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## G = C24.6D4order 128 = 27

### 6th 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 — C24.6D4
 Chief series C1 — C2 — C22 — C23 — C24 — C22×D4 — C24.3C22 — C24.6D4
 Lower central C1 — C2 — C22 — C2×C4 — C24.6D4
 Upper central C1 — C22 — C23 — C22×D4 — C24.6D4
 Jennings C1 — C2 — C22 — C22×D4 — C24.6D4

Generators and relations for C24.6D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=a, ab=ba, eae-1=ac=ca, ad=da, af=fa, bc=cb, ebe-1=fbf-1=bd=db, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=abe-1 >

Subgroups: 352 in 116 conjugacy classes, 32 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×9], C22 [×3], C22 [×16], C2×C4 [×2], C2×C4 [×17], D4 [×4], C23, C23 [×4], C23 [×8], C42, C22⋊C4 [×10], C4⋊C4, C22×C4, C22×C4 [×4], C2×D4 [×2], C2×D4 [×2], C2×D4 [×2], C24 [×2], C23⋊C4 [×4], C23⋊C4 [×2], C2×C42, C2×C22⋊C4 [×4], C2×C4⋊C4, C22×D4, C24.3C22, C2×C23⋊C4 [×2], C24.6D4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C23⋊C4 [×2], C23.9D4, C42⋊C4, C423C4, C24.6D4

Character table of C24.6D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P size 1 1 1 1 2 2 4 4 4 4 4 4 4 4 4 4 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 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 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 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 linear of order 2 ρ5 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 -1 -1 i i -i -i 1 1 -i -i i i linear of order 4 ρ6 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -i -i i i -1 -1 -i -i i i linear of order 4 ρ7 1 -1 1 -1 1 -1 1 -1 -1 1 -1 i 1 -i -i i -1 1 1 -1 i -i -i i i -i linear of order 4 ρ8 1 -1 1 -1 1 -1 -1 1 1 -1 -1 i 1 -i -i i -i i -i i -i i 1 -1 1 -1 linear of order 4 ρ9 1 -1 1 -1 1 -1 1 -1 -1 1 -1 i 1 -i -i i 1 -1 -1 1 i -i i -i -i i linear of order 4 ρ10 1 -1 1 -1 1 -1 -1 1 1 -1 -1 -i 1 i i -i i -i i -i i -i 1 -1 1 -1 linear of order 4 ρ11 1 -1 1 -1 1 -1 -1 1 1 -1 -1 -i 1 i i -i -i i -i i i -i -1 1 -1 1 linear of order 4 ρ12 1 -1 1 -1 1 -1 1 -1 -1 1 -1 -i 1 i i -i 1 -1 -1 1 -i i -i i i -i linear of order 4 ρ13 1 -1 1 -1 1 -1 -1 1 1 -1 -1 i 1 -i -i i i -i i -i -i i -1 1 -1 1 linear of order 4 ρ14 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 i i -i -i -1 -1 i i -i -i linear of order 4 ρ15 1 -1 1 -1 1 -1 1 -1 -1 1 -1 -i 1 i i -i -1 1 1 -1 -i i i -i -i i linear of order 4 ρ16 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 -1 -1 -i -i i i 1 1 i i -i -i linear of order 4 ρ17 2 2 2 2 2 2 2 2 -2 -2 -2 0 -2 0 0 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 -2 2 0 -2 0 0 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 2 -2 0 -2 0 0 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 2 2 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ21 4 4 -4 -4 0 0 0 0 0 0 0 2 0 2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C42⋊C4 ρ22 4 -4 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ23 4 4 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ24 4 4 -4 -4 0 0 0 0 0 0 0 -2 0 -2 2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C42⋊C4 ρ25 4 -4 -4 4 0 0 0 0 0 0 0 -2i 0 2i -2i 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C42⋊3C4 ρ26 4 -4 -4 4 0 0 0 0 0 0 0 2i 0 -2i 2i -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C42⋊3C4

Smallest permutation representation of C24.6D4
On 32 points
Generators in S32
```(1 11)(2 16)(3 12)(4 15)(5 9)(6 14)(7 13)(8 10)(17 21)(18 24)(19 20)(22 23)(25 32)(26 27)(28 29)(30 31)
(1 3)(2 7)(4 6)(5 8)(9 10)(11 12)(13 16)(14 15)(17 30)(18 27)(19 32)(20 25)(21 31)(22 28)(23 29)(24 26)
(1 4)(2 8)(3 6)(5 7)(9 13)(10 16)(11 15)(12 14)(17 22)(18 20)(19 24)(21 23)(25 27)(26 32)(28 30)(29 31)
(1 5)(2 6)(3 8)(4 7)(9 11)(10 12)(13 15)(14 16)(17 24)(18 21)(19 22)(20 23)(25 29)(26 30)(27 31)(28 32)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 30 11 31)(2 19 16 20)(3 24 12 18)(4 32 15 25)(5 26 9 27)(6 22 14 23)(7 28 13 29)(8 17 10 21)```

`G:=sub<Sym(32)| (1,11)(2,16)(3,12)(4,15)(5,9)(6,14)(7,13)(8,10)(17,21)(18,24)(19,20)(22,23)(25,32)(26,27)(28,29)(30,31), (1,3)(2,7)(4,6)(5,8)(9,10)(11,12)(13,16)(14,15)(17,30)(18,27)(19,32)(20,25)(21,31)(22,28)(23,29)(24,26), (1,4)(2,8)(3,6)(5,7)(9,13)(10,16)(11,15)(12,14)(17,22)(18,20)(19,24)(21,23)(25,27)(26,32)(28,30)(29,31), (1,5)(2,6)(3,8)(4,7)(9,11)(10,12)(13,15)(14,16)(17,24)(18,21)(19,22)(20,23)(25,29)(26,30)(27,31)(28,32), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,30,11,31)(2,19,16,20)(3,24,12,18)(4,32,15,25)(5,26,9,27)(6,22,14,23)(7,28,13,29)(8,17,10,21)>;`

`G:=Group( (1,11)(2,16)(3,12)(4,15)(5,9)(6,14)(7,13)(8,10)(17,21)(18,24)(19,20)(22,23)(25,32)(26,27)(28,29)(30,31), (1,3)(2,7)(4,6)(5,8)(9,10)(11,12)(13,16)(14,15)(17,30)(18,27)(19,32)(20,25)(21,31)(22,28)(23,29)(24,26), (1,4)(2,8)(3,6)(5,7)(9,13)(10,16)(11,15)(12,14)(17,22)(18,20)(19,24)(21,23)(25,27)(26,32)(28,30)(29,31), (1,5)(2,6)(3,8)(4,7)(9,11)(10,12)(13,15)(14,16)(17,24)(18,21)(19,22)(20,23)(25,29)(26,30)(27,31)(28,32), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,30,11,31)(2,19,16,20)(3,24,12,18)(4,32,15,25)(5,26,9,27)(6,22,14,23)(7,28,13,29)(8,17,10,21) );`

`G=PermutationGroup([(1,11),(2,16),(3,12),(4,15),(5,9),(6,14),(7,13),(8,10),(17,21),(18,24),(19,20),(22,23),(25,32),(26,27),(28,29),(30,31)], [(1,3),(2,7),(4,6),(5,8),(9,10),(11,12),(13,16),(14,15),(17,30),(18,27),(19,32),(20,25),(21,31),(22,28),(23,29),(24,26)], [(1,4),(2,8),(3,6),(5,7),(9,13),(10,16),(11,15),(12,14),(17,22),(18,20),(19,24),(21,23),(25,27),(26,32),(28,30),(29,31)], [(1,5),(2,6),(3,8),(4,7),(9,11),(10,12),(13,15),(14,16),(17,24),(18,21),(19,22),(20,23),(25,29),(26,30),(27,31),(28,32)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,30,11,31),(2,19,16,20),(3,24,12,18),(4,32,15,25),(5,26,9,27),(6,22,14,23),(7,28,13,29),(8,17,10,21)])`

Matrix representation of C24.6D4 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 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 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 2 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 2 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 -2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 -2 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 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 -1 -2 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 -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 0 0 1 -1 -1 0 0 0 0 0 0 1 1 0 0 0 0 -1 -1 0 1 0 0 0 0 1 1 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,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,-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,-1,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-2,1],[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,-1,0,0,0,0,0,0,0,0,-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,-1,1,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1],[0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,1,0,-1,1,0,0,0,0,-1,1,0,0,0,0,0,0,-1,1,1,0] >;`

C24.6D4 in GAP, Magma, Sage, TeX

`C_2^4._6D_4`
`% in TeX`

`G:=Group("C2^4.6D4");`
`// GroupNames label`

`G:=SmallGroup(128,125);`
`// by ID`

`G=gap.SmallGroup(128,125);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,352,1466,521,2804,4037]);`
`// Polycyclic`

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

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