non-abelian, soluble, monomial
Aliases: A4⋊C8, C4.4S4, C23.Dic3, (C2×A4).C4, C22⋊(C3⋊C8), (C4×A4).2C2, C2.1(A4⋊C4), (C22×C4).1S3, SmallGroup(96,65)
Series: Derived ►Chief ►Lower central ►Upper central
| A4 — A4⋊C8 |
Generators and relations for A4⋊C8
G = < a,b,c,d | a2=b2=c3=d8=1, cac-1=dad-1=ab=ba, cbc-1=a, bd=db, dcd-1=c-1 >
Character table of A4⋊C8
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | |
| size | 1 | 1 | 3 | 3 | 8 | 1 | 1 | 3 | 3 | 8 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -i | i | i | i | -i | -i | -i | i | -1 | -1 | linear of order 4 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | i | -i | -i | -i | i | i | i | -i | -1 | -1 | linear of order 4 |
| ρ5 | 1 | -1 | 1 | -1 | 1 | -i | i | i | -i | -1 | ζ85 | ζ83 | ζ87 | ζ87 | ζ8 | ζ8 | ζ85 | ζ83 | -i | i | linear of order 8 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | -i | i | i | -i | -1 | ζ8 | ζ87 | ζ83 | ζ83 | ζ85 | ζ85 | ζ8 | ζ87 | -i | i | linear of order 8 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | i | -i | -i | i | -1 | ζ87 | ζ8 | ζ85 | ζ85 | ζ83 | ζ83 | ζ87 | ζ8 | i | -i | linear of order 8 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | i | -i | -i | i | -1 | ζ83 | ζ85 | ζ8 | ζ8 | ζ87 | ζ87 | ζ83 | ζ85 | i | -i | linear of order 8 |
| ρ9 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ11 | 2 | -2 | 2 | -2 | -1 | -2i | 2i | 2i | -2i | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | i | -i | complex lifted from C3⋊C8 |
| ρ12 | 2 | -2 | 2 | -2 | -1 | 2i | -2i | -2i | 2i | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -i | i | complex lifted from C3⋊C8 |
| ρ13 | 3 | 3 | -1 | -1 | 0 | 3 | 3 | -1 | -1 | 0 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 0 | 0 | orthogonal lifted from S4 |
| ρ14 | 3 | 3 | -1 | -1 | 0 | 3 | 3 | -1 | -1 | 0 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 0 | 0 | orthogonal lifted from S4 |
| ρ15 | 3 | 3 | -1 | -1 | 0 | -3 | -3 | 1 | 1 | 0 | -i | i | -i | i | i | -i | i | -i | 0 | 0 | complex lifted from A4⋊C4 |
| ρ16 | 3 | 3 | -1 | -1 | 0 | -3 | -3 | 1 | 1 | 0 | i | -i | i | -i | -i | i | -i | i | 0 | 0 | complex lifted from A4⋊C4 |
| ρ17 | 3 | -3 | -1 | 1 | 0 | -3i | 3i | -i | i | 0 | ζ8 | ζ87 | ζ87 | ζ83 | ζ8 | ζ85 | ζ85 | ζ83 | 0 | 0 | complex faithful |
| ρ18 | 3 | -3 | -1 | 1 | 0 | -3i | 3i | -i | i | 0 | ζ85 | ζ83 | ζ83 | ζ87 | ζ85 | ζ8 | ζ8 | ζ87 | 0 | 0 | complex faithful |
| ρ19 | 3 | -3 | -1 | 1 | 0 | 3i | -3i | i | -i | 0 | ζ87 | ζ8 | ζ8 | ζ85 | ζ87 | ζ83 | ζ83 | ζ85 | 0 | 0 | complex faithful |
| ρ20 | 3 | -3 | -1 | 1 | 0 | 3i | -3i | i | -i | 0 | ζ83 | ζ85 | ζ85 | ζ8 | ζ83 | ζ87 | ζ87 | ζ8 | 0 | 0 | complex faithful |
►On 24 points - transitive group 24T89
Generators in S24
(1 5)(3 7)(9 13)(10 14)(11 15)(12 16)(18 22)(20 24)
(1 5)(2 6)(3 7)(4 8)(17 21)(18 22)(19 23)(20 24)
(1 19 11)(2 12 20)(3 21 13)(4 14 22)(5 23 15)(6 16 24)(7 17 9)(8 10 18)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
G:=sub<Sym(24)| (1,5)(3,7)(9,13)(10,14)(11,15)(12,16)(18,22)(20,24), (1,5)(2,6)(3,7)(4,8)(17,21)(18,22)(19,23)(20,24), (1,19,11)(2,12,20)(3,21,13)(4,14,22)(5,23,15)(6,16,24)(7,17,9)(8,10,18), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)>;
G:=Group( (1,5)(3,7)(9,13)(10,14)(11,15)(12,16)(18,22)(20,24), (1,5)(2,6)(3,7)(4,8)(17,21)(18,22)(19,23)(20,24), (1,19,11)(2,12,20)(3,21,13)(4,14,22)(5,23,15)(6,16,24)(7,17,9)(8,10,18), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24) );
G=PermutationGroup([[(1,5),(3,7),(9,13),(10,14),(11,15),(12,16),(18,22),(20,24)], [(1,5),(2,6),(3,7),(4,8),(17,21),(18,22),(19,23),(20,24)], [(1,19,11),(2,12,20),(3,21,13),(4,14,22),(5,23,15),(6,16,24),(7,17,9),(8,10,18)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)]])
G:=TransitiveGroup(24,89);
A4⋊C8 is a maximal subgroup of
C8×S4 C8⋊S4 A4⋊M4(2) A4⋊SD16 D4⋊S4 A4⋊2Q16 Q8⋊3S4 C12.12S4 A5⋊C8 C20.S4 Dic5.S4
A4⋊C8 is a maximal quotient of
C2.U2(𝔽3) A4⋊C16 C8.7S4 C12.S4 C12.12S4 C20.S4 Dic5.S4
Matrix representation of A4⋊C8 ►in GL3(𝔽73) generated by
| 72 | 0 | 0 |
| 0 | 72 | 0 |
| 72 | 0 | 1 |
| 72 | 0 | 0 |
| 72 | 1 | 0 |
| 0 | 0 | 72 |
| 1 | 71 | 0 |
| 0 | 72 | 1 |
| 0 | 72 | 0 |
| 63 | 0 | 20 |
| 0 | 63 | 10 |
| 0 | 0 | 10 |
G:=sub<GL(3,GF(73))| [72,0,72,0,72,0,0,0,1],[72,72,0,0,1,0,0,0,72],[1,0,0,71,72,72,0,1,0],[63,0,0,0,63,0,20,10,10] >;
A4⋊C8 in GAP, Magma, Sage, TeX
A_4\rtimes C_8
% in TeX
G:=Group("A4:C8"); // GroupNames label
G:=SmallGroup(96,65);
// by ID
G=gap.SmallGroup(96,65);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,2,12,31,387,1444,202,869,347]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^3=d^8=1,c*a*c^-1=d*a*d^-1=a*b=b*a,c*b*c^-1=a,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of A4⋊C8 in TeX
Character table of A4⋊C8 in TeX