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G = C164C4order 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

C1C8 — C164C4
C1C2C4C2×C4C2×C8C2×C16 — C164C4
C1C2C4C8 — C164C4
C1C22C2×C4C2×C8 — C164C4
C1C2C2C2C2C4C4C2×C8 — C164C4

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

8C4
8C4
4C2×C4
4C2×C4
2C4⋊C4
2C4⋊C4

Character table of C164C4

 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
ρ112222-2-2000000002-22-22-22-2    orthogonal lifted from D8
ρ122-2-222-200000000-2-222-222-2    symplectic lifted from Q16, Schur index 2
ρ132-2-222-20000000022-2-22-2-22    symplectic lifted from Q16, Schur index 2
ρ142-2-22-220000-22-2200000000    symplectic lifted from Q8, Schur index 2
ρ1522-2-2000000-222-2ζ16131611ζ16716ζ16131611ζ16716ζ165163ζ1615169ζ165163ζ1615169    complex lifted from SD32
ρ1622-2-2000000-222-2ζ165163ζ1615169ζ165163ζ1615169ζ16131611ζ16716ζ16131611ζ16716    complex lifted from SD32
ρ172-22-2000000-2-222ζ16131611ζ1615169ζ165163ζ16716ζ165163ζ1615169ζ16131611ζ16716    complex lifted from SD32
ρ182-22-2000000-2-222ζ165163ζ16716ζ16131611ζ1615169ζ16131611ζ16716ζ165163ζ1615169    complex lifted from SD32
ρ192-22-200000022-2-2ζ16716ζ16131611ζ1615169ζ165163ζ1615169ζ16131611ζ16716ζ165163    complex lifted from SD32
ρ202-22-200000022-2-2ζ1615169ζ165163ζ16716ζ16131611ζ16716ζ165163ζ1615169ζ16131611    complex lifted from SD32
ρ2122-2-20000002-2-22ζ16716ζ165163ζ16716ζ165163ζ1615169ζ16131611ζ1615169ζ16131611    complex lifted from SD32
ρ2222-2-20000002-2-22ζ1615169ζ16131611ζ1615169ζ16131611ζ16716ζ165163ζ16716ζ165163    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

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] >;

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

Export

Subgroup lattice of C164C4 in TeX
Character table of C164C4 in TeX

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