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G = C32⋊D8⋊C2order 288 = 25·32

3rd semidirect product of C32⋊D8 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C32⋊D8⋊C2
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — D6⋊S3 — C32⋊D8 — C32⋊D8⋊C2
 Lower central C32 — C3×C6 — C3⋊Dic3 — C32⋊D8⋊C2
 Upper central C1 — C2 — C4

Generators and relations for C32⋊D8⋊C2
G = < a,b,c,d,e | a3=b3=c8=d2=e2=1, ab=ba, cac-1=b, dad=eae=cbc-1=a-1, bd=db, ebe=b-1, dcd=c-1, ece=c5, ede=c4d >

Subgroups: 688 in 115 conjugacy classes, 23 normal (15 characteristic)
C1, C2, C2 [×4], C3 [×2], C4, C4 [×2], C22 [×6], S3 [×6], C6 [×5], C8 [×2], C2×C4 [×2], D4 [×5], Q8, C23, C32, Dic3 [×3], C12 [×3], D6 [×9], C2×C6 [×3], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3 [×3], C3⋊S3, C3×C6, Dic6, C4×S3 [×3], D12 [×2], C3⋊D4 [×4], C2×C12, C3×D4, C22×S3 [×2], C8⋊C22, C3×Dic3, C3⋊Dic3, C3×C12, S32 [×2], S3×C6 [×3], C2×C3⋊S3, C4○D12, S3×D4, C322C8 [×2], D6⋊S3, D6⋊S3 [×2], C3⋊D12, C322Q8, S3×C12, C3×D12, C4×C3⋊S3, C2×S32, C32⋊D8 [×2], C322SD16 [×2], C32⋊M4(2), D6.D6, D6⋊D6, C32⋊D8⋊C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, C8⋊C22, S3≀C2, C2×S3≀C2, C32⋊D8⋊C2

Character table of C32⋊D8⋊C2

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 6A 6B 6C 6D 6E 6F 8A 8B 12A 12B 12C 12D 12E size 1 1 12 12 12 18 4 4 2 12 18 4 4 12 12 24 24 36 36 4 4 8 12 12 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 0 0 0 2 2 2 -2 0 -2 2 2 0 0 0 0 0 0 -2 -2 -2 0 0 orthogonal lifted from D4 ρ10 2 2 0 0 0 -2 2 2 2 0 -2 2 2 0 0 0 0 0 0 2 2 2 0 0 orthogonal lifted from D4 ρ11 4 4 -2 -2 0 0 -2 1 4 0 0 1 -2 0 0 1 1 0 0 -2 -2 1 0 0 orthogonal lifted from S3≀C2 ρ12 4 4 2 -2 0 0 -2 1 -4 0 0 1 -2 0 0 -1 1 0 0 2 2 -1 0 0 orthogonal lifted from C2×S3≀C2 ρ13 4 -4 0 0 0 0 4 4 0 0 0 -4 -4 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ14 4 4 2 2 0 0 -2 1 4 0 0 1 -2 0 0 -1 -1 0 0 -2 -2 1 0 0 orthogonal lifted from S3≀C2 ρ15 4 4 0 0 -2 0 1 -2 -4 2 0 -2 1 1 1 0 0 0 0 -1 -1 2 -1 -1 orthogonal lifted from C2×S3≀C2 ρ16 4 4 0 0 2 0 1 -2 4 2 0 -2 1 -1 -1 0 0 0 0 1 1 -2 -1 -1 orthogonal lifted from S3≀C2 ρ17 4 4 -2 2 0 0 -2 1 -4 0 0 1 -2 0 0 1 -1 0 0 2 2 -1 0 0 orthogonal lifted from C2×S3≀C2 ρ18 4 4 0 0 -2 0 1 -2 4 -2 0 -2 1 1 1 0 0 0 0 1 1 -2 1 1 orthogonal lifted from S3≀C2 ρ19 4 4 0 0 2 0 1 -2 -4 -2 0 -2 1 -1 -1 0 0 0 0 -1 -1 2 1 1 orthogonal lifted from C2×S3≀C2 ρ20 4 -4 0 0 0 0 1 -2 0 0 0 2 -1 -√-3 √-3 0 0 0 0 -3i 3i 0 -√3 √3 complex faithful ρ21 4 -4 0 0 0 0 1 -2 0 0 0 2 -1 -√-3 √-3 0 0 0 0 3i -3i 0 √3 -√3 complex faithful ρ22 4 -4 0 0 0 0 1 -2 0 0 0 2 -1 √-3 -√-3 0 0 0 0 -3i 3i 0 √3 -√3 complex faithful ρ23 4 -4 0 0 0 0 1 -2 0 0 0 2 -1 √-3 -√-3 0 0 0 0 3i -3i 0 -√3 √3 complex faithful ρ24 8 -8 0 0 0 0 -4 2 0 0 0 -2 4 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C32⋊D8⋊C2
On 24 points - transitive group 24T658
Generators in S24
```(1 11 21)(2 12 22)(3 23 13)(4 24 14)(5 15 17)(6 16 18)(7 19 9)(8 20 10)
(1 11 21)(2 22 12)(3 23 13)(4 14 24)(5 15 17)(6 18 16)(7 19 9)(8 10 20)
(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 7)(3 6)(4 5)(9 12)(10 11)(13 16)(14 15)(17 24)(18 23)(19 22)(20 21)
(1 5)(3 7)(9 23)(10 20)(11 17)(12 22)(13 19)(14 24)(15 21)(16 18)```

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

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

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

`G:=TransitiveGroup(24,658);`

Matrix representation of C32⋊D8⋊C2 in GL8(ℤ)

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

`G:=sub<GL(8,Integers())| [0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1] >;`

C32⋊D8⋊C2 in GAP, Magma, Sage, TeX

`C_3^2\rtimes D_8\rtimes C_2`
`% in TeX`

`G:=Group("C3^2:D8:C2");`
`// GroupNames label`

`G:=SmallGroup(288,872);`
`// by ID`

`G=gap.SmallGroup(288,872);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,219,100,675,346,80,2693,2028,362,797,1203]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^3=b^3=c^8=d^2=e^2=1,a*b=b*a,c*a*c^-1=b,d*a*d=e*a*e=c*b*c^-1=a^-1,b*d=d*b,e*b*e=b^-1,d*c*d=c^-1,e*c*e=c^5,e*d*e=c^4*d>;`
`// generators/relations`

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