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G = C16⋊4C4  order 64 = 26

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

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

Aliases: C16⋊4C4, 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 C16⋊4C4
 G = < a,b | a16=b4=1, bab-1=a7 >

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⋊4C4

Character table of C16⋊4C4

 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
ρ112222-2-200000000√2-√2√2-√2√2-√2√2-√2    orthogonal lifted from D8
ρ122-2-222-200000000-√2-√2√2√2-√2√2√2-√2    symplectic lifted from Q16, Schur index 2
ρ132-2-222-200000000√2√2-√2-√2√2-√2-√2√2    symplectic lifted from Q16, Schur index 2
ρ142-2-22-220000-22-2200000000    symplectic lifted from Q8, Schur index 2
ρ1522-2-2000000-√2√2√2-√2ζ1613+ζ1611ζ167+ζ16ζ1613+ζ1611ζ167+ζ16ζ165+ζ163ζ1615+ζ169ζ165+ζ163ζ1615+ζ169    complex lifted from SD32
ρ1622-2-2000000-√2√2√2-√2ζ165+ζ163ζ1615+ζ169ζ165+ζ163ζ1615+ζ169ζ1613+ζ1611ζ167+ζ16ζ1613+ζ1611ζ167+ζ16    complex lifted from SD32
ρ172-22-2000000-√2-√2√2√2ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163ζ167+ζ16ζ165+ζ163ζ1615+ζ169ζ1613+ζ1611ζ167+ζ16    complex lifted from SD32
ρ182-22-2000000-√2-√2√2√2ζ165+ζ163ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169ζ1613+ζ1611ζ167+ζ16ζ165+ζ163ζ1615+ζ169    complex lifted from SD32
ρ192-22-2000000√2√2-√2-√2ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163ζ1615+ζ169ζ1613+ζ1611ζ167+ζ16ζ165+ζ163    complex lifted from SD32
ρ202-22-2000000√2√2-√2-√2ζ1615+ζ169ζ165+ζ163ζ167+ζ16ζ1613+ζ1611ζ167+ζ16ζ165+ζ163ζ1615+ζ169ζ1613+ζ1611    complex lifted from SD32
ρ2122-2-2000000√2-√2-√2√2ζ167+ζ16ζ165+ζ163ζ167+ζ16ζ165+ζ163ζ1615+ζ169ζ1613+ζ1611ζ1615+ζ169ζ1613+ζ1611    complex lifted from SD32
ρ2222-2-2000000√2-√2-√2√2ζ1615+ζ169ζ1613+ζ1611ζ1615+ζ169ζ1613+ζ1611ζ167+ζ16ζ165+ζ163ζ167+ζ16ζ165+ζ163    complex lifted from SD32

Smallest permutation representation of C16⋊4C4
►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 45 32 58)(2 36 17 49)(3 43 18 56)(4 34 19 63)(5 41 20 54)(6 48 21 61)(7 39 22 52)(8 46 23 59)(9 37 24 50)(10 44 25 57)(11 35 26 64)(12 42 27 55)(13 33 28 62)(14 40 29 53)(15 47 30 60)(16 38 31 51)
 
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), (1,45,32,58)(2,36,17,49)(3,43,18,56)(4,34,19,63)(5,41,20,54)(6,48,21,61)(7,39,22,52)(8,46,23,59)(9,37,24,50)(10,44,25,57)(11,35,26,64)(12,42,27,55)(13,33,28,62)(14,40,29,53)(15,47,30,60)(16,38,31,51)>;
 
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), (1,45,32,58)(2,36,17,49)(3,43,18,56)(4,34,19,63)(5,41,20,54)(6,48,21,61)(7,39,22,52)(8,46,23,59)(9,37,24,50)(10,44,25,57)(11,35,26,64)(12,42,27,55)(13,33,28,62)(14,40,29,53)(15,47,30,60)(16,38,31,51) );
 
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)], [(1,45,32,58),(2,36,17,49),(3,43,18,56),(4,34,19,63),(5,41,20,54),(6,48,21,61),(7,39,22,52),(8,46,23,59),(9,37,24,50),(10,44,25,57),(11,35,26,64),(12,42,27,55),(13,33,28,62),(14,40,29,53),(15,47,30,60),(16,38,31,51)]])
 

C16⋊4C4 is a maximal subgroup of
 D16⋊3C4  C23.25D8  M5(2)⋊1C4  C4×SD32  Q32⋊4C4  D16⋊4C4  C16⋊8D4  C16⋊D4  C16.D4  Q16⋊Q8  D8⋊Q8  D8.Q8  Q16.Q8  C23.49D8  C23.19D8  C23.50D8  C23.20D8  C16⋊3Q8  C16⋊Q8
 C16p⋊C4: C8.Q16  C48⋊6C4  C80⋊14C4  C80⋊3C4  C112⋊6C4 ...
 C8p.Q8: C16.5Q8  C6.SD32  C10.SD32  C8.5Dic14 ...
C16⋊4C4 is a maximal quotient of
 C8.7C42
 C16p⋊C4: C8.Q16  C48⋊6C4  C80⋊14C4  C80⋊3C4  C112⋊6C4 ...
 C2p.SD32: C16⋊4C8  C6.SD32  C10.SD32  C8.5Dic14 ...

Matrix representation of C16⋊4C4 ►in GL3(𝔽17) generated by

100
0815
016
,
400
0138
004
G:=sub<GL(3,GF(17))| [1,0,0,0,8,1,0,15,6],[4,0,0,0,13,0,0,8,4] >;
 

C16⋊4C4 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
 

Export

Subgroup lattice of C16⋊4C4 in TeX
Character table of C16⋊4C4 in TeX

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