p-group, metacyclic, nilpotent (class 4), monomial
Aliases: C8.4Q8, C16.1C4, C4.19D8, C22.1Q16, C4.9(C4⋊C4), (C2×C16).5C2, C8.16(C2×C4), (C2×C4).65D4, C2.5(C2.D8), C8.C4.3C2, (C2×C8).87C22, SmallGroup(64,49)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.4Q8
G = < a,b,c | a8=1, b4=a2, c2=ab2, ab=ba, cac-1=a3, cbc-1=a6b3 >
Character table of C8.4Q8
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -i | -i | i | i | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | i | i | -i | -i | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -i | i | -i | i | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | i | -i | i | -i | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | √2 | -√2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ11 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | -√2 | √2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ12 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ13 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | -√2 | √2 | √2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ14 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | √2 | -√2 | -√2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ15 | 2 | -2 | 0 | -2i | 2i | 0 | √-2 | -√-2 | -√2 | √2 | 0 | 0 | 0 | 0 | ζ1615+ζ169 | ζ167-ζ16 | ζ165-ζ163 | ζ1613+ζ1611 | ζ167+ζ16 | ζ165+ζ163 | -ζ165+ζ163 | -ζ167+ζ16 | complex faithful |
| ρ16 | 2 | -2 | 0 | 2i | -2i | 0 | -√-2 | √-2 | -√2 | √2 | 0 | 0 | 0 | 0 | ζ167+ζ16 | ζ167-ζ16 | ζ165-ζ163 | ζ165+ζ163 | ζ1615+ζ169 | ζ1613+ζ1611 | -ζ165+ζ163 | -ζ167+ζ16 | complex faithful |
| ρ17 | 2 | -2 | 0 | -2i | 2i | 0 | -√-2 | √-2 | √2 | -√2 | 0 | 0 | 0 | 0 | ζ165+ζ163 | ζ165-ζ163 | -ζ167+ζ16 | ζ1615+ζ169 | ζ1613+ζ1611 | ζ167+ζ16 | ζ167-ζ16 | -ζ165+ζ163 | complex faithful |
| ρ18 | 2 | -2 | 0 | 2i | -2i | 0 | -√-2 | √-2 | -√2 | √2 | 0 | 0 | 0 | 0 | ζ1615+ζ169 | -ζ167+ζ16 | -ζ165+ζ163 | ζ1613+ζ1611 | ζ167+ζ16 | ζ165+ζ163 | ζ165-ζ163 | ζ167-ζ16 | complex faithful |
| ρ19 | 2 | -2 | 0 | -2i | 2i | 0 | √-2 | -√-2 | -√2 | √2 | 0 | 0 | 0 | 0 | ζ167+ζ16 | -ζ167+ζ16 | -ζ165+ζ163 | ζ165+ζ163 | ζ1615+ζ169 | ζ1613+ζ1611 | ζ165-ζ163 | ζ167-ζ16 | complex faithful |
| ρ20 | 2 | -2 | 0 | 2i | -2i | 0 | √-2 | -√-2 | √2 | -√2 | 0 | 0 | 0 | 0 | ζ1613+ζ1611 | ζ165-ζ163 | -ζ167+ζ16 | ζ167+ζ16 | ζ165+ζ163 | ζ1615+ζ169 | ζ167-ζ16 | -ζ165+ζ163 | complex faithful |
| ρ21 | 2 | -2 | 0 | 2i | -2i | 0 | √-2 | -√-2 | √2 | -√2 | 0 | 0 | 0 | 0 | ζ165+ζ163 | -ζ165+ζ163 | ζ167-ζ16 | ζ1615+ζ169 | ζ1613+ζ1611 | ζ167+ζ16 | -ζ167+ζ16 | ζ165-ζ163 | complex faithful |
| ρ22 | 2 | -2 | 0 | -2i | 2i | 0 | -√-2 | √-2 | √2 | -√2 | 0 | 0 | 0 | 0 | ζ1613+ζ1611 | -ζ165+ζ163 | ζ167-ζ16 | ζ167+ζ16 | ζ165+ζ163 | ζ1615+ζ169 | -ζ167+ζ16 | ζ165-ζ163 | complex faithful |
►On 32 points
(1 11 5 15 9 3 13 7)(2 12 6 16 10 4 14 8)(17 19 21 23 25 27 29 31)(18 20 22 24 26 28 30 32)
(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)
(1 27 13 31 9 19 5 23)(2 26 14 30 10 18 6 22)(3 25 15 29 11 17 7 21)(4 24 16 28 12 32 8 20)
G:=sub<Sym(32)| (1,11,5,15,9,3,13,7)(2,12,6,16,10,4,14,8)(17,19,21,23,25,27,29,31)(18,20,22,24,26,28,30,32), (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), (1,27,13,31,9,19,5,23)(2,26,14,30,10,18,6,22)(3,25,15,29,11,17,7,21)(4,24,16,28,12,32,8,20)>;
G:=Group( (1,11,5,15,9,3,13,7)(2,12,6,16,10,4,14,8)(17,19,21,23,25,27,29,31)(18,20,22,24,26,28,30,32), (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), (1,27,13,31,9,19,5,23)(2,26,14,30,10,18,6,22)(3,25,15,29,11,17,7,21)(4,24,16,28,12,32,8,20) );
G=PermutationGroup([[(1,11,5,15,9,3,13,7),(2,12,6,16,10,4,14,8),(17,19,21,23,25,27,29,31),(18,20,22,24,26,28,30,32)], [(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)], [(1,27,13,31,9,19,5,23),(2,26,14,30,10,18,6,22),(3,25,15,29,11,17,7,21),(4,24,16,28,12,32,8,20)]])
C8.4Q8 is a maximal subgroup of
D16.C4 M6(2)⋊C2 C16.18D4 M5(2).1C4 C8○D16 D16⋊5C4 D4.3D8 D4.4D8 D4.5D8
C16p.C4: C32.C4 C48.C4 C80.6C4 C16.F5 C80.2C4 C112.C4 ...
C8p.Q8: C8.Q16 C24.7Q8 C40.7Q8 C8.7Dic14 ...
C8.4Q8 is a maximal quotient of
C16⋊4C8
C4.D8p: C16⋊3C8 C48.C4 C80.6C4 C112.C4 ...
C4p.(C4⋊C4): C8.8C42 C24.7Q8 C40.7Q8 C16.F5 C80.2C4 C8.7Dic14 ...
Matrix representation of C8.4Q8 ►in GL2(𝔽17) generated by
| 9 | 0 |
| 0 | 15 |
| 5 | 0 |
| 0 | 7 |
| 0 | 1 |
| 4 | 0 |
G:=sub<GL(2,GF(17))| [9,0,0,15],[5,0,0,7],[0,4,1,0] >;
C8.4Q8 in GAP, Magma, Sage, TeX
C_8._4Q_8
% in TeX
G:=Group("C8.4Q8"); // GroupNames label
G:=SmallGroup(64,49);
// by ID
G=gap.SmallGroup(64,49);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,127,362,230,117,1444,88]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^4=a^2,c^2=a*b^2,a*b=b*a,c*a*c^-1=a^3,c*b*c^-1=a^6*b^3>;
// generators/relations
Export
Subgroup lattice of C8.4Q8 in TeX
Character table of C8.4Q8 in TeX