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G = C24.39D4order 128 = 27

39th non-split extension by C24 of D4 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C24.39D4
 Chief series C1 — C2 — C22 — C23 — C2×D4 — C22×D4 — C22.29C24 — C24.39D4
 Lower central C1 — C2 — C22 — C2×C4 — C24.39D4
 Upper central C1 — C2 — C23 — C22×D4 — C24.39D4
 Jennings C1 — C2 — C22 — C2×D4 — C24.39D4

Generators and relations for C24.39D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=b, ab=ba, ac=ca, eae-1=ad=da, af=fa, ebe-1=bc=cb, bd=db, bf=fb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=be-1 >

Subgroups: 428 in 143 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2 [×8], C4 [×9], C22, C22 [×2], C22 [×18], C2×C4 [×2], C2×C4 [×15], D4 [×11], Q8, C23, C23 [×4], C23 [×7], C42 [×2], C22⋊C4 [×11], C4⋊C4, C22×C4, C22×C4 [×3], C2×D4 [×3], C2×D4 [×2], C2×D4 [×7], C2×Q8, C4○D4 [×2], C24 [×2], C23⋊C4 [×4], C23⋊C4 [×2], C2×C22⋊C4 [×2], C42⋊C2, C22≀C2 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4 [×2], C22×D4, C2×C4○D4, C42⋊C4 [×2], C423C4 [×2], C2×C23⋊C4 [×2], C22.29C24, C24.39D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C23⋊C4 [×2], C2×C22⋊C4, C2×C23⋊C4, C24.39D4

Character table of C24.39D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M size 1 1 2 2 2 4 4 4 4 8 4 4 8 8 8 8 8 8 8 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 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 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 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 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 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 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 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 linear of order 2 ρ9 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -i i -1 1 i i -i i -i -1 -i linear of order 4 ρ10 1 1 1 1 1 -1 -1 -1 -1 1 1 1 i -i -1 -1 -i i i i -i 1 -i linear of order 4 ρ11 1 1 -1 1 -1 1 1 -1 -1 -1 1 -1 -i -i 1 -1 i i i -i i 1 -i linear of order 4 ρ12 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 i i 1 1 -i i -i -i i -1 -i linear of order 4 ρ13 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -i i -1 -1 i -i -i -i i 1 i linear of order 4 ρ14 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 i -i -1 1 -i -i i -i i -1 i linear of order 4 ρ15 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -i -i 1 1 i -i i i -i -1 i linear of order 4 ρ16 1 1 -1 1 -1 1 1 -1 -1 -1 1 -1 i i 1 -1 -i -i -i i -i 1 i linear of order 4 ρ17 2 2 2 2 2 -2 2 -2 2 0 -2 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 2 -2 -2 2 2 -2 0 -2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 2 2 -2 2 -2 0 -2 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 2 -2 2 -2 -2 2 0 -2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ22 4 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ23 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C24.39D4
On 16 points - transitive group 16T234
Generators in S16
```(1 8)(2 11)(3 6)(4 9)(5 15)(7 13)(10 14)(12 16)
(1 8)(2 5)(3 12)(4 9)(6 16)(7 13)(10 14)(11 15)
(2 15)(4 13)(5 11)(7 9)
(1 14)(2 15)(3 16)(4 13)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 9 8 4)(2 3 5 12)(6 15 16 11)(7 10 13 14)```

`G:=sub<Sym(16)| (1,8)(2,11)(3,6)(4,9)(5,15)(7,13)(10,14)(12,16), (1,8)(2,5)(3,12)(4,9)(6,16)(7,13)(10,14)(11,15), (2,15)(4,13)(5,11)(7,9), (1,14)(2,15)(3,16)(4,13)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,9,8,4)(2,3,5,12)(6,15,16,11)(7,10,13,14)>;`

`G:=Group( (1,8)(2,11)(3,6)(4,9)(5,15)(7,13)(10,14)(12,16), (1,8)(2,5)(3,12)(4,9)(6,16)(7,13)(10,14)(11,15), (2,15)(4,13)(5,11)(7,9), (1,14)(2,15)(3,16)(4,13)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,9,8,4)(2,3,5,12)(6,15,16,11)(7,10,13,14) );`

`G=PermutationGroup([(1,8),(2,11),(3,6),(4,9),(5,15),(7,13),(10,14),(12,16)], [(1,8),(2,5),(3,12),(4,9),(6,16),(7,13),(10,14),(11,15)], [(2,15),(4,13),(5,11),(7,9)], [(1,14),(2,15),(3,16),(4,13),(5,11),(6,12),(7,9),(8,10)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,9,8,4),(2,3,5,12),(6,15,16,11),(7,10,13,14)])`

`G:=TransitiveGroup(16,234);`

On 16 points - transitive group 16T250
Generators in S16
```(1 4)(2 8)(3 5)(6 7)(9 15)(10 14)(11 13)(12 16)
(1 9)(2 10)(3 16)(4 15)(5 12)(6 11)(7 13)(8 14)
(2 5)(3 8)(10 12)(14 16)
(1 6)(2 5)(3 8)(4 7)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 10 9 2)(3 7 16 13)(4 14 15 8)(5 6 12 11)```

`G:=sub<Sym(16)| (1,4)(2,8)(3,5)(6,7)(9,15)(10,14)(11,13)(12,16), (1,9)(2,10)(3,16)(4,15)(5,12)(6,11)(7,13)(8,14), (2,5)(3,8)(10,12)(14,16), (1,6)(2,5)(3,8)(4,7)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,10,9,2)(3,7,16,13)(4,14,15,8)(5,6,12,11)>;`

`G:=Group( (1,4)(2,8)(3,5)(6,7)(9,15)(10,14)(11,13)(12,16), (1,9)(2,10)(3,16)(4,15)(5,12)(6,11)(7,13)(8,14), (2,5)(3,8)(10,12)(14,16), (1,6)(2,5)(3,8)(4,7)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,10,9,2)(3,7,16,13)(4,14,15,8)(5,6,12,11) );`

`G=PermutationGroup([(1,4),(2,8),(3,5),(6,7),(9,15),(10,14),(11,13),(12,16)], [(1,9),(2,10),(3,16),(4,15),(5,12),(6,11),(7,13),(8,14)], [(2,5),(3,8),(10,12),(14,16)], [(1,6),(2,5),(3,8),(4,7),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,10,9,2),(3,7,16,13),(4,14,15,8),(5,6,12,11)])`

`G:=TransitiveGroup(16,250);`

On 16 points - transitive group 16T297
Generators in S16
```(1 4)(2 3)(5 6)(7 8)(9 13)(10 12)(11 15)(14 16)
(1 5)(2 7)(3 8)(4 6)(9 10)(11 14)(12 13)(15 16)
(1 3)(2 4)(5 8)(6 7)(9 13)(10 12)(11 15)(14 16)
(1 2)(3 4)(5 7)(6 8)(9 15)(10 16)(11 13)(12 14)
(3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 16 5 15)(2 10 7 9)(3 12 8 13)(4 14 6 11)```

`G:=sub<Sym(16)| (1,4)(2,3)(5,6)(7,8)(9,13)(10,12)(11,15)(14,16), (1,5)(2,7)(3,8)(4,6)(9,10)(11,14)(12,13)(15,16), (1,3)(2,4)(5,8)(6,7)(9,13)(10,12)(11,15)(14,16), (1,2)(3,4)(5,7)(6,8)(9,15)(10,16)(11,13)(12,14), (3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16,5,15)(2,10,7,9)(3,12,8,13)(4,14,6,11)>;`

`G:=Group( (1,4)(2,3)(5,6)(7,8)(9,13)(10,12)(11,15)(14,16), (1,5)(2,7)(3,8)(4,6)(9,10)(11,14)(12,13)(15,16), (1,3)(2,4)(5,8)(6,7)(9,13)(10,12)(11,15)(14,16), (1,2)(3,4)(5,7)(6,8)(9,15)(10,16)(11,13)(12,14), (3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16,5,15)(2,10,7,9)(3,12,8,13)(4,14,6,11) );`

`G=PermutationGroup([(1,4),(2,3),(5,6),(7,8),(9,13),(10,12),(11,15),(14,16)], [(1,5),(2,7),(3,8),(4,6),(9,10),(11,14),(12,13),(15,16)], [(1,3),(2,4),(5,8),(6,7),(9,13),(10,12),(11,15),(14,16)], [(1,2),(3,4),(5,7),(6,8),(9,15),(10,16),(11,13),(12,14)], [(3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,16,5,15),(2,10,7,9),(3,12,8,13),(4,14,6,11)])`

`G:=TransitiveGroup(16,297);`

On 16 points - transitive group 16T310
Generators in S16
```(1 5)(4 8)(9 11)(14 16)
(1 11)(2 12)(3 15)(4 16)(5 9)(6 10)(7 13)(8 14)
(2 6)(4 8)(10 12)(14 16)
(1 5)(2 6)(3 7)(4 8)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 14 11 8)(2 7 12 13)(3 10 15 6)(4 5 16 9)```

`G:=sub<Sym(16)| (1,5)(4,8)(9,11)(14,16), (1,11)(2,12)(3,15)(4,16)(5,9)(6,10)(7,13)(8,14), (2,6)(4,8)(10,12)(14,16), (1,5)(2,6)(3,7)(4,8)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,14,11,8)(2,7,12,13)(3,10,15,6)(4,5,16,9)>;`

`G:=Group( (1,5)(4,8)(9,11)(14,16), (1,11)(2,12)(3,15)(4,16)(5,9)(6,10)(7,13)(8,14), (2,6)(4,8)(10,12)(14,16), (1,5)(2,6)(3,7)(4,8)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,14,11,8)(2,7,12,13)(3,10,15,6)(4,5,16,9) );`

`G=PermutationGroup([(1,5),(4,8),(9,11),(14,16)], [(1,11),(2,12),(3,15),(4,16),(5,9),(6,10),(7,13),(8,14)], [(2,6),(4,8),(10,12),(14,16)], [(1,5),(2,6),(3,7),(4,8),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,14,11,8),(2,7,12,13),(3,10,15,6),(4,5,16,9)])`

`G:=TransitiveGroup(16,310);`

Matrix representation of C24.39D4 in GL8(ℤ)

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

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

C24.39D4 in GAP, Magma, Sage, TeX

`C_2^4._{39}D_4`
`% in TeX`

`G:=Group("C2^4.39D4");`
`// GroupNames label`

`G:=SmallGroup(128,859);`
`// by ID`

`G=gap.SmallGroup(128,859);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,1123,1018,248,1971,375,4037]);`
`// Polycyclic`

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

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