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## G = C42.28C22order 64 = 26

### 28th non-split extension by C42 of C22 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.28C22
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C8⋊C4 — C42.28C22
 Lower central C1 — C2 — C2×C4 — C42.28C22
 Upper central C1 — C22 — C42 — C42.28C22
 Jennings C1 — C2 — C2 — C2×C4 — C42.28C22

Generators and relations for C42.28C22
G = < a,b,c,d | a4=b4=c2=1, d2=b, ab=ba, cac=a-1b2, dad-1=ab2, cbc=b-1, bd=db, dcd-1=a2b-1c >

Character table of C42.28C22

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D size 1 1 1 1 8 2 2 4 4 8 8 8 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 2 2 2 0 -2 -2 -2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 -2 -2 2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 -2 2 0 0 0 0 0 2i 0 -2i 0 complex lifted from C4○D4 ρ12 2 -2 2 -2 0 2 -2 0 0 0 0 0 0 -2i 0 2i complex lifted from C4○D4 ρ13 2 -2 2 -2 0 2 -2 0 0 0 0 0 0 2i 0 -2i complex lifted from C4○D4 ρ14 2 -2 2 -2 0 -2 2 0 0 0 0 0 -2i 0 2i 0 complex lifted from C4○D4 ρ15 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ16 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C42.28C22
On 32 points
Generators in S32
```(1 20 25 15)(2 17 26 12)(3 22 27 9)(4 19 28 14)(5 24 29 11)(6 21 30 16)(7 18 31 13)(8 23 32 10)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)
(2 28)(3 7)(4 26)(6 32)(8 30)(9 22)(10 12)(11 20)(13 18)(14 16)(15 24)(17 23)(19 21)(27 31)
(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)```

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

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

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

Matrix representation of C42.28C22 in GL6(𝔽17)

 16 9 0 0 0 0 13 1 0 0 0 0 0 0 8 3 13 13 0 0 14 9 0 4 0 0 5 5 12 7 0 0 0 12 14 5
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 4 4 16 15 0 0 13 0 1 1
,
 1 0 0 0 0 0 4 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 4 4 16 15 0 0 0 13 0 1
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 2 15 5 0 0 0 5 5 12 7 0 0 8 3 13 13 0 0 3 0 2 14

`G:=sub<GL(6,GF(17))| [16,13,0,0,0,0,9,1,0,0,0,0,0,0,8,14,5,0,0,0,3,9,5,12,0,0,13,0,12,14,0,0,13,4,7,5],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,4,13,0,0,1,0,4,0,0,0,0,0,16,1,0,0,0,0,15,1],[1,4,0,0,0,0,0,16,0,0,0,0,0,0,1,0,4,0,0,0,0,16,4,13,0,0,0,0,16,0,0,0,0,0,15,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,5,8,3,0,0,15,5,3,0,0,0,5,12,13,2,0,0,0,7,13,14] >;`

C42.28C22 in GAP, Magma, Sage, TeX

`C_4^2._{28}C_2^2`
`% in TeX`

`G:=Group("C4^2.28C2^2");`
`// GroupNames label`

`G:=SmallGroup(64,170);`
`// by ID`

`G=gap.SmallGroup(64,170);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,121,103,362,332,50,963,117,1444,88]);`
`// Polycyclic`

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

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