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## G = C42.29C22order 64 = 26

### 29th non-split extension by C42 of C22 acting faithfully

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

Aliases: C42.29C22, C8⋊C411C2, (C2×C4).42D4, C41D4.5C2, C42.C23C2, D4⋊C419C2, C4.17(C4○D4), C4⋊C4.20C22, (C2×C8).55C22, C2.21(C8⋊C22), (C2×C4).115C23, (C2×D4).27C22, C22.111(C2×D4), C2.13(C4.4D4), SmallGroup(64,171)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.29C22
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C8⋊C4 — C42.29C22
 Lower central C1 — C2 — C2×C4 — C42.29C22
 Upper central C1 — C22 — C42 — C42.29C22
 Jennings C1 — C2 — C2 — C2×C4 — C42.29C22

Generators and relations for C42.29C22
G = < a,b,c,d | a4=b4=c2=1, d2=b, ab=ba, cac=a-1, dad-1=ab2, cbc=b-1, bd=db, dcd-1=a2b-1c >

Character table of C42.29C22

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D size 1 1 1 1 8 8 2 2 4 4 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 0 -2 -2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 -2 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 0 -2 2 0 0 0 0 -2i 0 2i 0 complex lifted from C4○D4 ρ12 2 -2 2 -2 0 0 2 -2 0 0 0 0 0 -2i 0 2i complex lifted from C4○D4 ρ13 2 -2 2 -2 0 0 2 -2 0 0 0 0 0 2i 0 -2i complex lifted from C4○D4 ρ14 2 -2 2 -2 0 0 -2 2 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 orthogonal lifted from C8⋊C22

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

C42.29C22 is a maximal subgroup of
C4.C4≀C2  C8⋊C45C4
C42.D2p: C42.239D4  C42.240D4  C42.244D4  C42.255D4  C42.257D4  C42.275D4  C42.277D4  C42.286D4 ...
C4⋊C4.D2p: C42.2C23  C42.3C23  C42.4C23  C42.366C23  C42.386C23  C42.388C23  C42.406C23  C42.408C23 ...
C42.29C22 is a maximal quotient of
C42.24Q8  C2.(C82D4)  C4⋊C47D4  C428C4⋊C2  (C2×C4).23Q16
C4⋊C4.D2p: C4⋊C4.84D4  C4⋊C4.D6  C42.70D6  C4⋊C4.D10  C42.70D10  C4⋊C4.D14  C42.70D14 ...
C42.D2p: C42.112D4  C42.124D4  C42.19D6  C42.72D6  C42.19D10  C42.72D10  C42.19D14  C42.72D14 ...

Matrix representation of C42.29C22 in GL6(𝔽17)

 1 8 0 0 0 0 4 16 0 0 0 0 0 0 0 6 5 12 0 0 11 0 5 5 0 0 12 12 0 11 0 0 5 12 6 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0
,
 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 0 0 0 1 0 0 0 0 1 0
,
 13 2 0 0 0 0 1 4 0 0 0 0 0 0 12 5 11 0 0 0 12 12 0 11 0 0 0 6 5 12 0 0 11 0 5 5

`G:=sub<GL(6,GF(17))| [1,4,0,0,0,0,8,16,0,0,0,0,0,0,0,11,12,5,0,0,6,0,12,12,0,0,5,5,0,6,0,0,12,5,11,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[1,4,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[13,1,0,0,0,0,2,4,0,0,0,0,0,0,12,12,0,11,0,0,5,12,6,0,0,0,11,0,5,5,0,0,0,11,12,5] >;`

C42.29C22 in GAP, Magma, Sage, TeX

`C_4^2._{29}C_2^2`
`% in TeX`

`G:=Group("C4^2.29C2^2");`
`// GroupNames label`

`G:=SmallGroup(64,171);`
`// by ID`

`G=gap.SmallGroup(64,171);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,121,295,362,332,50,963,117,1444,88]);`
`// 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,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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