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G = C16⋊3C4  order 64 = 26

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

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

Aliases: C16⋊3C4, 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 C16⋊3C4
 G = < a,b | a16=b4=1, bab-1=a-1 >

C1
C2
C2
C2
C4
C4
C22
8C4
8C4
C8
C8
C2×C4
4C2×C4
4C2×C4
C16
C16
C2×C8
2C4⋊C4
2C4⋊C4
C2×C16
C2.D8
C2.D8
C16⋊3C4

Character table of C16⋊3C4

 class 12A2B2C4A4B4C4D4E4F8A8B8C8D16A16B16C16D16E16F16G16H
 size 1111228888222222222222
ρ11111111111111111111111    trivial
ρ2111111-11-111111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111111-1-1-1-1111111111111    linear of order 2
ρ41111111-11-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ51-1-11-11i-i-ii1-11-11-1-1111-1-1    linear of order 4
ρ61-1-11-11-i-iii1-11-1-111-1-1-111    linear of order 4
ρ71-1-11-11-iii-i1-11-11-1-1111-1-1    linear of order 4
ρ81-1-11-11ii-i-i1-11-1-111-1-1-111    linear of order 4
ρ92222220000-2-2-2-200000000    orthogonal lifted from D4
ρ102222-2-200000000-√2√2-√2√2-√2√2-√2√2    orthogonal lifted from D8
ρ112-22-2000000√2√2-√2-√2-ζ165+ζ163-ζ1615+ζ169ζ165-ζ163ζ1615-ζ169ζ165-ζ163-ζ1615+ζ169-ζ165+ζ163ζ1615-ζ169    orthogonal lifted from D16
ρ122-22-2000000-√2-√2√2√2-ζ1615+ζ169ζ165-ζ163ζ1615-ζ169-ζ165+ζ163ζ1615-ζ169ζ165-ζ163-ζ1615+ζ169-ζ165+ζ163    orthogonal lifted from D16
ρ132222-2-200000000√2-√2√2-√2√2-√2√2-√2    orthogonal lifted from D8
ρ142-22-2000000√2√2-√2-√2ζ165-ζ163ζ1615-ζ169-ζ165+ζ163-ζ1615+ζ169-ζ165+ζ163ζ1615-ζ169ζ165-ζ163-ζ1615+ζ169    orthogonal lifted from D16
ρ152-22-2000000-√2-√2√2√2ζ1615-ζ169-ζ165+ζ163-ζ1615+ζ169ζ165-ζ163-ζ1615+ζ169-ζ165+ζ163ζ1615-ζ169ζ165-ζ163    orthogonal lifted from D16
ρ162-2-222-200000000-√2-√2√2√2-√2√2√2-√2    symplectic lifted from Q16, Schur index 2
ρ172-2-222-200000000√2√2-√2-√2√2-√2-√2√2    symplectic lifted from Q16, Schur index 2
ρ182-2-22-220000-22-2200000000    symplectic lifted from Q8, Schur index 2
ρ1922-2-2000000-√2√2√2-√2-ζ1615+ζ169-ζ165+ζ163-ζ1615+ζ169-ζ165+ζ163ζ1615-ζ169ζ165-ζ163ζ1615-ζ169ζ165-ζ163    symplectic lifted from Q32, Schur index 2
ρ2022-2-2000000√2-√2-√2√2ζ165-ζ163-ζ1615+ζ169ζ165-ζ163-ζ1615+ζ169-ζ165+ζ163ζ1615-ζ169-ζ165+ζ163ζ1615-ζ169    symplectic lifted from Q32, Schur index 2
ρ2122-2-2000000√2-√2-√2√2-ζ165+ζ163ζ1615-ζ169-ζ165+ζ163ζ1615-ζ169ζ165-ζ163-ζ1615+ζ169ζ165-ζ163-ζ1615+ζ169    symplectic lifted from Q32, Schur index 2
ρ2222-2-2000000-√2√2√2-√2ζ1615-ζ169ζ165-ζ163ζ1615-ζ169ζ165-ζ163-ζ1615+ζ169-ζ165+ζ163-ζ1615+ζ169-ζ165+ζ163    symplectic lifted from Q32, Schur index 2

Smallest permutation representation of C16⋊3C4
►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)]])
 

C16⋊3C4 is a maximal subgroup of
 D16⋊2C4  Q32⋊2C4  C23.25D8  M5(2)⋊1C4  C4×D16  C4×Q32  SD32⋊3C4  C16⋊7D4  C16.19D4  C16⋊2D4  D8⋊1Q8  C4.Q32  D8.Q8  Q16.Q8  C22.D16  C23.19D8  C23.51D8  C23.20D8  C16⋊2Q8  C16⋊Q8
 C16p⋊C4: C32⋊3C4  C32⋊4C4  C48⋊5C4  C80⋊13C4  C80⋊2C4  C112⋊5C4 ...
 C8p.Q8: C16.5Q8  C6.6D16  C40.2Q8  C8.4Dic14 ...
C16⋊3C4 is a maximal quotient of
 C16⋊3C8  C8.7C42
 C16p⋊C4: C32⋊3C4  C32⋊4C4  C48⋊5C4  C80⋊13C4  C80⋊2C4  C112⋊5C4 ...
 C8p.Q8: C32.C4  C6.6D16  C40.2Q8  C8.4Dic14 ...

Matrix representation of C16⋊3C4 ►in GL3(𝔽17) generated by

1600
01311
0613
,
400
0016
0160
G:=sub<GL(3,GF(17))| [16,0,0,0,13,6,0,11,13],[4,0,0,0,0,16,0,16,0] >;
 

C16⋊3C4 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
 

Export

Subgroup lattice of C16⋊3C4 in TeX
Character table of C16⋊3C4 in TeX

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