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## G = C8.1Q16order 128 = 27

### 1st non-split extension by C8 of Q16 acting via Q16/C4=C22

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

Aliases: C8.1Q16, C8.28SD16, (C2×C8).80D4, C8.C8.2C2, C42.47(C2×C4), C42.C2.3C4, C8.5Q8.5C2, (C4×C8).135C22, C4.4(Q8⋊C4), C2.3(C4.6Q16), C22.15(C4.D4), (C2×C4).61(C22⋊C4), SmallGroup(128,98)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — C8.1Q16
 Chief series C1 — C2 — C22 — C2×C4 — C2×C8 — C4×C8 — C8.5Q8 — C8.1Q16
 Lower central C1 — C2 — C2×C4 — C42 — C8.1Q16
 Upper central C1 — C2 — C2×C4 — C4×C8 — C8.1Q16
 Jennings C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C4×C8 — C8.1Q16

Generators and relations for C8.1Q16
G = < a,b,c | a8=1, b8=a4, c2=a2b4, bab-1=a3, cac-1=a-1, cbc-1=a-1b7 >

Character table of C8.1Q16

 class 1 2A 2B 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 8E 8F 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 2 2 2 4 4 16 16 2 2 2 2 4 4 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 -i -i i i i i -i -i linear of order 4 ρ6 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 ρ7 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 ρ8 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 ρ9 2 2 2 2 2 -2 -2 0 0 -2 2 -2 2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 -2 -2 0 0 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 2 -2 0 0 0 0 -2 0 -2 0 2 0 √2 0 -√2 0 0 √2 -√2 0 symplectic lifted from Q16, Schur index 2 ρ12 2 2 -2 -2 2 0 0 0 0 0 -2 0 -2 0 2 0 √2 0 -√2 √2 0 0 -√2 symplectic lifted from Q16, Schur index 2 ρ13 2 2 -2 -2 2 0 0 0 0 0 -2 0 -2 0 2 0 -√2 0 √2 -√2 0 0 √2 symplectic lifted from Q16, Schur index 2 ρ14 2 2 -2 2 -2 0 0 0 0 -2 0 -2 0 2 0 -√2 0 √2 0 0 -√2 √2 0 symplectic lifted from Q16, Schur index 2 ρ15 2 2 -2 2 -2 0 0 0 0 2 0 2 0 -2 0 √-2 0 √-2 0 0 -√-2 -√-2 0 complex lifted from SD16 ρ16 2 2 -2 -2 2 0 0 0 0 0 2 0 2 0 -2 0 √-2 0 √-2 -√-2 0 0 -√-2 complex lifted from SD16 ρ17 2 2 -2 2 -2 0 0 0 0 2 0 2 0 -2 0 -√-2 0 -√-2 0 0 √-2 √-2 0 complex lifted from SD16 ρ18 2 2 -2 -2 2 0 0 0 0 0 2 0 2 0 -2 0 -√-2 0 -√-2 √-2 0 0 √-2 complex lifted from SD16 ρ19 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 C4.D4 ρ20 4 -4 0 0 0 -2i 2i 0 0 2√2 2√-2 -2√2 -2√-2 0 0 0 0 0 0 0 0 0 0 complex faithful ρ21 4 -4 0 0 0 -2i 2i 0 0 -2√2 -2√-2 2√2 2√-2 0 0 0 0 0 0 0 0 0 0 complex faithful ρ22 4 -4 0 0 0 2i -2i 0 0 -2√2 2√-2 2√2 -2√-2 0 0 0 0 0 0 0 0 0 0 complex faithful ρ23 4 -4 0 0 0 2i -2i 0 0 2√2 -2√-2 -2√2 2√-2 0 0 0 0 0 0 0 0 0 0 complex faithful

Smallest permutation representation of C8.1Q16
On 32 points
Generators in S32
```(1 22 13 18 9 30 5 26)(2 19 6 23 10 27 14 31)(3 24 15 20 11 32 7 28)(4 21 8 25 12 29 16 17)
(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)
(2 25 10 17)(3 7)(4 31 12 23)(5 13)(6 21 14 29)(8 27 16 19)(11 15)(18 30)(22 26)(24 32)```

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

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

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

Matrix representation of C8.1Q16 in GL4(𝔽17) generated by

 3 14 0 0 3 3 0 0 0 0 14 14 0 0 3 14
,
 0 0 1 0 0 0 0 1 12 5 0 0 12 12 0 0
,
 1 0 0 0 0 16 0 0 0 0 13 0 0 0 0 4
`G:=sub<GL(4,GF(17))| [3,3,0,0,14,3,0,0,0,0,14,3,0,0,14,14],[0,0,12,12,0,0,5,12,1,0,0,0,0,1,0,0],[1,0,0,0,0,16,0,0,0,0,13,0,0,0,0,4] >;`

C8.1Q16 in GAP, Magma, Sage, TeX

`C_8._1Q_{16}`
`% in TeX`

`G:=Group("C8.1Q16");`
`// GroupNames label`

`G:=SmallGroup(128,98);`
`// by ID`

`G=gap.SmallGroup(128,98);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,232,422,387,520,794,192,2804,1411,172,4037]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=1,b^8=a^4,c^2=a^2*b^4,b*a*b^-1=a^3,c*a*c^-1=a^-1,c*b*c^-1=a^-1*b^7>;`
`// generators/relations`

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