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## G = C23.2D8order 128 = 27

### 2nd non-split extension by C23 of D8 acting via D8/C2=D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — C23.2D8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×M4(2) — C8.D4 — C23.2D8
 Lower central C1 — C2 — C2×C4 — C2×C8 — C23.2D8
 Upper central C1 — C2 — C2×C4 — C2×M4(2) — C23.2D8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C2×C4 — C2×M4(2) — C23.2D8

Generators and relations for C23.2D8
G = < a,b,c,d,e | a2=b2=c2=1, d8=c, e2=a, dad-1=ab=ba, ac=ca, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=ad7 >

Character table of C23.2D8

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 16A 16B 16C 16D size 1 1 2 4 2 2 4 8 8 8 8 16 16 4 4 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 trivial ρ2 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 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 linear of order 2 ρ5 1 1 1 -1 1 1 -1 i i -i -i 1 -1 -1 -1 1 i -i -i i linear of order 4 ρ6 1 1 1 -1 1 1 -1 -i -i i i 1 -1 -1 -1 1 -i i i -i linear of order 4 ρ7 1 1 1 -1 1 1 -1 i i -i -i -1 1 -1 -1 1 -i i i -i linear of order 4 ρ8 1 1 1 -1 1 1 -1 -i -i i i -1 1 -1 -1 1 i -i -i i linear of order 4 ρ9 2 2 2 -2 2 2 -2 0 0 0 0 0 0 2 2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 2 0 0 0 0 0 0 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 -√2 -√2 √2 √2 orthogonal lifted from D8 ρ12 2 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 orthogonal lifted from D8 ρ13 2 2 -2 0 2 -2 0 1+i -1-i 1-i -1+i 0 0 -2i 2i 0 0 0 0 0 complex lifted from C4≀C2 ρ14 2 2 -2 0 2 -2 0 1-i -1+i 1+i -1-i 0 0 2i -2i 0 0 0 0 0 complex lifted from C4≀C2 ρ15 2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 -√-2 √-2 -√-2 √-2 complex lifted from SD16 ρ16 2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 √-2 -√-2 √-2 -√-2 complex lifted from SD16 ρ17 2 2 -2 0 2 -2 0 -1+i 1-i -1-i 1+i 0 0 2i -2i 0 0 0 0 0 complex lifted from C4≀C2 ρ18 2 2 -2 0 2 -2 0 -1-i 1+i -1+i 1-i 0 0 -2i 2i 0 0 0 0 0 complex lifted from C4≀C2 ρ19 4 4 -4 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ20 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of C23.2D8 in GL8(𝔽17)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0
,
 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 8 5 12 8 12 9 9 5 12 8 9 12 8 12 12 9 12 8 9 12 9 5 5 8 9 12 5 9 12 9 9 5 8 12 12 9 8 5 12 8 5 8 8 12 12 8 9 12 12 9 9 5 12 8 9 12 8 12 12 9 9 12 5 9
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0

`G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[8,12,12,9,8,5,12,8,5,8,8,12,12,8,9,12,12,9,9,5,12,8,9,12,8,12,12,9,9,12,5,9,12,8,9,12,8,12,12,9,9,12,5,9,5,8,8,12,9,12,5,9,12,9,9,5,5,9,8,5,8,12,12,9],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0] >;`

C23.2D8 in GAP, Magma, Sage, TeX

`C_2^3._2D_8`
`% in TeX`

`G:=Group("C2^3.2D8");`
`// GroupNames label`

`G:=SmallGroup(128,72);`
`// by ID`

`G=gap.SmallGroup(128,72);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,387,520,1690,521,1411,172,4037,2028,124]);`
`// Polycyclic`

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

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