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## G = C42.78C22order 64 = 26

### 21st non-split extension by C42 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 — C2×C4 — C42.78C22
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C42.78C22
 Lower central C1 — C2 — C2×C4 — C42.78C22
 Upper central C1 — C22 — C42 — C42.78C22
 Jennings C1 — C2 — C2 — C2×C4 — C42.78C22

Generators and relations for C42.78C22
G = < a,b,c,d | a4=b4=c2=1, d2=b, ab=ba, cac=a-1b2, ad=da, cbc=b-1, bd=db, dcd-1=a2bc >

Character table of C42.78C22

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 8 2 2 2 2 2 2 8 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 -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 2 2 2 2 0 2 2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 -2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 0 0 2 0 -2 0 0 0 0 0 2i -2i 2i 0 -2i 0 0 complex lifted from C4○D4 ρ12 2 -2 2 -2 0 0 0 -2 0 2 0 0 0 0 -2i 0 0 0 -2i 0 2i 2i complex lifted from C4○D4 ρ13 2 -2 2 -2 0 0 0 2 0 -2 0 0 0 0 0 -2i 2i -2i 0 2i 0 0 complex lifted from C4○D4 ρ14 2 -2 2 -2 0 0 0 -2 0 2 0 0 0 0 2i 0 0 0 2i 0 -2i -2i complex lifted from C4○D4 ρ15 2 -2 -2 2 0 -2i 2i 0 0 0 0 0 0 0 -√2 -√-2 √-2 √-2 √2 -√-2 -√2 √2 complex lifted from C4○D8 ρ16 2 -2 -2 2 0 -2i 2i 0 0 0 0 0 0 0 √2 √-2 -√-2 -√-2 -√2 √-2 √2 -√2 complex lifted from C4○D8 ρ17 2 -2 -2 2 0 2i -2i 0 0 0 0 0 0 0 -√2 √-2 -√-2 -√-2 √2 √-2 -√2 √2 complex lifted from C4○D8 ρ18 2 2 -2 -2 0 0 0 0 -2i 0 2i 0 0 0 -√-2 -√2 -√2 √2 √-2 √2 √-2 -√-2 complex lifted from C4○D8 ρ19 2 2 -2 -2 0 0 0 0 2i 0 -2i 0 0 0 √-2 -√2 -√2 √2 -√-2 √2 -√-2 √-2 complex lifted from C4○D8 ρ20 2 -2 -2 2 0 2i -2i 0 0 0 0 0 0 0 √2 -√-2 √-2 √-2 -√2 -√-2 √2 -√2 complex lifted from C4○D8 ρ21 2 2 -2 -2 0 0 0 0 2i 0 -2i 0 0 0 -√-2 √2 √2 -√2 √-2 -√2 √-2 -√-2 complex lifted from C4○D8 ρ22 2 2 -2 -2 0 0 0 0 -2i 0 2i 0 0 0 √-2 √2 √2 -√2 -√-2 -√2 -√-2 √-2 complex lifted from C4○D8

Smallest permutation representation of C42.78C22
On 32 points
Generators in S32
```(1 19 25 15)(2 20 26 16)(3 21 27 9)(4 22 28 10)(5 23 29 11)(6 24 30 12)(7 17 31 13)(8 18 32 14)
(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 21)(10 16)(11 19)(12 14)(13 17)(15 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)| (1,19,25,15)(2,20,26,16)(3,21,27,9)(4,22,28,10)(5,23,29,11)(6,24,30,12)(7,17,31,13)(8,18,32,14), (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)(9,21)(10,16)(11,19)(12,14)(13,17)(15,23)(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( (1,19,25,15)(2,20,26,16)(3,21,27,9)(4,22,28,10)(5,23,29,11)(6,24,30,12)(7,17,31,13)(8,18,32,14), (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)(9,21)(10,16)(11,19)(12,14)(13,17)(15,23)(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([[(1,19,25,15),(2,20,26,16),(3,21,27,9),(4,22,28,10),(5,23,29,11),(6,24,30,12),(7,17,31,13),(8,18,32,14)], [(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),(9,21),(10,16),(11,19),(12,14),(13,17),(15,23),(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)]])`

C42.78C22 is a maximal subgroup of
(C4×C8)⋊C4
C42.D2p: C42.355D4  C42.242D4  C42.244D4  C42.308D4  C42.260D4  C42.269D4  C42.270D4  C42.284D4 ...
C4⋊C4.D2p: C42.366C23  C42.367C23  C42.390C23  C42.391C23  C42.406C23  C42.407C23  C42.408C23  C42.409C23 ...
C42.78C22 is a maximal quotient of
C42.56Q8  C2.(C88D4)  C2.(C87D4)  C428C4⋊C2  (C2×Q8).109D4  C4⋊C4.Q8
C42.D2p: C42.433D4  C42.437D4  C42.264D6  C42.213D6  C42.216D6  C42.264D10  C42.213D10  C42.216D10 ...
C4⋊C4.D2p: C4⋊C4.84D4  C4⋊C4.85D4  C4⋊C4.94D4  (C2×C8).200D6  Q8⋊C4⋊S3  (C8×Dic5)⋊C2  Q8⋊Dic5⋊C2  (C8×Dic7)⋊C2 ...

Matrix representation of C42.78C22 in GL4(𝔽17) generated by

 0 13 0 0 13 0 0 0 0 0 4 9 0 0 4 13
,
 1 0 0 0 0 1 0 0 0 0 16 2 0 0 16 1
,
 1 0 0 0 0 16 0 0 0 0 1 0 0 0 1 16
,
 0 1 0 0 1 0 0 0 0 0 0 11 0 0 3 11
`G:=sub<GL(4,GF(17))| [0,13,0,0,13,0,0,0,0,0,4,4,0,0,9,13],[1,0,0,0,0,1,0,0,0,0,16,16,0,0,2,1],[1,0,0,0,0,16,0,0,0,0,1,1,0,0,0,16],[0,1,0,0,1,0,0,0,0,0,0,3,0,0,11,11] >;`

C42.78C22 in GAP, Magma, Sage, TeX

`C_4^2._{78}C_2^2`
`% in TeX`

`G:=Group("C4^2.78C2^2");`
`// GroupNames label`

`G:=SmallGroup(64,169);`
`// by ID`

`G=gap.SmallGroup(64,169);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,121,295,362,50,963,117,1444,88]);`
`// 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,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=a^2*b*c>;`
`// generators/relations`

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