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## G = C42.70D4order 128 = 27

### 52nd non-split extension by C42 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.70D4
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4.4D4 — C42.70D4
 Lower central C1 — C22 — C2×C4 — C42.70D4
 Upper central C1 — C22 — C2×C42 — C42.70D4
 Jennings C1 — C22 — C22 — C42 — C42.70D4

Generators and relations for C42.70D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=ab2, dad-1=a-1b2, cbc-1=a2b-1, bd=db, dcd-1=a2b-1c3 >

Subgroups: 308 in 124 conjugacy classes, 44 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×8], C22, C22 [×2], C22 [×12], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], D4 [×4], Q8 [×4], C23, C23 [×8], C42 [×2], C42 [×2], C22⋊C4 [×8], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4, C2×D4 [×2], C2×D4 [×3], C2×Q8 [×2], C2×Q8 [×3], C24, C8⋊C4 [×4], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C2×C22⋊C4 [×2], C4.4D4 [×4], C4.4D4 [×2], C22×D4, C22×Q8, C42.C22 [×4], C42.6C4 [×2], C2×C4.4D4, C42.70D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4.D4 [×2], C2×C22⋊C4, C2×C4.D4, C42⋊C22 [×2], C42.70D4

Character table of C42.70D4

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 2 2 8 8 2 2 2 2 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 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 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 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 linear of order 2 ρ9 1 1 1 1 -1 -1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 -i i i -i i i -i -i linear of order 4 ρ10 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 -1 -1 i -i i -i i -i i -i linear of order 4 ρ11 1 1 1 1 -1 -1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 i -i -i i -i -i i i linear of order 4 ρ12 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 -1 -1 -i i -i i -i i -i i linear of order 4 ρ13 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -1 1 -1 1 1 i -i i i -i i -i -i linear of order 4 ρ14 1 1 1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 1 1 -1 1 -i i i i -i -i i -i linear of order 4 ρ15 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -1 1 -1 1 1 -i i -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 -1 1 i -i -i -i i i -i i linear of order 4 ρ17 2 2 2 2 -2 -2 0 0 2 2 -2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 -2 -2 0 0 -2 -2 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 2 2 0 0 -2 -2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 2 2 0 0 2 2 -2 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 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 C4.D4 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 C4.D4 ρ23 4 -4 -4 4 0 0 0 0 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C42⋊C22 ρ24 4 -4 4 -4 0 0 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C42⋊C22 ρ25 4 -4 -4 4 0 0 0 0 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C42⋊C22 ρ26 4 -4 4 -4 0 0 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C42⋊C22

Smallest permutation representation of C42.70D4
On 32 points
Generators in S32
```(1 9 23 30)(2 27 24 14)(3 11 17 32)(4 29 18 16)(5 13 19 26)(6 31 20 10)(7 15 21 28)(8 25 22 12)
(1 32 19 15)(2 29 20 12)(3 26 21 9)(4 31 22 14)(5 28 23 11)(6 25 24 16)(7 30 17 13)(8 27 18 10)
(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 27 32 18 19 10 15 8)(2 21 29 9 20 3 12 26)(4 5 31 28 22 23 14 11)(6 17 25 13 24 7 16 30)```

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

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

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

Matrix representation of C42.70D4 in GL8(𝔽17)

 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 13 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0
,
 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 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
,
 3 10 14 10 0 0 0 0 10 14 10 3 0 0 0 0 3 7 14 7 0 0 0 0 7 14 7 3 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 13 0 0 0 0 0 0 0 0 4 0 0
,
 3 10 14 10 0 0 0 0 7 3 7 14 0 0 0 0 3 7 14 7 0 0 0 0 10 3 10 14 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

`G:=sub<GL(8,GF(17))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[3,10,3,7,0,0,0,0,10,14,7,14,0,0,0,0,14,10,14,7,0,0,0,0,10,3,7,3,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0],[3,7,3,10,0,0,0,0,10,3,7,3,0,0,0,0,14,7,14,10,0,0,0,0,10,14,7,14,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;`

C42.70D4 in GAP, Magma, Sage, TeX

`C_4^2._{70}D_4`
`% in TeX`

`G:=Group("C4^2.70D4");`
`// GroupNames label`

`G:=SmallGroup(128,265);`
`// by ID`

`G=gap.SmallGroup(128,265);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,1123,1018,248,1971,102]);`
`// Polycyclic`

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

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