p-group, metacyclic, nilpotent (class 3), monomial
Aliases: C8.1C8, C8.6Q8, C8.29D4, C42.8C4, M5(2).4C2, C22.5M4(2), (C2×C8).9C4, C4.8(C2×C8), C2.5(C4⋊C8), (C4×C8).10C2, C4.20(C4⋊C4), (C2×C8).96C22, (C2×C4).68(C2×C4), SmallGroup(64,45)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.C8
G = < a,b | a8=1, b8=a4, bab-1=a3 >
Character table of C8.C8
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -i | -i | -i | i | -i | i | i | i | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | i | i | i | -i | i | -i | -i | -i | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | i | i | -i | i | -i | -i | i | -i | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -i | -i | i | -i | i | i | -i | i | linear of order 4 |
| ρ9 | 1 | 1 | -1 | -1 | -1 | -i | 1 | i | i | -i | i | i | -i | -i | 1 | -1 | -i | -1 | 1 | i | ζ83 | ζ87 | ζ8 | ζ87 | ζ85 | ζ85 | ζ83 | ζ8 | linear of order 8 |
| ρ10 | 1 | 1 | -1 | -1 | -1 | -i | 1 | i | i | -i | i | i | -i | -i | 1 | -1 | -i | -1 | 1 | i | ζ87 | ζ83 | ζ85 | ζ83 | ζ8 | ζ8 | ζ87 | ζ85 | linear of order 8 |
| ρ11 | 1 | 1 | -1 | -1 | -1 | -i | 1 | i | i | -i | -i | -i | i | i | -1 | 1 | i | 1 | -1 | -i | ζ8 | ζ85 | ζ87 | ζ8 | ζ83 | ζ87 | ζ85 | ζ83 | linear of order 8 |
| ρ12 | 1 | 1 | -1 | -1 | -1 | -i | 1 | i | i | -i | -i | -i | i | i | -1 | 1 | i | 1 | -1 | -i | ζ85 | ζ8 | ζ83 | ζ85 | ζ87 | ζ83 | ζ8 | ζ87 | linear of order 8 |
| ρ13 | 1 | 1 | -1 | -1 | -1 | i | 1 | -i | -i | i | -i | -i | i | i | 1 | -1 | i | -1 | 1 | -i | ζ85 | ζ8 | ζ87 | ζ8 | ζ83 | ζ83 | ζ85 | ζ87 | linear of order 8 |
| ρ14 | 1 | 1 | -1 | -1 | -1 | i | 1 | -i | -i | i | i | i | -i | -i | -1 | 1 | -i | 1 | -1 | i | ζ87 | ζ83 | ζ8 | ζ87 | ζ85 | ζ8 | ζ83 | ζ85 | linear of order 8 |
| ρ15 | 1 | 1 | -1 | -1 | -1 | i | 1 | -i | -i | i | -i | -i | i | i | 1 | -1 | i | -1 | 1 | -i | ζ8 | ζ85 | ζ83 | ζ85 | ζ87 | ζ87 | ζ8 | ζ83 | linear of order 8 |
| ρ16 | 1 | 1 | -1 | -1 | -1 | i | 1 | -i | -i | i | i | i | -i | -i | -1 | 1 | -i | 1 | -1 | i | ζ83 | ζ87 | ζ85 | ζ83 | ζ8 | ζ85 | ζ87 | ζ8 | linear of order 8 |
| ρ17 | 2 | 2 | -2 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ19 | 2 | 2 | 2 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ20 | 2 | 2 | 2 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ21 | 2 | -2 | 0 | 2i | -2i | 1+i | 0 | 1-i | -1+i | -1-i | 2ζ83 | 2ζ87 | 2ζ85 | 2ζ8 | -√2 | -√-2 | 0 | √-2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 2 | -2 | 0 | 2i | -2i | 1+i | 0 | 1-i | -1+i | -1-i | 2ζ87 | 2ζ83 | 2ζ8 | 2ζ85 | √2 | √-2 | 0 | -√-2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 2 | -2 | 0 | -2i | 2i | -1+i | 0 | -1-i | 1+i | 1-i | 2ζ8 | 2ζ85 | 2ζ87 | 2ζ83 | -√2 | √-2 | 0 | -√-2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 2 | -2 | 0 | -2i | 2i | 1-i | 0 | 1+i | -1-i | -1+i | 2ζ85 | 2ζ8 | 2ζ83 | 2ζ87 | -√2 | √-2 | 0 | -√-2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ25 | 2 | -2 | 0 | 2i | -2i | -1-i | 0 | -1+i | 1-i | 1+i | 2ζ87 | 2ζ83 | 2ζ8 | 2ζ85 | -√2 | -√-2 | 0 | √-2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ26 | 2 | -2 | 0 | 2i | -2i | -1-i | 0 | -1+i | 1-i | 1+i | 2ζ83 | 2ζ87 | 2ζ85 | 2ζ8 | √2 | √-2 | 0 | -√-2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ27 | 2 | -2 | 0 | -2i | 2i | -1+i | 0 | -1-i | 1+i | 1-i | 2ζ85 | 2ζ8 | 2ζ83 | 2ζ87 | √2 | -√-2 | 0 | √-2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ28 | 2 | -2 | 0 | -2i | 2i | 1-i | 0 | 1+i | -1-i | -1+i | 2ζ8 | 2ζ85 | 2ζ87 | 2ζ83 | √2 | -√-2 | 0 | √-2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 16 points - transitive group 16T124
(1 7 13 3 9 15 5 11)(2 4 6 8 10 12 14 16)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
G:=sub<Sym(16)| (1,7,13,3,9,15,5,11)(2,4,6,8,10,12,14,16), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)>;
G:=Group( (1,7,13,3,9,15,5,11)(2,4,6,8,10,12,14,16), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16) );
G=PermutationGroup([[(1,7,13,3,9,15,5,11),(2,4,6,8,10,12,14,16)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)]])
G:=TransitiveGroup(16,124);
C8.C8 is a maximal subgroup of
C8≀C2 C8.32D8 C8.24D8 C8.25D8 C8.29D8 C8.1Q16 M4(2).1C8 C8.5M4(2) C8.19M4(2) C8.3D8 D8⋊3D4 C8.5D8 D8⋊3Q8 D8.2Q8 C42.9F5
C8p.C8: C16.C8 C16.3C8 C24.1C8 C40.7C8 C40.1C8 C56.16Q8 ...
C8p.D4: C16○D8 D8.C8 C24.97D4 C40.9Q8 C56.9Q8 ...
C8.C8 is a maximal quotient of
C8⋊2C16
C8.D4p: C8.36D8 C24.1C8 C40.7C8 C56.16Q8 ...
C2p.(C4⋊C8): C42.7C8 C24.97D4 C40.9Q8 C42.9F5 C40.1C8 C56.9Q8 ...
Matrix representation of C8.C8 ►in GL2(𝔽17) generated by
| 9 | 0 |
| 0 | 15 |
| 0 | 1 |
| 2 | 0 |
G:=sub<GL(2,GF(17))| [9,0,0,15],[0,2,1,0] >;
C8.C8 in GAP, Magma, Sage, TeX
C_8.C_8
% in TeX
G:=Group("C8.C8"); // GroupNames label
G:=SmallGroup(64,45);
// by ID
G=gap.SmallGroup(64,45);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,31,650,158,69,88]);
// Polycyclic
G:=Group<a,b|a^8=1,b^8=a^4,b*a*b^-1=a^3>;
// generators/relations
Export
Subgroup lattice of C8.C8 in TeX
Character table of C8.C8 in TeX