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## G = C23.32D8order 128 = 27

### 3rd non-split extension by C23 of D8 acting via D8/D4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — C23.32D8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C8.18D4 — C23.32D8
 Lower central C1 — C2 — C2×C4 — C2×C8 — C23.32D8
 Upper central C1 — C22 — C22×C4 — C22×C8 — C23.32D8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C2×C4 — C22×C8 — C23.32D8

Generators and relations for C23.32D8
G = < a,b,c,d,e | a2=b2=c2=1, d8=c, e2=a, ab=ba, ac=ca, dad-1=abc, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=ad7 >

Character table of C23.32D8

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E 8F 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 2 2 2 2 4 8 8 8 8 16 16 2 2 2 2 4 4 4 4 4 4 4 4 4 4 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 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 -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 -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 1 1 1 linear of order 2 ρ5 1 1 1 1 -1 -1 1 1 -1 -i i -i i -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 -i i -i i 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 i -i i -i 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 i -i i -i -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 2 -2 0 0 0 0 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 2 2 0 0 0 0 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 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 -√2 -√2 -√2 √2 √2 -√2 √2 √2 orthogonal lifted from D8 ρ12 2 2 2 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 √2 √2 √2 -√2 -√2 √2 -√2 -√2 orthogonal lifted from D8 ρ13 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 -√2 √2 ζ165-ζ163 ζ165-ζ163 -ζ165+ζ163 ζ1615-ζ169 ζ1615-ζ169 -ζ165+ζ163 -ζ1615+ζ169 -ζ1615+ζ169 symplectic lifted from Q32, Schur index 2 ρ14 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 -√2 √2 -ζ165+ζ163 -ζ165+ζ163 ζ165-ζ163 -ζ1615+ζ169 -ζ1615+ζ169 ζ165-ζ163 ζ1615-ζ169 ζ1615-ζ169 symplectic lifted from Q32, Schur index 2 ρ15 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 √2 -√2 ζ1615-ζ169 ζ1615-ζ169 -ζ1615+ζ169 -ζ165+ζ163 -ζ165+ζ163 -ζ1615+ζ169 ζ165-ζ163 ζ165-ζ163 symplectic lifted from Q32, Schur index 2 ρ16 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 √2 -√2 -ζ1615+ζ169 -ζ1615+ζ169 ζ1615-ζ169 ζ165-ζ163 ζ165-ζ163 ζ1615-ζ169 -ζ165+ζ163 -ζ165+ζ163 symplectic lifted from Q32, Schur index 2 ρ17 2 -2 2 -2 0 0 -2 2 0 -1+i 1+i 1-i -1-i 0 0 2i -2i -2i 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4≀C2 ρ18 2 -2 2 -2 0 0 -2 2 0 1+i -1+i -1-i 1-i 0 0 -2i 2i 2i -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4≀C2 ρ19 2 2 2 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 √-2 -√-2 √-2 -√-2 √-2 -√-2 √-2 -√-2 complex lifted from SD16 ρ20 2 2 2 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 -√-2 √-2 -√-2 √-2 -√-2 √-2 -√-2 √-2 complex lifted from SD16 ρ21 2 -2 2 -2 0 0 -2 2 0 1-i -1-i -1+i 1+i 0 0 2i -2i -2i 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4≀C2 ρ22 2 -2 2 -2 0 0 -2 2 0 -1-i 1-i 1+i -1+i 0 0 -2i 2i 2i -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4≀C2 ρ23 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 -√2 √2 ζ1615+ζ169 ζ167+ζ16 ζ167+ζ16 ζ165+ζ163 ζ1613+ζ1611 ζ1615+ζ169 ζ165+ζ163 ζ1613+ζ1611 complex lifted from SD32 ρ24 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 -√2 √2 ζ167+ζ16 ζ1615+ζ169 ζ1615+ζ169 ζ1613+ζ1611 ζ165+ζ163 ζ167+ζ16 ζ1613+ζ1611 ζ165+ζ163 complex lifted from SD32 ρ25 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 √2 -√2 ζ1613+ζ1611 ζ165+ζ163 ζ165+ζ163 ζ1615+ζ169 ζ167+ζ16 ζ1613+ζ1611 ζ1615+ζ169 ζ167+ζ16 complex lifted from SD32 ρ26 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 √2 -√2 ζ165+ζ163 ζ1613+ζ1611 ζ1613+ζ1611 ζ167+ζ16 ζ1615+ζ169 ζ165+ζ163 ζ167+ζ16 ζ1615+ζ169 complex lifted from SD32 ρ27 4 -4 4 -4 0 0 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ28 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 -2√-2 -2√-2 2√-2 2√-2 0 0 0 0 0 0 0 0 0 0 complex lifted from D8⋊2C4 ρ29 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 2√-2 2√-2 -2√-2 -2√-2 0 0 0 0 0 0 0 0 0 0 complex lifted from D8⋊2C4

Smallest permutation representation of C23.32D8
On 32 points
Generators in S32
```(2 25)(4 27)(6 29)(8 31)(10 17)(12 19)(14 21)(16 23)
(1 32)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(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 13)(2 27 25 4)(3 18)(5 9)(6 23 29 16)(7 30)(8 14 31 21)(10 19 17 12)(11 26)(15 22)(20 24)(28 32)```

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

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

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

Matrix representation of C23.32D8 in GL4(𝔽17) generated by

 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 16
,
 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 11 2 0 0 16 9 0 0 0 0 0 4 0 0 16 0
,
 16 15 0 0 0 1 0 0 0 0 16 0 0 0 0 4
`G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[11,16,0,0,2,9,0,0,0,0,0,16,0,0,4,0],[16,0,0,0,15,1,0,0,0,0,16,0,0,0,0,4] >;`

C23.32D8 in GAP, Magma, Sage, TeX

`C_2^3._{32}D_8`
`% in TeX`

`G:=Group("C2^3.32D8");`
`// GroupNames label`

`G:=SmallGroup(128,80);`
`// by ID`

`G=gap.SmallGroup(128,80);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,387,520,1690,2804,1411,172,4037,2028,124]);`
`// Polycyclic`

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

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