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## G = C42.10D4order 128 = 27

### 10th non-split extension by C42 of D4 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — C42.10D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C42⋊C22 — C42.10D4
 Lower central C1 — C2 — C22×C4 — C42.10D4
 Upper central C1 — C4 — C22×C4 — C42.10D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.10D4

Generators and relations for C42.10D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b-1, ab=ba, cac-1=a-1b-1, dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=b2c3 >

Subgroups: 208 in 105 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×4], C8 [×5], C2×C4 [×2], C2×C4 [×4], C2×C4 [×7], D4 [×4], Q8 [×2], C23, C23, C42 [×4], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×5], M4(2) [×6], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C22⋊C8 [×2], C4≀C2 [×4], C4⋊C8 [×4], C8.C4 [×2], C42⋊C2 [×2], C22×C8, C2×M4(2), C2×M4(2) [×2], C2×C4○D4, C4.9C42, (C22×C8)⋊C2, C42⋊C22 [×2], C42.6C22 [×2], M4(2).C4, C42.10D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, C2×D4 [×3], C2×Q8, C4○D4 [×3], C22⋊Q8 [×3], C22.D4 [×3], C41D4, C23.4Q8, C42.10D4

Character table of C42.10D4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J size 1 1 2 2 2 8 1 1 2 2 2 8 8 8 8 8 4 4 4 4 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 2 2 -2 -2 2 0 2 2 -2 2 -2 0 0 0 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -2 -2 0 2 2 2 -2 -2 2 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 2 -2 0 2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 2 0 -2 0 0 orthogonal lifted from D4 ρ12 2 2 -2 2 -2 0 2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 -2 0 2 0 0 orthogonal lifted from D4 ρ13 2 2 2 -2 -2 0 2 2 2 -2 -2 -2 0 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 -2 -2 2 0 2 2 -2 2 -2 0 0 0 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 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 ρ16 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 ρ17 2 2 -2 -2 2 0 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 -2i 2i complex lifted from C4○D4 ρ18 2 2 2 2 2 0 -2 -2 -2 -2 -2 0 0 0 0 0 2i 2i -2i -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ19 2 2 -2 -2 2 0 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 2i -2i complex lifted from C4○D4 ρ20 2 2 2 -2 -2 0 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 -2i 0 2i 0 0 0 complex lifted from C4○D4 ρ21 2 2 2 2 2 0 -2 -2 -2 -2 -2 0 0 0 0 0 -2i -2i 2i 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 2 -2 -2 0 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 2i 0 -2i 0 0 0 complex lifted from C4○D4 ρ23 4 -4 0 0 0 0 -4i 4i 0 0 0 0 0 0 0 0 2ζ8 2ζ85 2ζ83 2ζ87 0 0 0 0 0 0 complex faithful ρ24 4 -4 0 0 0 0 4i -4i 0 0 0 0 0 0 0 0 2ζ87 2ζ83 2ζ85 2ζ8 0 0 0 0 0 0 complex faithful ρ25 4 -4 0 0 0 0 -4i 4i 0 0 0 0 0 0 0 0 2ζ85 2ζ8 2ζ87 2ζ83 0 0 0 0 0 0 complex faithful ρ26 4 -4 0 0 0 0 4i -4i 0 0 0 0 0 0 0 0 2ζ83 2ζ87 2ζ8 2ζ85 0 0 0 0 0 0 complex faithful

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

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

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

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

Matrix representation of C42.10D4 in GL4(𝔽17) generated by

 1 0 10 7 2 16 16 1 0 0 0 13 0 0 13 0
,
 4 0 13 4 0 4 0 0 0 0 13 0 0 0 0 13
,
 15 2 14 3 0 0 0 8 0 2 0 0 4 15 15 2
,
 8 9 9 8 0 9 0 0 0 0 0 8 0 0 8 0
`G:=sub<GL(4,GF(17))| [1,2,0,0,0,16,0,0,10,16,0,13,7,1,13,0],[4,0,0,0,0,4,0,0,13,0,13,0,4,0,0,13],[15,0,0,4,2,0,2,15,14,0,0,15,3,8,0,2],[8,0,0,0,9,9,0,0,9,0,0,8,8,0,8,0] >;`

C42.10D4 in GAP, Magma, Sage, TeX

`C_4^2._{10}D_4`
`% in TeX`

`G:=Group("C4^2.10D4");`
`// GroupNames label`

`G:=SmallGroup(128,830);`
`// by ID`

`G=gap.SmallGroup(128,830);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,176,422,387,1018,248,1411,4037,2028,1027,124]);`
`// Polycyclic`

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

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