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G = C81C8order 64 = 26

1st semidirect product of C8 and C8 acting via C8/C4=C2

p-group, metacyclic, nilpotent (class 3), monomial

Aliases: C81C8, C4.17D8, C4.9Q16, C4.5M4(2), C42.65C22, C4⋊C8.5C2, C4.7(C2×C8), (C2×C8).8C4, (C4×C8).5C2, C2.4(C4⋊C8), (C2×C4).9Q8, (C2×C4).139D4, C2.1(C2.D8), C2.2(C8.C4), C22.13(C4⋊C4), (C2×C4).60(C2×C4), SmallGroup(64,16)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C81C8
C1C2C22C2×C4C42C4×C8 — C81C8
C1C2C4 — C81C8
C1C2×C4C42 — C81C8
C1C22C22C42 — C81C8

Generators and relations for C81C8
 G = < a,b | a8=b8=1, bab-1=a-1 >

2C4
2C8
4C8
4C8
2C2×C8
2C2×C8

Character table of C81C8

 class 12A2B2C4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J8K8L8M8N8O8P
 size 1111111122222222222244444444
ρ11111111111111111111111111111    trivial
ρ2111111111111-1-1-1-1-1-1-1-1-11-11-11-11    linear of order 2
ρ3111111111111-1-1-1-1-1-1-1-11-11-11-11-1    linear of order 2
ρ411111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-1-1-1-1-1-111-1-11-1-1111-ii-iii-ii-i    linear of order 4
ρ61111-1-1-1-1-1-11111-111-1-1-1iiii-i-i-i-i    linear of order 4
ρ71111-1-1-1-1-1-111-1-11-1-1111i-ii-i-ii-ii    linear of order 4
ρ81111-1-1-1-1-1-11111-111-1-1-1-i-i-i-iiiii    linear of order 4
ρ91-11-1i-ii-ii-i1-1-11i1-1-ii-iζ87ζ83ζ83ζ87ζ85ζ8ζ8ζ85    linear of order 8
ρ101-11-1i-ii-ii-i1-1-11i1-1-ii-iζ83ζ87ζ87ζ83ζ8ζ85ζ85ζ8    linear of order 8
ρ111-11-1i-ii-ii-i1-11-1-i-11i-iiζ83ζ83ζ87ζ87ζ8ζ8ζ85ζ85    linear of order 8
ρ121-11-1i-ii-ii-i1-11-1-i-11i-iiζ87ζ87ζ83ζ83ζ85ζ85ζ8ζ8    linear of order 8
ρ131-11-1-ii-ii-ii1-1-11-i1-1i-iiζ8ζ85ζ85ζ8ζ83ζ87ζ87ζ83    linear of order 8
ρ141-11-1-ii-ii-ii1-11-1i-11-ii-iζ85ζ85ζ8ζ8ζ87ζ87ζ83ζ83    linear of order 8
ρ151-11-1-ii-ii-ii1-1-11-i1-1i-iiζ85ζ8ζ8ζ85ζ87ζ83ζ83ζ87    linear of order 8
ρ161-11-1-ii-ii-ii1-11-1i-11-ii-iζ8ζ8ζ85ζ85ζ83ζ83ζ87ζ87    linear of order 8
ρ1722222222-2-2-2-20000000000000000    orthogonal lifted from D4
ρ1822-2-2-222-200002-2-22-2-22200000000    orthogonal lifted from D8
ρ1922-2-2-222-20000-222-222-2-200000000    orthogonal lifted from D8
ρ2022-2-22-2-2200002-222-22-2-200000000    symplectic lifted from Q16, Schur index 2
ρ212222-2-2-2-222-2-20000000000000000    symplectic lifted from Q8, Schur index 2
ρ2222-2-22-2-220000-22-2-22-22200000000    symplectic lifted from Q16, Schur index 2
ρ232-22-2-2i2i-2i2i2i-2i-220000000000000000    complex lifted from M4(2)
ρ242-2-222i2i-2i-2i000022--2-2-2-2-2--200000000    complex lifted from C8.C4
ρ252-2-22-2i-2i2i2i0000-2-2--222-2-2--200000000    complex lifted from C8.C4
ρ262-2-22-2i-2i2i2i000022-2-2-2--2--2-200000000    complex lifted from C8.C4
ρ272-2-222i2i-2i-2i0000-2-2-222--2--2-200000000    complex lifted from C8.C4
ρ282-22-22i-2i2i-2i-2i2i-220000000000000000    complex lifted from M4(2)

Smallest permutation representation of C81C8
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 18 47 64 55 31 34 16)(2 17 48 63 56 30 35 15)(3 24 41 62 49 29 36 14)(4 23 42 61 50 28 37 13)(5 22 43 60 51 27 38 12)(6 21 44 59 52 26 39 11)(7 20 45 58 53 25 40 10)(8 19 46 57 54 32 33 9)

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

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

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

C81C8 is a maximal subgroup of
C4.16D16  Q161C8  C4.10D16  C4.6Q32  C42.42Q8  M4(2)⋊1C8  C87M4(2)  C42.43Q8  C81M4(2)  C42.91D4  C42.Q8  C42.92D4  C42.21Q8  C8×D8  C8×Q16  SD16⋊C8  C89D8  C89Q16  D4.M4(2)  Q82M4(2)  C86D8  C86Q16  C8⋊M4(2)  C87D8  C813SD16  Q81Q16  C810SD16  C87Q16  D4.1Q16  Q8.1Q16  D4.2D8  Q8.2D8  D4.Q16  Q8.2Q16  C8⋊D8  C8⋊SD16  C83SD16  C8⋊Q16  C42.248C23  C42.249C23  C42.250C23  C42.251C23  Dic5.13D8
 C8p⋊C8: C163C8  C164C8  C241C8  C405C8  C401C8  C561C8 ...
 C4p.D8: C4.D16  C8.27D8  C8.28D8  C12.53D8  C20.53D8  C28.53D8 ...
C81C8 is a maximal quotient of
C42.385D4  C163C8  C164C8  C401C8  Dic5.13D8
 C4p.D8: C8.36D8  C16.3C8  C241C8  C12.53D8  C405C8  C20.53D8  C561C8  C28.53D8 ...

Matrix representation of C81C8 in GL3(𝔽17) generated by

1600
020
009
,
800
001
010
G:=sub<GL(3,GF(17))| [16,0,0,0,2,0,0,0,9],[8,0,0,0,0,1,0,1,0] >;

C81C8 in GAP, Magma, Sage, TeX

C_8\rtimes_1C_8
% in TeX

G:=Group("C8:1C8");
// GroupNames label

G:=SmallGroup(64,16);
// by ID

G=gap.SmallGroup(64,16);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,127,362,86,117]);
// Polycyclic

G:=Group<a,b|a^8=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C81C8 in TeX
Character table of C81C8 in TeX

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