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## G = C2.D8order 32 = 25

### 2nd central extension by C2 of D8

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

Aliases: C81C4, C2.2D8, C4.2Q8, C2.2Q16, C22.11D4, C4⋊C4.3C2, (C2×C8).3C2, C4.7(C2×C4), C2.4(C4⋊C4), (C2×C4).18C22, SmallGroup(32,14)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2.D8
G = < a,b,c | a2=b8=1, c2=a, ab=ba, ac=ca, cbc-1=b-1 >

Character table of C2.D8

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D size 1 1 1 1 2 2 4 4 4 4 2 2 2 2 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 -1 1 -1 1 -1 i -i i -i -1 1 -1 1 linear of order 4 ρ6 1 -1 1 -1 1 -1 -i -i i i 1 -1 1 -1 linear of order 4 ρ7 1 -1 1 -1 1 -1 i i -i -i 1 -1 1 -1 linear of order 4 ρ8 1 -1 1 -1 1 -1 -i i -i i -1 1 -1 1 linear of order 4 ρ9 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 0 0 0 0 0 0 -√2 -√2 √2 √2 orthogonal lifted from D8 ρ11 2 -2 -2 2 0 0 0 0 0 0 √2 √2 -√2 -√2 orthogonal lifted from D8 ρ12 2 2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 symplectic lifted from Q16, Schur index 2 ρ13 2 2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 symplectic lifted from Q16, Schur index 2 ρ14 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2

Smallest permutation representation of C2.D8
Regular action on 32 points
Generators in S32
```(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)
(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 9 23 31)(2 16 24 30)(3 15 17 29)(4 14 18 28)(5 13 19 27)(6 12 20 26)(7 11 21 25)(8 10 22 32)```

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

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

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

C2.D8 is a maximal subgroup of
C2.D16  C2.Q32  C23.25D4  M4(2)⋊C4  C4×D8  C4×Q16  SD16⋊C4  C87D4  C8.18D4  C8⋊D4  C4.Q16  D4.Q8  Q8.Q8  C22.D8  C23.19D4  C23.48D4  C23.20D4  C8.5Q8  C8⋊Q8  C62.7D4  C4.2PSU3(𝔽2)  C3⋊S3.4D8
C8p⋊C4: C163C4  C164C4  C241C4  C405C4  D5.D8  C561C4  C44.5Q8  C1045C4 ...
C2p.D8: D4⋊Q8  C82Q8  C6.Q16  C10.D8  C28.Q8  C44.Q8  C26.D8 ...
C2.D8 is a maximal quotient of
C22.4Q16  C62.7D4  C4.2PSU3(𝔽2)  C3⋊S3.4D8
C8p⋊C4: C163C4  C164C4  C241C4  C405C4  D5.D8  C561C4  C44.5Q8  C1045C4 ...
C2p.D8: C81C8  C8.4Q8  C6.Q16  C10.D8  C28.Q8  C44.Q8  C26.D8 ...

Matrix representation of C2.D8 in GL3(𝔽17) generated by

 16 0 0 0 1 0 0 0 1
,
 16 0 0 0 9 0 0 0 2
,
 4 0 0 0 0 2 0 9 0
`G:=sub<GL(3,GF(17))| [16,0,0,0,1,0,0,0,1],[16,0,0,0,9,0,0,0,2],[4,0,0,0,0,9,0,2,0] >;`

C2.D8 in GAP, Magma, Sage, TeX

`C_2.D_8`
`% in TeX`

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

`G:=SmallGroup(32,14);`
`// by ID`

`G=gap.SmallGroup(32,14);`
`# by ID`

`G:=PCGroup([5,-2,2,-2,2,-2,40,61,106,302,72]);`
`// Polycyclic`

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

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