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## G = C42.32Q8order 128 = 27

### 32nd non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — C42.32Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×M4(2) — M4(2)⋊4C4 — C42.32Q8
 Lower central C1 — C2 — C22×C4 — C42.32Q8
 Upper central C1 — C4 — C22×C4 — C42.32Q8
 Jennings C1 — C2 — C2 — C22×C4 — C42.32Q8

Generators and relations for C42.32Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b2c2, ab=ba, cac-1=a-1, dad-1=ab-1, cbc-1=dbd-1=b-1, dcd-1=b2c3 >

Subgroups: 152 in 79 conjugacy classes, 36 normal (18 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22, C8 [×5], C2×C4 [×2], C2×C4 [×4], C2×C4 [×6], C23, C42 [×2], C42 [×3], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×3], M4(2) [×8], C22×C4, C22×C4, C4⋊C8 [×2], C8.C4 [×2], C2×C42, C42⋊C2 [×2], C2×M4(2) [×2], C2×M4(2) [×2], C4.9C42, C426C4 [×2], M4(2)⋊4C4 [×2], C4⋊M4(2), M4(2).C4, C42.32Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, C2×D4, C2×Q8, C4○D4 [×5], C22⋊Q8, C22.D4, C4.4D4, C42.C2 [×2], C422C2 [×2], C23.83C23, C42.32Q8

Character table of C42.32Q8

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 2 2 2 1 1 2 2 2 4 4 4 4 8 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 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 2 2 -2 2 -2 0 0 0 0 0 0 0 0 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -2 -2 2 2 -2 2 -2 0 0 0 0 0 0 0 0 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 -2 -2 -2 -2 2 -2 2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ12 2 2 2 -2 -2 -2 -2 2 -2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ13 2 2 -2 2 -2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 -2i 0 0 0 2i 0 complex lifted from C4○D4 ρ14 2 2 2 2 2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 2i 0 0 -2i complex lifted from C4○D4 ρ15 2 2 -2 -2 2 2 2 2 -2 -2 0 0 0 0 0 2i -2i 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ16 2 2 -2 2 -2 2 2 -2 -2 2 0 0 0 0 2i 0 0 -2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ17 2 2 2 2 2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 -2i 0 0 2i complex lifted from C4○D4 ρ18 2 2 -2 -2 2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 -2i 0 2i 0 0 complex lifted from C4○D4 ρ19 2 2 -2 -2 2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 2i 0 -2i 0 0 complex lifted from C4○D4 ρ20 2 2 -2 2 -2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 2i 0 0 0 -2i 0 complex lifted from C4○D4 ρ21 2 2 -2 2 -2 2 2 -2 -2 2 0 0 0 0 -2i 0 0 2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 -2 -2 2 2 2 2 -2 -2 0 0 0 0 0 -2i 2i 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 4 -4 0 0 0 -4i 4i 0 0 0 -2 2 -2i 2i 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ24 4 -4 0 0 0 4i -4i 0 0 0 2 -2 -2i 2i 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ25 4 -4 0 0 0 -4i 4i 0 0 0 2 -2 2i -2i 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ26 4 -4 0 0 0 4i -4i 0 0 0 -2 2 2i -2i 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C42.32Q8
On 16 points - transitive group 16T334
Generators in S16
```(9 11 13 15)(10 16 14 12)
(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 13 7 15 5 9 3 11)(2 12 8 14 6 16 4 10)```

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

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

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

`G:=TransitiveGroup(16,334);`

Matrix representation of C42.32Q8 in GL4(𝔽5) generated by

 4 0 0 4 0 0 1 0 0 2 4 0 1 0 0 4
,
 2 0 0 0 0 3 0 0 0 0 3 0 0 0 0 2
,
 0 2 2 0 2 0 0 2 3 0 0 0 0 0 2 0
,
 0 4 3 0 0 0 0 2 1 0 0 4 0 4 0 0
`G:=sub<GL(4,GF(5))| [4,0,0,1,0,0,2,0,0,1,4,0,4,0,0,4],[2,0,0,0,0,3,0,0,0,0,3,0,0,0,0,2],[0,2,3,0,2,0,0,0,2,0,0,2,0,2,0,0],[0,0,1,0,4,0,0,4,3,0,0,0,0,2,4,0] >;`

C42.32Q8 in GAP, Magma, Sage, TeX

`C_4^2._{32}Q_8`
`% in TeX`

`G:=Group("C4^2.32Q8");`
`// GroupNames label`

`G:=SmallGroup(128,834);`
`// by ID`

`G=gap.SmallGroup(128,834);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,176,422,387,58,1018,248,1411,4037,1027]);`
`// Polycyclic`

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

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