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## G = C16⋊4C4order 64 = 26

### 2nd semidirect product of C16 and C4 acting via C4/C2=C2

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

Aliases: C164C4, C8.3Q8, C4.2Q16, C2.3SD32, C22.11D8, C4.8(C4⋊C4), (C2×C16).6C2, C8.15(C2×C4), (C2×C4).64D4, C2.D8.3C2, C2.4(C2.D8), (C2×C8).72C22, SmallGroup(64,48)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C16⋊4C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C16 — C16⋊4C4
 Lower central C1 — C2 — C4 — C8 — C16⋊4C4
 Upper central C1 — C22 — C2×C4 — C2×C8 — C16⋊4C4
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C16⋊4C4

Generators and relations for C164C4
G = < a,b | a16=b4=1, bab-1=a7 >

Character table of C164C4

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 2 2 8 8 8 8 2 2 2 2 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 i -i -i i 1 -1 1 -1 1 -1 -1 1 1 1 -1 -1 linear of order 4 ρ6 1 -1 -1 1 -1 1 -i -i i i 1 -1 1 -1 -1 1 1 -1 -1 -1 1 1 linear of order 4 ρ7 1 -1 -1 1 -1 1 -i i i -i 1 -1 1 -1 1 -1 -1 1 1 1 -1 -1 linear of order 4 ρ8 1 -1 -1 1 -1 1 i i -i -i 1 -1 1 -1 -1 1 1 -1 -1 -1 1 1 linear of order 4 ρ9 2 2 2 2 2 2 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ11 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ12 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 -√2 -√2 √2 √2 -√2 √2 √2 -√2 symplectic lifted from Q16, Schur index 2 ρ13 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 √2 -√2 -√2 √2 symplectic lifted from Q16, Schur index 2 ρ14 2 -2 -2 2 -2 2 0 0 0 0 -2 2 -2 2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ15 2 2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 ζ1613+ζ1611 ζ167+ζ16 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 ζ165+ζ163 ζ1615+ζ169 complex lifted from SD32 ρ16 2 2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 ζ165+ζ163 ζ1615+ζ169 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 ζ1613+ζ1611 ζ167+ζ16 complex lifted from SD32 ρ17 2 -2 2 -2 0 0 0 0 0 0 -√2 -√2 √2 √2 ζ1613+ζ1611 ζ1615+ζ169 ζ165+ζ163 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 complex lifted from SD32 ρ18 2 -2 2 -2 0 0 0 0 0 0 -√2 -√2 √2 √2 ζ165+ζ163 ζ167+ζ16 ζ1613+ζ1611 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 complex lifted from SD32 ρ19 2 -2 2 -2 0 0 0 0 0 0 √2 √2 -√2 -√2 ζ167+ζ16 ζ1613+ζ1611 ζ1615+ζ169 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 complex lifted from SD32 ρ20 2 -2 2 -2 0 0 0 0 0 0 √2 √2 -√2 -√2 ζ1615+ζ169 ζ165+ζ163 ζ167+ζ16 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 complex lifted from SD32 ρ21 2 2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 ζ167+ζ16 ζ165+ζ163 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 ζ1615+ζ169 ζ1613+ζ1611 complex lifted from SD32 ρ22 2 2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 ζ1615+ζ169 ζ1613+ζ1611 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 ζ167+ζ16 ζ165+ζ163 complex lifted from SD32

Smallest permutation representation of C164C4
Regular action on 64 points
Generators in S64
```(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)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)
(1 59 48 26)(2 50 33 17)(3 57 34 24)(4 64 35 31)(5 55 36 22)(6 62 37 29)(7 53 38 20)(8 60 39 27)(9 51 40 18)(10 58 41 25)(11 49 42 32)(12 56 43 23)(13 63 44 30)(14 54 45 21)(15 61 46 28)(16 52 47 19)```

`G:=sub<Sym(64)| (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,59,48,26)(2,50,33,17)(3,57,34,24)(4,64,35,31)(5,55,36,22)(6,62,37,29)(7,53,38,20)(8,60,39,27)(9,51,40,18)(10,58,41,25)(11,49,42,32)(12,56,43,23)(13,63,44,30)(14,54,45,21)(15,61,46,28)(16,52,47,19)>;`

`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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,59,48,26)(2,50,33,17)(3,57,34,24)(4,64,35,31)(5,55,36,22)(6,62,37,29)(7,53,38,20)(8,60,39,27)(9,51,40,18)(10,58,41,25)(11,49,42,32)(12,56,43,23)(13,63,44,30)(14,54,45,21)(15,61,46,28)(16,52,47,19) );`

`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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)], [(1,59,48,26),(2,50,33,17),(3,57,34,24),(4,64,35,31),(5,55,36,22),(6,62,37,29),(7,53,38,20),(8,60,39,27),(9,51,40,18),(10,58,41,25),(11,49,42,32),(12,56,43,23),(13,63,44,30),(14,54,45,21),(15,61,46,28),(16,52,47,19)])`

C164C4 is a maximal subgroup of
D163C4  C23.25D8  M5(2)⋊1C4  C4×SD32  Q324C4  D164C4  C168D4  C16⋊D4  C16.D4  Q16⋊Q8  D8⋊Q8  D8.Q8  Q16.Q8  C23.49D8  C23.19D8  C23.50D8  C23.20D8  C163Q8  C16⋊Q8
C16p⋊C4: C8.Q16  C486C4  C8014C4  C803C4  C1126C4 ...
C8p.Q8: C16.5Q8  C6.SD32  C10.SD32  C8.5Dic14 ...
C164C4 is a maximal quotient of
C8.7C42
C16p⋊C4: C8.Q16  C486C4  C8014C4  C803C4  C1126C4 ...
C2p.SD32: C164C8  C6.SD32  C10.SD32  C8.5Dic14 ...

Matrix representation of C164C4 in GL3(𝔽17) generated by

 1 0 0 0 8 15 0 1 6
,
 4 0 0 0 13 8 0 0 4
`G:=sub<GL(3,GF(17))| [1,0,0,0,8,1,0,15,6],[4,0,0,0,13,0,0,8,4] >;`

C164C4 in GAP, Magma, Sage, TeX

`C_{16}\rtimes_4C_4`
`% in TeX`

`G:=Group("C16:4C4");`
`// GroupNames label`

`G:=SmallGroup(64,48);`
`// by ID`

`G=gap.SmallGroup(64,48);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,319,362,230,1444,88]);`
`// Polycyclic`

`G:=Group<a,b|a^16=b^4=1,b*a*b^-1=a^7>;`
`// generators/relations`

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