Copied to
clipboard

## G = D8.C4order 64 = 26

### 1st non-split extension by D8 of C4 acting via C4/C2=C2

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

Aliases: D8.1C4, C8.21D4, C4.18D8, Q16.1C4, C22.1SD16, (C2×C16)⋊4C2, C8.9(C2×C4), C4○D8.1C2, (C2×C4).62D4, C8.C41C2, C4.3(C22⋊C4), (C2×C8).86C22, C2.8(D4⋊C4), SmallGroup(64,40)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for D8.C4
G = < a,b,c | a8=b2=1, c4=a4, bab=cac-1=a-1, cbc-1=a5b >

Character table of D8.C4

 class 1 2A 2B 2C 4A 4B 4C 4D 8A 8B 8C 8D 8E 8F 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 2 8 1 1 2 8 2 2 2 2 8 8 2 2 2 2 2 2 2 2 ρ1 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 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 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 linear of order 2 ρ5 1 1 -1 1 -1 -1 1 -1 -1 1 1 -1 -i i 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 -i i -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 i -i 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 i -i -i i i -i -i -i i i linear of order 4 ρ9 2 2 -2 0 -2 -2 2 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 0 2 2 2 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 0 2 2 -2 0 0 0 0 0 0 0 √2 -√2 √2 -√2 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ12 2 2 -2 0 2 2 -2 0 0 0 0 0 0 0 -√2 √2 -√2 √2 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ13 2 2 2 0 -2 -2 -2 0 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ14 2 2 2 0 -2 -2 -2 0 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ15 2 -2 0 0 2i -2i 0 0 √-2 √2 -√2 -√-2 0 0 ζ1615+ζ165 ζ1615+ζ1613 ζ1611+ζ16 ζ163+ζ16 ζ1613+ζ167 ζ1611+ζ169 ζ169+ζ163 ζ167+ζ165 complex faithful ρ16 2 -2 0 0 2i -2i 0 0 -√-2 -√2 √2 √-2 0 0 ζ1611+ζ169 ζ1611+ζ16 ζ167+ζ165 ζ1615+ζ165 ζ163+ζ16 ζ1613+ζ167 ζ1615+ζ1613 ζ169+ζ163 complex faithful ρ17 2 -2 0 0 2i -2i 0 0 √-2 √2 -√2 -√-2 0 0 ζ1613+ζ167 ζ167+ζ165 ζ169+ζ163 ζ1611+ζ169 ζ1615+ζ165 ζ163+ζ16 ζ1611+ζ16 ζ1615+ζ1613 complex faithful ρ18 2 -2 0 0 -2i 2i 0 0 -√-2 √2 -√2 √-2 0 0 ζ1611+ζ16 ζ163+ζ16 ζ1615+ζ165 ζ1615+ζ1613 ζ169+ζ163 ζ167+ζ165 ζ1613+ζ167 ζ1611+ζ169 complex faithful ρ19 2 -2 0 0 -2i 2i 0 0 -√-2 √2 -√2 √-2 0 0 ζ169+ζ163 ζ1611+ζ169 ζ1613+ζ167 ζ167+ζ165 ζ1611+ζ16 ζ1615+ζ1613 ζ1615+ζ165 ζ163+ζ16 complex faithful ρ20 2 -2 0 0 2i -2i 0 0 -√-2 -√2 √2 √-2 0 0 ζ163+ζ16 ζ169+ζ163 ζ1615+ζ1613 ζ1613+ζ167 ζ1611+ζ169 ζ1615+ζ165 ζ167+ζ165 ζ1611+ζ16 complex faithful ρ21 2 -2 0 0 -2i 2i 0 0 √-2 -√2 √2 -√-2 0 0 ζ167+ζ165 ζ1615+ζ165 ζ1611+ζ169 ζ1611+ζ16 ζ1615+ζ1613 ζ169+ζ163 ζ163+ζ16 ζ1613+ζ167 complex faithful ρ22 2 -2 0 0 -2i 2i 0 0 √-2 -√2 √2 -√-2 0 0 ζ1615+ζ1613 ζ1613+ζ167 ζ163+ζ16 ζ169+ζ163 ζ167+ζ165 ζ1611+ζ16 ζ1611+ζ169 ζ1615+ζ165 complex faithful

Smallest permutation representation of D8.C4
On 32 points
Generators in S32
```(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 8)(2 7)(3 6)(4 5)(9 13)(10 12)(14 16)(17 24)(18 23)(19 22)(20 21)(25 29)(26 28)(30 32)
(1 12 21 32 5 16 17 28)(2 11 22 31 6 15 18 27)(3 10 23 30 7 14 19 26)(4 9 24 29 8 13 20 25)```

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

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

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

D8.C4 is a maximal subgroup of
C23.20SD16  Q16.10D4  Q16.D4  D8.3D4  D8.12D4  D8.F5  Q16.F5
C4p.D8: C8○D16  D165C4  Dic12.C4  D24.1C4  C24.41D4  D40.5C4  D40.3C4  C20.58D8 ...
D8.C4 is a maximal quotient of
C4.16D16  Q161C8  C23.12SD16  C23.13SD16  C8.16Q16  C8.9C42  D8.F5  Q16.F5
C4p.D8: C8.30D8  Dic12.C4  D24.1C4  C24.41D4  D40.5C4  D40.3C4  C20.58D8  Dic28.C4 ...

Matrix representation of D8.C4 in GL2(𝔽17) generated by

 14 3 14 14
,
 14 3 3 3
,
 15 14 14 2
`G:=sub<GL(2,GF(17))| [14,14,3,14],[14,3,3,3],[15,14,14,2] >;`

D8.C4 in GAP, Magma, Sage, TeX

`D_8.C_4`
`% in TeX`

`G:=Group("D8.C4");`
`// GroupNames label`

`G:=SmallGroup(64,40);`
`// by ID`

`G=gap.SmallGroup(64,40);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,362,188,230,117,1444,730,88]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=b^2=1,c^4=a^4,b*a*b=c*a*c^-1=a^-1,c*b*c^-1=a^5*b>;`
`// generators/relations`

Export

׿
×
𝔽