and relations for C428D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=a-1, dad=a-1b-1, cbc-1=b-1, bd=db, dcd=c-1 >

Subgroups: 560 in 168 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2 [×6], C3, C4 [×2], C4 [×4], C22, C22 [×11], S3 [×2], C6, C6 [×4], C8 [×2], C2×C4, C2×C4 [×5], D4 [×16], Q8 [×2], C23 [×2], C23 [×3], Dic3 [×2], C12 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6 [×7], C42, M4(2) [×2], D8 [×2], SD16 [×2], C2×D4 [×2], C2×D4 [×6], C4○D4 [×4], C3⋊C8 [×2], Dic6 [×2], C4×S3 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×6], C2×C12, C2×C12, C3×D4 [×8], C22×S3 [×2], C22×C6 [×2], C22×C6, C4.D4, C4≀C2 [×2], C41D4, C8⋊C22 [×2], 2+ 1+4, C4.Dic3 [×2], D4⋊S3 [×2], D4.S3 [×2], C4×C12, C4○D12 [×2], S3×D4 [×2], D42S3 [×2], C2×C3⋊D4 [×2], C6×D4 [×2], C6×D4 [×2], D44D4, C424S3 [×2], C12.D4, D126C22 [×2], C3×C41D4, D46D6, C428D6
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, D44D4, C232D6, C428D6

Character table of C428D6

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 8A 8B 12A 12B 12C 12D 12E 12F size 1 1 2 4 4 8 12 12 2 2 2 4 4 12 12 2 2 2 8 8 8 8 24 24 4 4 4 4 4 4 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 0 0 0 0 2 2 2 -2 0 0 -2 0 -2 2 -2 0 0 0 0 0 0 -2 0 0 0 2 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 0 0 0 2 -2 -2 0 0 0 0 2 2 2 -2 0 0 2 0 0 -2 0 0 0 -2 0 orthogonal lifted from D4 ρ11 2 2 -2 0 0 0 0 -2 2 2 -2 0 0 2 0 -2 2 -2 0 0 0 0 0 0 -2 0 0 0 2 0 orthogonal lifted from D4 ρ12 2 2 -2 0 0 0 -2 0 2 -2 2 0 0 0 2 -2 2 -2 0 0 0 0 0 0 2 0 0 0 -2 0 orthogonal lifted from D4 ρ13 2 2 2 2 2 2 0 0 -1 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ14 2 2 -2 0 0 0 2 0 2 -2 2 0 0 0 -2 -2 2 -2 0 0 0 0 0 0 2 0 0 0 -2 0 orthogonal lifted from D4 ρ15 2 2 2 2 2 -2 0 0 -1 2 2 -2 -2 0 0 -1 -1 -1 -1 1 1 -1 0 0 -1 1 1 1 -1 1 orthogonal lifted from D6 ρ16 2 2 2 -2 2 0 0 0 2 -2 -2 0 0 0 0 2 2 2 2 0 0 -2 0 0 -2 0 0 0 -2 0 orthogonal lifted from D4 ρ17 2 2 2 -2 -2 -2 0 0 -1 2 2 2 2 0 0 -1 -1 -1 1 1 1 1 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from D6 ρ18 2 2 2 -2 -2 2 0 0 -1 2 2 -2 -2 0 0 -1 -1 -1 1 -1 -1 1 0 0 -1 1 1 1 -1 1 orthogonal lifted from D6 ρ19 2 2 2 2 -2 0 0 0 -1 -2 -2 0 0 0 0 -1 -1 -1 1 -√-3 √-3 -1 0 0 1 -√-3 √-3 √-3 1 -√-3 complex lifted from C3⋊D4 ρ20 2 2 2 2 -2 0 0 0 -1 -2 -2 0 0 0 0 -1 -1 -1 1 √-3 -√-3 -1 0 0 1 √-3 -√-3 -√-3 1 √-3 complex lifted from C3⋊D4 ρ21 2 2 2 -2 2 0 0 0 -1 -2 -2 0 0 0 0 -1 -1 -1 -1 -√-3 √-3 1 0 0 1 √-3 -√-3 -√-3 1 √-3 complex lifted from C3⋊D4 ρ22 2 2 2 -2 2 0 0 0 -1 -2 -2 0 0 0 0 -1 -1 -1 -1 √-3 -√-3 1 0 0 1 -√-3 √-3 √-3 1 -√-3 complex lifted from C3⋊D4 ρ23 4 -4 0 0 0 0 0 0 4 0 0 2 -2 0 0 0 -4 0 0 0 0 0 0 0 0 2 -2 2 0 -2 orthogonal lifted from D4⋊4D4 ρ24 4 4 -4 0 0 0 0 0 -2 -4 4 0 0 0 0 2 -2 2 0 0 0 0 0 0 -2 0 0 0 2 0 orthogonal lifted from S3×D4 ρ25 4 4 -4 0 0 0 0 0 -2 4 -4 0 0 0 0 2 -2 2 0 0 0 0 0 0 2 0 0 0 -2 0 orthogonal lifted from S3×D4 ρ26 4 -4 0 0 0 0 0 0 4 0 0 -2 2 0 0 0 -4 0 0 0 0 0 0 0 0 -2 2 -2 0 2 orthogonal lifted from D4⋊4D4 ρ27 4 -4 0 0 0 0 0 0 -2 0 0 2 -2 0 0 2√-3 2 -2√-3 0 0 0 0 0 0 0 -1-√-3 1-√-3 -1+√-3 0 1+√-3 complex faithful ρ28 4 -4 0 0 0 0 0 0 -2 0 0 2 -2 0 0 -2√-3 2 2√-3 0 0 0 0 0 0 0 -1+√-3 1+√-3 -1-√-3 0 1-√-3 complex faithful ρ29 4 -4 0 0 0 0 0 0 -2 0 0 -2 2 0 0 2√-3 2 -2√-3 0 0 0 0 0 0 0 1+√-3 -1+√-3 1-√-3 0 -1-√-3 complex faithful ρ30 4 -4 0 0 0 0 0 0 -2 0 0 -2 2 0 0 -2√-3 2 2√-3 0 0 0 0 0 0 0 1-√-3 -1-√-3 1+√-3 0 -1+√-3 complex faithful

Permutation representations of C428D6
On 24 points - transitive group 24T355
Generators in S24
```(1 14 17 4)(2 5 18 15)(3 16 13 6)
(1 4 17 14)(2 15 18 5)(3 6 13 16)(7 10 23 20)(8 21 24 11)(9 12 19 22)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 9)(2 8)(3 7)(4 12)(5 11)(6 10)(13 23)(14 22)(15 21)(16 20)(17 19)(18 24)```

`G:=sub<Sym(24)| (1,14,17,4)(2,5,18,15)(3,16,13,6), (1,4,17,14)(2,15,18,5)(3,6,13,16)(7,10,23,20)(8,21,24,11)(9,12,19,22), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,9)(2,8)(3,7)(4,12)(5,11)(6,10)(13,23)(14,22)(15,21)(16,20)(17,19)(18,24)>;`

`G:=Group( (1,14,17,4)(2,5,18,15)(3,16,13,6), (1,4,17,14)(2,15,18,5)(3,6,13,16)(7,10,23,20)(8,21,24,11)(9,12,19,22), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,9)(2,8)(3,7)(4,12)(5,11)(6,10)(13,23)(14,22)(15,21)(16,20)(17,19)(18,24) );`

`G=PermutationGroup([(1,14,17,4),(2,5,18,15),(3,16,13,6)], [(1,4,17,14),(2,15,18,5),(3,6,13,16),(7,10,23,20),(8,21,24,11),(9,12,19,22)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,9),(2,8),(3,7),(4,12),(5,11),(6,10),(13,23),(14,22),(15,21),(16,20),(17,19),(18,24)])`

`G:=TransitiveGroup(24,355);`

Matrix representation of C428D6 in GL4(𝔽7) generated by

 2 3 0 1 1 0 1 5 4 4 4 6 0 0 0 6
,
 4 1 6 4 2 6 4 4 4 4 4 6 5 2 1 0
,
 0 4 1 0 1 4 3 6 6 6 6 2 0 0 0 4
,
 4 6 3 2 5 1 6 1 1 6 3 3 4 4 3 6
`G:=sub<GL(4,GF(7))| [2,1,4,0,3,0,4,0,0,1,4,0,1,5,6,6],[4,2,4,5,1,6,4,2,6,4,4,1,4,4,6,0],[0,1,6,0,4,4,6,0,1,3,6,0,0,6,2,4],[4,5,1,4,6,1,6,4,3,6,3,3,2,1,3,6] >;`

C428D6 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_8D_6`
`% in TeX`

`G:=Group("C4^2:8D6");`
`// GroupNames label`

`G:=SmallGroup(192,636);`
`// by ID`

`G=gap.SmallGroup(192,636);`
`# by ID`

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

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