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G = C163C4order 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

C1C8 — C163C4
C1C2C4C2×C4C2×C8C2×C16 — C163C4
C1C2C4C8 — C163C4
C1C22C2×C4C2×C8 — C163C4
C1C2C2C2C2C4C4C2×C8 — C163C4

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

8C4
8C4
4C2×C4
4C2×C4
2C4⋊C4
2C4⋊C4

Character table of C163C4

 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
ρ102222-2-200000000-22-22-22-22    orthogonal lifted from D8
ρ112-22-200000022-2-21651631615169ζ165163ζ1615169ζ1651631615169165163ζ1615169    orthogonal lifted from D16
ρ122-22-2000000-2-2221615169ζ165163ζ1615169165163ζ1615169ζ1651631615169165163    orthogonal lifted from D16
ρ132222-2-2000000002-22-22-22-2    orthogonal lifted from D8
ρ142-22-200000022-2-2ζ165163ζ16151691651631615169165163ζ1615169ζ1651631615169    orthogonal lifted from D16
ρ152-22-2000000-2-222ζ16151691651631615169ζ1651631615169165163ζ1615169ζ165163    orthogonal lifted from D16
ρ162-2-222-200000000-2-222-222-2    symplectic lifted from Q16, Schur index 2
ρ172-2-222-20000000022-2-22-2-22    symplectic lifted from Q16, Schur index 2
ρ182-2-22-220000-22-2200000000    symplectic lifted from Q8, Schur index 2
ρ1922-2-2000000-222-216151691651631615169165163ζ1615169ζ165163ζ1615169ζ165163    symplectic lifted from Q32, Schur index 2
ρ2022-2-20000002-2-22ζ1651631615169ζ1651631615169165163ζ1615169165163ζ1615169    symplectic lifted from Q32, Schur index 2
ρ2122-2-20000002-2-22165163ζ1615169165163ζ1615169ζ1651631615169ζ1651631615169    symplectic lifted from Q32, Schur index 2
ρ2222-2-2000000-222-2ζ1615169ζ165163ζ1615169ζ16516316151691651631615169165163    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 56 17 42)(2 55 18 41)(3 54 19 40)(4 53 20 39)(5 52 21 38)(6 51 22 37)(7 50 23 36)(8 49 24 35)(9 64 25 34)(10 63 26 33)(11 62 27 48)(12 61 28 47)(13 60 29 46)(14 59 30 45)(15 58 31 44)(16 57 32 43)

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

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

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

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

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] >;

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

Export

Subgroup lattice of C163C4 in TeX
Character table of C163C4 in TeX

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