Copied to
clipboard

## G = M4(2).8C22order 64 = 26

### 3rd non-split extension by M4(2) of C22 acting via C22/C2=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — M4(2).8C22
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — M4(2).8C22
 Lower central C1 — C2 — C22 — M4(2).8C22
 Upper central C1 — C4 — C22×C4 — M4(2).8C22
 Jennings C1 — C2 — C2 — C2×C4 — M4(2).8C22

Generators and relations for M4(2).8C22
G = < a,b,c,d | a8=b2=c2=1, d2=a2, bab=dad-1=a5, cac=ab, bc=cb, dbd-1=a4b, dcd-1=a4bc >

Subgroups: 121 in 75 conjugacy classes, 39 normal (13 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4.D4, C4.10D4, C2×M4(2), C2×C4○D4, M4(2).8C22
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, M4(2).8C22

Character table of M4(2).8C22

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 2 2 2 4 4 1 1 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 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 -1 -1 -1 1 1 -1 1 -1 1 -1 linear of order 2 ρ6 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 ρ7 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 ρ8 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 ρ9 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -1 i -i -i i i -i i -i linear of order 4 ρ10 1 1 -1 -1 1 -1 1 1 1 -1 1 -1 1 -1 -i -i i i i i -i -i linear of order 4 ρ11 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -1 -i i i -i -i i -i i linear of order 4 ρ12 1 1 -1 -1 1 -1 1 1 1 -1 1 -1 1 -1 i i -i -i -i -i i i linear of order 4 ρ13 1 1 -1 -1 1 1 -1 1 1 -1 1 -1 -1 1 -i -i -i -i i i i i linear of order 4 ρ14 1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 1 i -i i -i i -i -i i linear of order 4 ρ15 1 1 -1 -1 1 1 -1 1 1 -1 1 -1 -1 1 i i i i -i -i -i -i linear of order 4 ρ16 1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 1 -i i -i i -i i i -i linear of order 4 ρ17 2 2 -2 2 -2 0 0 2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 -2 -2 0 0 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 -2 -2 0 0 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 2 -2 0 0 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 -4 0 0 0 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ22 4 -4 0 0 0 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of M4(2).8C22
On 16 points - transitive group 16T71
Generators in S16
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 6)(4 8)(10 14)(12 16)
(1 11)(2 16)(3 9)(4 14)(5 15)(6 12)(7 13)(8 10)
(1 8 3 2 5 4 7 6)(9 12 11 14 13 16 15 10)```

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

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

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

`G:=TransitiveGroup(16,71);`

On 16 points - transitive group 16T86
Generators in S16
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 6)(4 8)(10 14)(12 16)
(1 7)(2 4)(3 5)(6 8)(9 15)(10 12)(11 13)(14 16)
(1 10 3 12 5 14 7 16)(2 15 4 9 6 11 8 13)```

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

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

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

`G:=TransitiveGroup(16,86);`

Matrix representation of M4(2).8C22 in GL4(𝔽5) generated by

 0 1 0 0 0 0 2 0 0 0 0 1 2 0 0 0
,
 1 0 0 0 0 4 0 0 0 0 1 0 0 0 0 4
,
 4 0 0 0 0 4 0 0 0 0 1 0 0 0 0 1
,
 0 3 0 0 0 0 4 0 0 0 0 3 4 0 0 0
`G:=sub<GL(4,GF(5))| [0,0,0,2,1,0,0,0,0,2,0,0,0,0,1,0],[1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[0,0,0,4,3,0,0,0,0,4,0,0,0,0,3,0] >;`

M4(2).8C22 in GAP, Magma, Sage, TeX

`M_4(2)._8C_2^2`
`% in TeX`

`G:=Group("M4(2).8C2^2");`
`// GroupNames label`

`G:=SmallGroup(64,94);`
`// by ID`

`G=gap.SmallGroup(64,94);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,158,963,730,88]);`
`// Polycyclic`

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

Export

׿
×
𝔽