Copied to
clipboard

## G = C62.9D4order 288 = 25·32

### 9th non-split extension by C62 of D4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C3⋊S3 — C62.9D4
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×S32 — S32⋊C4 — C62.9D4
 Lower central C32 — C2×C3⋊S3 — C62.9D4
 Upper central C1 — C2 — C22

Generators and relations for C62.9D4
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a3b, dad=a-1b3, cbc-1=a2b3, bd=db, dcd=b3c-1 >

Subgroups: 744 in 132 conjugacy classes, 25 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, D4, C23, C32, Dic3, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22.D4, C3×Dic3, C32⋊C4, S32, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C2×C4, S3×D4, C6.D6, C6.D6, C6.D6, C3⋊D12, C6×Dic3, C3×C3⋊D4, C2×C32⋊C4, C2×S32, C22×C3⋊S3, S32⋊C4, C3⋊S3.Q8, C62⋊C4, C2×C6.D6, Dic3⋊D6, C62.9D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22.D4, S3≀C2, C2×S3≀C2, C62.9D4

Character table of C62.9D4

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 4A 4B 4C 4D 4E 4F 4G 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E size 1 1 2 9 9 12 18 4 4 6 6 6 6 12 36 36 4 4 4 4 8 24 12 12 12 12 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 2 -2 -2 0 -2 2 2 0 0 0 0 0 0 0 2 2 2 2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 -2 -2 0 2 2 2 0 0 0 0 0 0 0 2 -2 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 0 -2 2 0 0 2 2 0 2i -2i 0 0 0 0 -2 0 0 -2 0 0 0 -2i 2i 0 0 complex lifted from C4○D4 ρ12 2 -2 0 -2 2 0 0 2 2 0 -2i 2i 0 0 0 0 -2 0 0 -2 0 0 0 2i -2i 0 0 complex lifted from C4○D4 ρ13 2 -2 0 2 -2 0 0 2 2 2i 0 0 -2i 0 0 0 -2 0 0 -2 0 0 2i 0 0 -2i 0 complex lifted from C4○D4 ρ14 2 -2 0 2 -2 0 0 2 2 -2i 0 0 2i 0 0 0 -2 0 0 -2 0 0 -2i 0 0 2i 0 complex lifted from C4○D4 ρ15 4 4 -4 0 0 -2 0 -2 1 0 0 0 0 2 0 0 1 2 2 -2 -1 1 0 0 0 0 -1 orthogonal lifted from C2×S3≀C2 ρ16 4 4 -4 0 0 0 0 1 -2 2 -2 -2 2 0 0 0 -2 -1 -1 1 2 0 -1 1 1 -1 0 orthogonal lifted from C2×S3≀C2 ρ17 4 4 -4 0 0 0 0 1 -2 -2 2 2 -2 0 0 0 -2 -1 -1 1 2 0 1 -1 -1 1 0 orthogonal lifted from C2×S3≀C2 ρ18 4 4 4 0 0 2 0 -2 1 0 0 0 0 2 0 0 1 -2 -2 -2 1 -1 0 0 0 0 -1 orthogonal lifted from S3≀C2 ρ19 4 4 4 0 0 0 0 1 -2 2 2 2 2 0 0 0 -2 1 1 1 -2 0 -1 -1 -1 -1 0 orthogonal lifted from S3≀C2 ρ20 4 4 4 0 0 0 0 1 -2 -2 -2 -2 -2 0 0 0 -2 1 1 1 -2 0 1 1 1 1 0 orthogonal lifted from S3≀C2 ρ21 4 4 -4 0 0 2 0 -2 1 0 0 0 0 -2 0 0 1 2 2 -2 -1 -1 0 0 0 0 1 orthogonal lifted from C2×S3≀C2 ρ22 4 4 4 0 0 -2 0 -2 1 0 0 0 0 -2 0 0 1 -2 -2 -2 1 1 0 0 0 0 1 orthogonal lifted from S3≀C2 ρ23 4 -4 0 0 0 0 0 1 -2 -2i 2i -2i 2i 0 0 0 2 3 -3 -1 0 0 i i -i -i 0 complex faithful ρ24 4 -4 0 0 0 0 0 1 -2 -2i -2i 2i 2i 0 0 0 2 -3 3 -1 0 0 i -i i -i 0 complex faithful ρ25 4 -4 0 0 0 0 0 1 -2 2i 2i -2i -2i 0 0 0 2 -3 3 -1 0 0 -i i -i i 0 complex faithful ρ26 4 -4 0 0 0 0 0 1 -2 2i -2i 2i -2i 0 0 0 2 3 -3 -1 0 0 -i -i i i 0 complex faithful ρ27 8 -8 0 0 0 0 0 -4 2 0 0 0 0 0 0 0 -2 0 0 4 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C62.9D4
On 24 points - transitive group 24T595
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 10 11 8 9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 22 4 19)(2 20 6 21)(3 24 5 23)(7 13 10 17)(8 16 9 14)(11 15 12 18)
(1 11)(2 7)(3 9)(4 12)(5 8)(6 10)(13 24)(14 20)(15 22)(16 21)(17 23)(18 19)```

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

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

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

`G:=TransitiveGroup(24,595);`

On 24 points - transitive group 24T599
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 5 3 4 2 6)(7 11 9 10 8 12)(13 18 17 16 15 14)(19 24 23 22 21 20)
(1 22 4 19)(2 20 6 21)(3 24 5 23)(7 13 10 16)(8 17 12 18)(9 15 11 14)
(1 13)(2 15)(3 17)(4 16)(5 18)(6 14)(7 19)(8 21)(9 23)(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), (1,5,3,4,2,6)(7,11,9,10,8,12)(13,18,17,16,15,14)(19,24,23,22,21,20), (1,22,4,19)(2,20,6,21)(3,24,5,23)(7,13,10,16)(8,17,12,18)(9,15,11,14), (1,13)(2,15)(3,17)(4,16)(5,18)(6,14)(7,19)(8,21)(9,23)(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), (1,5,3,4,2,6)(7,11,9,10,8,12)(13,18,17,16,15,14)(19,24,23,22,21,20), (1,22,4,19)(2,20,6,21)(3,24,5,23)(7,13,10,16)(8,17,12,18)(9,15,11,14), (1,13)(2,15)(3,17)(4,16)(5,18)(6,14)(7,19)(8,21)(9,23)(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)], [(1,5,3,4,2,6),(7,11,9,10,8,12),(13,18,17,16,15,14),(19,24,23,22,21,20)], [(1,22,4,19),(2,20,6,21),(3,24,5,23),(7,13,10,16),(8,17,12,18),(9,15,11,14)], [(1,13),(2,15),(3,17),(4,16),(5,18),(6,14),(7,19),(8,21),(9,23),(10,22),(11,24),(12,20)]])`

`G:=TransitiveGroup(24,599);`

On 24 points - transitive group 24T646
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 11 9 4 5 7)(2 12 10 3 6 8)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)
(1 19 4 17)(2 14 3 22)(5 23 7 13)(6 18 8 24)(9 21 11 15)(10 16 12 20)
(1 4)(5 11)(7 9)(13 23)(15 21)(16 18)(17 19)(20 24)```

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

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

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

`G:=TransitiveGroup(24,646);`

Matrix representation of C62.9D4 in GL4(𝔽5) generated by

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

C62.9D4 in GAP, Magma, Sage, TeX

`C_6^2._9D_4`
`% in TeX`

`G:=Group("C6^2.9D4");`
`// GroupNames label`

`G:=SmallGroup(288,881);`
`// by ID`

`G=gap.SmallGroup(288,881);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,120,422,2693,2028,362,797,1203]);`
`// 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,d*a*d=a^-1*b^3,c*b*c^-1=a^2*b^3,b*d=d*b,d*c*d=b^3*c^-1>;`
`// generators/relations`

Export

׿
×
𝔽