Copied to
clipboard

## G = C62⋊D4order 288 = 25·32

### 2nd semidirect product of C62 and D4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C3⋊S3 — C62⋊D4
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×S32 — C2×S3≀C2 — C62⋊D4
 Lower central C32 — C2×C3⋊S3 — C62⋊D4
 Upper central C1 — C2 — C22

Generators and relations for C62⋊D4
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a3b4, dad=a-1b3, cbc-1=a2b3, bd=db, dcd=c-1 >

Subgroups: 920 in 148 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2 [×6], C3 [×2], C4 [×5], C22, C22 [×10], S3 [×8], C6 [×6], C2×C4 [×6], D4 [×6], C23 [×3], C32, Dic3 [×2], C12 [×2], D6 [×14], C2×C6 [×4], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3 [×2], C3⋊S3 [×2], C3⋊S3, C3×C6, C3×C6, C4×S3 [×2], D12 [×2], C3⋊D4 [×4], C3×D4 [×2], C22×S3 [×4], C4⋊D4, C3×Dic3 [×2], C32⋊C4 [×2], C32⋊C4, S32 [×4], S3×C6 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, S3×D4 [×2], C6.D6 [×2], C3⋊D12 [×2], C3×C3⋊D4 [×2], S3≀C2 [×2], C2×C32⋊C4 [×2], C2×C32⋊C4 [×2], C2×S32 [×2], C22×C3⋊S3, S32⋊C4 [×2], C3⋊S3.Q8, Dic3⋊D6 [×2], C2×S3≀C2, C22×C32⋊C4, C62⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, C2×D4 [×2], C4○D4, C4⋊D4, S3≀C2, C2×S3≀C2, C62⋊D4

Character table of C62⋊D4

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 12A 12B size 1 1 2 9 9 12 12 18 4 4 12 12 18 18 18 18 4 4 8 8 24 24 24 24 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 2 -2 -2 0 0 -2 2 2 0 0 0 0 0 0 2 2 2 2 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 0 -2 2 0 0 0 2 2 0 0 0 -2 0 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 0 -2 2 0 0 0 2 2 0 0 0 2 0 -2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 -2 -2 0 0 2 2 2 0 0 0 0 0 0 2 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ13 2 -2 0 2 -2 0 0 0 2 2 0 0 2i 0 -2i 0 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ14 2 -2 0 2 -2 0 0 0 2 2 0 0 -2i 0 2i 0 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ15 4 4 4 0 0 0 2 0 1 -2 0 2 0 0 0 0 -2 1 -2 1 -1 0 -1 0 orthogonal lifted from S3≀C2 ρ16 4 4 -4 0 0 0 2 0 1 -2 0 -2 0 0 0 0 -2 1 2 -1 -1 0 1 0 orthogonal lifted from C2×S3≀C2 ρ17 4 4 4 0 0 -2 0 0 -2 1 -2 0 0 0 0 0 1 -2 1 -2 0 1 0 1 orthogonal lifted from S3≀C2 ρ18 4 4 4 0 0 2 0 0 -2 1 2 0 0 0 0 0 1 -2 1 -2 0 -1 0 -1 orthogonal lifted from S3≀C2 ρ19 4 4 -4 0 0 -2 0 0 -2 1 2 0 0 0 0 0 1 -2 -1 2 0 1 0 -1 orthogonal lifted from C2×S3≀C2 ρ20 4 4 -4 0 0 2 0 0 -2 1 -2 0 0 0 0 0 1 -2 -1 2 0 -1 0 1 orthogonal lifted from C2×S3≀C2 ρ21 4 4 4 0 0 0 -2 0 1 -2 0 -2 0 0 0 0 -2 1 -2 1 1 0 1 0 orthogonal lifted from S3≀C2 ρ22 4 4 -4 0 0 0 -2 0 1 -2 0 2 0 0 0 0 -2 1 2 -1 1 0 -1 0 orthogonal lifted from C2×S3≀C2 ρ23 8 -8 0 0 0 0 0 0 -4 2 0 0 0 0 0 0 -2 4 0 0 0 0 0 0 orthogonal faithful ρ24 8 -8 0 0 0 0 0 0 2 -4 0 0 0 0 0 0 4 -2 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C62⋊D4
On 24 points - transitive group 24T592
Generators in S24
```(7 8)(9 10)(11 12)(13 14 15)(16 17 18)(19 20 21 22 23 24)
(1 5 3 4 2 6)(7 9 12 8 10 11)(13 18)(14 16)(15 17)(19 22)(20 23)(21 24)
(1 15)(2 13 3 14)(4 17)(5 18 6 16)(7 20 12 22)(8 23 11 19)(9 21)(10 24)
(1 9)(2 11)(3 8)(4 10)(5 12)(6 7)(13 23)(14 19)(15 21)(16 22)(17 24)(18 20)```

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

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

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

`G:=TransitiveGroup(24,592);`

On 24 points - transitive group 24T593
Generators in S24
```(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 8 2 9 3 7)(4 12 6 11 5 10)(13 14 15 16 17 18)(19 24 23 22 21 20)
(1 11 9 4)(2 10 8 5)(3 12 7 6)(13 20 18 19)(14 21 17 24)(15 22 16 23)
(1 14)(2 16)(3 18)(4 21)(5 23)(6 19)(7 13)(8 15)(9 17)(10 22)(11 24)(12 20)```

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

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

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

`G:=TransitiveGroup(24,593);`

On 24 points - transitive group 24T640
Generators in S24
```(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 7 9 3 6)(2 11 8 10 4 5)(13 24)(14 19)(15 20)(16 21)(17 22)(18 23)
(1 17 9 22)(2 14 10 19)(3 15 6 24)(4 18 5 21)(7 13 12 20)(8 16 11 23)
(2 10)(4 11)(5 8)(13 20)(15 24)(16 18)(17 22)(21 23)```

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

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

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

`G:=TransitiveGroup(24,640);`

On 24 points - transitive group 24T647
Generators in S24
```(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 6 11 3 7 9)(2 5 12 4 8 10)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 18 11 14)(2 15 12 17)(3 24 9 20)(4 21 10 23)(5 19)(6 22)(7 16)(8 13)
(1 3)(6 7)(9 11)(14 24)(15 17)(16 22)(18 20)(21 23)```

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

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

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

`G:=TransitiveGroup(24,647);`

Matrix representation of C62⋊D4 in GL6(ℤ)

 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 0 0 0 0 0 0 -1 1 0 0 0 0 -1 0
,
 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0
,
 0 -1 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

`G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,-1,0,0,0,0,-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] >;`

C62⋊D4 in GAP, Magma, Sage, TeX

`C_6^2\rtimes D_4`
`% in TeX`

`G:=Group("C6^2:D4");`
`// GroupNames label`

`G:=SmallGroup(288,890);`
`// by ID`

`G=gap.SmallGroup(288,890);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,422,219,2693,2028,362,797,1203]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^3*b^4,d*a*d=a^-1*b^3,c*b*c^-1=a^2*b^3,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

Export

׿
×
𝔽