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G = C16⋊3C4order 64 = 26

1st semidirect product of C16 and C4 acting via C4/C2=C2

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

Aliases: C163C4, C8.2Q8, C2.2D16, C4.1Q16, C2.2Q32, C22.10D8, C4.7(C4⋊C4), (C2×C16).3C2, C8.14(C2×C4), (C2×C4).63D4, C2.D8.2C2, C2.3(C2.D8), (C2×C8).71C22, SmallGroup(64,47)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C16⋊3C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C16 — C16⋊3C4
 Lower central C1 — C2 — C4 — C8 — C16⋊3C4
 Upper central C1 — C22 — C2×C4 — C2×C8 — C16⋊3C4
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C16⋊3C4

Generators and relations for C163C4
G = < a,b | a16=b4=1, bab-1=a-1 >

Character table of C163C4

 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 0 0 0 0 0 0 √2 √2 -√2 -√2 -ζ165+ζ163 -ζ1615+ζ169 ζ165-ζ163 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 orthogonal lifted from D16 ρ12 2 -2 2 -2 0 0 0 0 0 0 -√2 -√2 √2 √2 -ζ1615+ζ169 ζ165-ζ163 ζ1615-ζ169 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 orthogonal lifted from D16 ρ13 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 ρ14 2 -2 2 -2 0 0 0 0 0 0 √2 √2 -√2 -√2 ζ165-ζ163 ζ1615-ζ169 -ζ165+ζ163 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 orthogonal lifted from D16 ρ15 2 -2 2 -2 0 0 0 0 0 0 -√2 -√2 √2 √2 ζ1615-ζ169 -ζ165+ζ163 -ζ1615+ζ169 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 orthogonal lifted from D16 ρ16 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 ρ17 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 ρ18 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 ρ19 2 2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 -ζ1615+ζ169 -ζ165+ζ163 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 ζ1615-ζ169 ζ165-ζ163 symplectic lifted from Q32, Schur index 2 ρ20 2 2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 ζ165-ζ163 -ζ1615+ζ169 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 -ζ165+ζ163 ζ1615-ζ169 symplectic lifted from Q32, Schur index 2 ρ21 2 2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 -ζ165+ζ163 ζ1615-ζ169 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 ζ165-ζ163 -ζ1615+ζ169 symplectic lifted from Q32, Schur index 2 ρ22 2 2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 ζ1615-ζ169 ζ165-ζ163 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 -ζ1615+ζ169 -ζ165+ζ163 symplectic lifted from Q32, Schur index 2

Smallest permutation representation of C163C4
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 21 36 51)(2 20 37 50)(3 19 38 49)(4 18 39 64)(5 17 40 63)(6 32 41 62)(7 31 42 61)(8 30 43 60)(9 29 44 59)(10 28 45 58)(11 27 46 57)(12 26 47 56)(13 25 48 55)(14 24 33 54)(15 23 34 53)(16 22 35 52)```

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

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

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

C163C4 is a maximal subgroup of
D162C4  Q322C4  C23.25D8  M5(2)⋊1C4  C4×D16  C4×Q32  SD323C4  C167D4  C16.19D4  C162D4  D81Q8  C4.Q32  D8.Q8  Q16.Q8  C22.D16  C23.19D8  C23.51D8  C23.20D8  C162Q8  C16⋊Q8
C16p⋊C4: C323C4  C324C4  C485C4  C8013C4  C802C4  C1125C4 ...
C8p.Q8: C16.5Q8  C6.6D16  C40.2Q8  C8.4Dic14 ...
C163C4 is a maximal quotient of
C163C8  C8.7C42
C16p⋊C4: C323C4  C324C4  C485C4  C8013C4  C802C4  C1125C4 ...
C8p.Q8: C32.C4  C6.6D16  C40.2Q8  C8.4Dic14 ...

Matrix representation of C163C4 in GL3(𝔽17) generated by

 16 0 0 0 13 11 0 6 13
,
 4 0 0 0 0 16 0 16 0
`G:=sub<GL(3,GF(17))| [16,0,0,0,13,6,0,11,13],[4,0,0,0,0,16,0,16,0] >;`

C163C4 in GAP, Magma, Sage, TeX

`C_{16}\rtimes_3C_4`
`% in TeX`

`G:=Group("C16:3C4");`
`// GroupNames label`

`G:=SmallGroup(64,47);`
`// by ID`

`G=gap.SmallGroup(64,47);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,127,362,230,1444,88]);`
`// Polycyclic`

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

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