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## G = C23.31D4order 64 = 26

### 2nd non-split extension by C23 of D4 acting via D4/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C23.31D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C22⋊Q8 — C23.31D4
 Lower central C1 — C22 — C2×C4 — C23.31D4
 Upper central C1 — C22 — C22×C4 — C23.31D4
 Jennings C1 — C2 — C22 — C22×C4 — C23.31D4

Generators and relations for C23.31D4
G = < a,b,c,d,e | a2=b2=c2=1, d4=c, e2=a, ab=ba, ac=ca, dad-1=abc, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=abcd3 >

Character table of C23.31D4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D size 1 1 1 1 2 2 2 2 4 4 4 4 4 8 8 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 1 1 1 -1 -1 -1 -1 -i i 1 i -i -1 1 i i -i -i linear of order 4 ρ6 1 1 1 1 -1 -1 -1 -1 i -i 1 -i i 1 -1 i i -i -i linear of order 4 ρ7 1 1 1 1 -1 -1 -1 -1 i -i 1 -i i -1 1 -i -i i i linear of order 4 ρ8 1 1 1 1 -1 -1 -1 -1 -i i 1 i -i 1 -1 -i -i i i linear of order 4 ρ9 2 2 2 2 -2 -2 2 2 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 -2 -2 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 -√2 √2 √2 -√2 symplectic lifted from Q16, Schur index 2 ρ12 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 √2 -√2 -√2 √2 symplectic lifted from Q16, Schur index 2 ρ13 2 2 -2 -2 0 0 -2i 2i 1-i 1+i 0 -1-i -1+i 0 0 0 0 0 0 complex lifted from C4≀C2 ρ14 2 2 -2 -2 0 0 -2i 2i -1+i -1-i 0 1+i 1-i 0 0 0 0 0 0 complex lifted from C4≀C2 ρ15 2 2 -2 -2 0 0 2i -2i 1+i 1-i 0 -1+i -1-i 0 0 0 0 0 0 complex lifted from C4≀C2 ρ16 2 2 -2 -2 0 0 2i -2i -1-i -1+i 0 1-i 1+i 0 0 0 0 0 0 complex lifted from C4≀C2 ρ17 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 √-2 -√-2 √-2 -√-2 complex lifted from SD16 ρ18 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 -√-2 √-2 -√-2 √-2 complex lifted from SD16 ρ19 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4

Permutation representations of C23.31D4
On 16 points - transitive group 16T161
Generators in S16
```(2 15)(4 9)(6 11)(8 13)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 4 15 9)(3 12)(6 8 11 13)(7 16)```

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

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

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

`G:=TransitiveGroup(16,161);`

Matrix representation of C23.31D4 in GL4(𝔽17) generated by

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

C23.31D4 in GAP, Magma, Sage, TeX

`C_2^3._{31}D_4`
`% in TeX`

`G:=Group("C2^3.31D4");`
`// GroupNames label`

`G:=SmallGroup(64,9);`
`// by ID`

`G=gap.SmallGroup(64,9);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,103,362,332,158,681]);`
`// Polycyclic`

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

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