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## G = C24⋊6D6order 192 = 26·3

### 1st semidirect product of C24 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — C24⋊6D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C2×C3⋊D4 — D4⋊6D6 — C24⋊6D6
 Lower central C3 — C6 — C22×C6 — C24⋊6D6
 Upper central C1 — C2 — C23 — C22≀C2

Generators and relations for C246D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=f2=1, ab=ba, eae-1=ac=ca, ad=da, faf=abcd, bc=cb, ebe-1=fbf=bd=db, fcf=cd=dc, ce=ec, de=ed, df=fd, fef=e-1 >

Subgroups: 688 in 198 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2 [×8], C3, C4 [×6], C22, C22 [×2], C22 [×18], S3 [×2], C6, C6 [×6], C2×C4, C2×C4 [×8], D4 [×15], Q8, C23 [×2], C23 [×8], Dic3 [×4], C12 [×2], D6 [×5], C2×C6, C2×C6 [×2], C2×C6 [×13], C22⋊C4, C22⋊C4 [×5], C2×D4, C2×D4 [×8], C4○D4 [×3], C24, Dic6, C4×S3 [×2], D12, C2×Dic3 [×2], C2×Dic3 [×3], C3⋊D4 [×10], C2×C12, C2×C12, C3×D4 [×4], C22×S3 [×2], C22×S3, C22×C6 [×2], C22×C6 [×5], C23⋊C4 [×3], C22≀C2, C22≀C2 [×2], 2+ 1+4, C6.D4 [×2], C6.D4 [×2], C3×C22⋊C4, C3×C22⋊C4, C4○D12, S3×D4 [×2], D42S3 [×2], C2×C3⋊D4 [×2], C2×C3⋊D4 [×3], C6×D4, C6×D4, C23×C6, C2≀C22, C23.6D6 [×2], C23.7D6, C244S3 [×2], C3×C22≀C2, D46D6, C246D6
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, C2≀C22, C232D6, C246D6

Character table of C246D6

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 12A 12B 12C size 1 1 2 2 2 4 4 4 12 12 2 4 8 12 12 24 24 2 2 2 4 4 4 4 4 4 8 8 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 0 0 -2 2 0 0 2 0 0 0 2 -2 -2 2 0 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -2 -2 0 0 -2 0 0 2 2 0 0 0 0 0 2 2 2 -2 0 0 -2 0 0 -2 0 2 0 orthogonal lifted from D4 ρ11 2 2 -2 -2 2 0 0 0 2 0 2 0 0 0 -2 0 0 2 -2 -2 -2 0 0 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 2 -2 0 0 0 0 2 2 0 0 -2 0 0 0 2 -2 -2 2 0 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 2 -2 -2 2 0 0 -1 2 -2 0 0 0 0 -1 -1 -1 -1 1 1 -1 1 1 -1 1 -1 1 orthogonal lifted from D6 ρ14 2 2 2 2 2 2 2 -2 0 0 -1 -2 -2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ15 2 2 2 2 2 -2 -2 -2 0 0 -1 -2 2 0 0 0 0 -1 -1 -1 -1 1 1 -1 1 1 1 -1 1 -1 orthogonal lifted from D6 ρ16 2 2 2 2 2 2 2 2 0 0 -1 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ17 2 2 2 -2 -2 0 0 2 0 0 2 -2 0 0 0 0 0 2 2 2 -2 0 0 -2 0 0 2 0 -2 0 orthogonal lifted from D4 ρ18 2 2 -2 -2 2 0 0 0 -2 0 2 0 0 0 2 0 0 2 -2 -2 -2 0 0 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 -2 -2 0 0 -2 0 0 -1 2 0 0 0 0 0 -1 -1 -1 1 -√-3 √-3 1 -√-3 √-3 1 -√-3 -1 √-3 complex lifted from C3⋊D4 ρ20 2 2 2 -2 -2 0 0 -2 0 0 -1 2 0 0 0 0 0 -1 -1 -1 1 √-3 -√-3 1 √-3 -√-3 1 √-3 -1 -√-3 complex lifted from C3⋊D4 ρ21 2 2 2 -2 -2 0 0 2 0 0 -1 -2 0 0 0 0 0 -1 -1 -1 1 -√-3 √-3 1 -√-3 √-3 -1 √-3 1 -√-3 complex lifted from C3⋊D4 ρ22 2 2 2 -2 -2 0 0 2 0 0 -1 -2 0 0 0 0 0 -1 -1 -1 1 √-3 -√-3 1 √-3 -√-3 -1 -√-3 1 √-3 complex lifted from C3⋊D4 ρ23 4 4 -4 -4 4 0 0 0 0 0 -2 0 0 0 0 0 0 -2 2 2 2 0 0 -2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 -4 0 0 0 2 -2 0 0 0 4 0 0 0 0 0 0 -4 0 0 0 -2 -2 0 2 2 0 0 0 0 orthogonal lifted from C2≀C22 ρ25 4 4 -4 4 -4 0 0 0 0 0 -2 0 0 0 0 0 0 -2 2 2 -2 0 0 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ26 4 -4 0 0 0 -2 2 0 0 0 4 0 0 0 0 0 0 -4 0 0 0 2 2 0 -2 -2 0 0 0 0 orthogonal lifted from C2≀C22 ρ27 4 -4 0 0 0 -2 2 0 0 0 -2 0 0 0 0 0 0 2 -2√-3 2√-3 0 -1-√-3 -1+√-3 0 1+√-3 1-√-3 0 0 0 0 complex faithful ρ28 4 -4 0 0 0 -2 2 0 0 0 -2 0 0 0 0 0 0 2 2√-3 -2√-3 0 -1+√-3 -1-√-3 0 1-√-3 1+√-3 0 0 0 0 complex faithful ρ29 4 -4 0 0 0 2 -2 0 0 0 -2 0 0 0 0 0 0 2 -2√-3 2√-3 0 1+√-3 1-√-3 0 -1-√-3 -1+√-3 0 0 0 0 complex faithful ρ30 4 -4 0 0 0 2 -2 0 0 0 -2 0 0 0 0 0 0 2 2√-3 -2√-3 0 1-√-3 1+√-3 0 -1+√-3 -1-√-3 0 0 0 0 complex faithful

Permutation representations of C246D6
On 24 points - transitive group 24T287
Generators in S24
```(8 19)(10 21)(12 23)
(2 13)(4 15)(6 17)(8 19)(10 21)(12 23)
(7 24)(8 19)(9 20)(10 21)(11 22)(12 23)
(1 18)(2 13)(3 14)(4 15)(5 16)(6 17)(7 24)(8 19)(9 20)(10 21)(11 22)(12 23)
(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 23)(2 22)(3 21)(4 20)(5 19)(6 24)(7 17)(8 16)(9 15)(10 14)(11 13)(12 18)```

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

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

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

`G:=TransitiveGroup(24,287);`

On 24 points - transitive group 24T357
Generators in S24
```(7 14)(8 18)(9 16)(10 17)(11 15)(12 13)
(1 20)(2 24)(3 22)(4 23)(5 21)(6 19)(7 14)(8 18)(9 16)(10 17)(11 15)(12 13)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 23)(14 22)(15 21)(16 20)(17 19)(18 24)```

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

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

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

`G:=TransitiveGroup(24,357);`

On 24 points - transitive group 24T364
Generators in S24
```(1 19)(2 16)(3 21)(4 18)(5 23)(6 14)(7 20)(8 17)(9 22)(10 13)(11 24)(12 15)
(1 15)(2 7)(3 17)(4 9)(5 13)(6 11)(8 21)(10 23)(12 19)(14 24)(16 20)(18 22)
(1 12)(2 7)(3 8)(4 9)(5 10)(6 11)(13 23)(14 24)(15 19)(16 20)(17 21)(18 22)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 16)(8 17)(9 18)(10 13)(11 14)(12 15)
(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 3)(4 6)(7 16)(8 15)(9 14)(10 13)(11 18)(12 17)(19 21)(22 24)```

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

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

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

`G:=TransitiveGroup(24,364);`

Matrix representation of C246D6 in GL4(𝔽7) generated by

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

C246D6 in GAP, Magma, Sage, TeX

`C_2^4\rtimes_6D_6`
`% in TeX`

`G:=Group("C2^4:6D6");`
`// GroupNames label`

`G:=SmallGroup(192,591);`
`// by ID`

`G=gap.SmallGroup(192,591);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,570,1684,6278]);`
`// Polycyclic`

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

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