metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C24.1C4, C4.18D12, C12.34D4, C8.1Dic3, C22.2Dic6, (C2×C8).5S3, C6.7(C4⋊C4), (C2×C6).3Q8, (C2×C24).7C2, (C2×C4).70D6, C3⋊1(C8.C4), C12.35(C2×C4), C4.8(C2×Dic3), C2.5(C4⋊Dic3), C4.Dic3.1C2, (C2×C12).97C22, SmallGroup(96,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.Dic3
G = < a,b,c | a8=1, b6=a4, c2=a4b3, ab=ba, cac-1=a-1, cbc-1=b5 >
Character table of C8.Dic3
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 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 | 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 | -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 | 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 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | i | i | -i | -i | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | i | -i | -i | i | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -i | -i | i | i | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | i | i | -i | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | -2 | 2 | 2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | -1 | 2 | 2 | -2 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | -√3 | √3 | -√3 | √3 | -√3 | -√3 | √3 | √3 | orthogonal lifted from D12 |
| ρ12 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | -2 | -1 | 2 | 2 | -2 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | √3 | -√3 | √3 | -√3 | √3 | √3 | -√3 | -√3 | orthogonal lifted from D12 |
| ρ14 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ15 | 2 | 2 | -2 | -1 | -2 | -2 | 2 | -1 | 1 | 1 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ16 | 2 | 2 | -2 | -1 | -2 | -2 | 2 | -1 | 1 | 1 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ17 | 2 | 2 | 2 | -1 | -2 | -2 | -2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | √3 | √3 | -√3 | -√3 | √3 | -√3 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ18 | 2 | 2 | 2 | -1 | -2 | -2 | -2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -√3 | -√3 | √3 | √3 | -√3 | √3 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ19 | 2 | -2 | 0 | 2 | -2i | 2i | 0 | -2 | 0 | 0 | -√-2 | √-2 | -√2 | √2 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | √2 | √-2 | √-2 | √2 | -√2 | -√-2 | -√-2 | -√2 | complex lifted from C8.C4 |
| ρ20 | 2 | -2 | 0 | 2 | 2i | -2i | 0 | -2 | 0 | 0 | -√-2 | √-2 | √2 | -√2 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | -√2 | √-2 | √-2 | -√2 | √2 | -√-2 | -√-2 | √2 | complex lifted from C8.C4 |
| ρ21 | 2 | -2 | 0 | 2 | -2i | 2i | 0 | -2 | 0 | 0 | √-2 | -√-2 | √2 | -√2 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | -√2 | -√-2 | -√-2 | -√2 | √2 | √-2 | √-2 | √2 | complex lifted from C8.C4 |
| ρ22 | 2 | -2 | 0 | 2 | 2i | -2i | 0 | -2 | 0 | 0 | √-2 | -√-2 | -√2 | √2 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | √2 | -√-2 | -√-2 | √2 | -√2 | √-2 | √-2 | -√2 | complex lifted from C8.C4 |
| ρ23 | 2 | -2 | 0 | -1 | 2i | -2i | 0 | 1 | -√-3 | √-3 | -√-2 | √-2 | √2 | -√2 | 0 | 0 | 0 | 0 | -√3 | ζ2 | √3 | ζ2 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ32+ζ87-ζ85ζ32 | ζ87ζ3+ζ87-ζ85ζ3 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ83ζ3+ζ83-ζ8ζ3 | ζ83ζ32+ζ83-ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | complex faithful |
| ρ24 | 2 | -2 | 0 | -1 | -2i | 2i | 0 | 1 | -√-3 | √-3 | √-2 | -√-2 | √2 | -√2 | 0 | 0 | 0 | 0 | √3 | ζ2 | -√3 | ζ2 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ83-ζ8ζ3 | ζ83ζ32+ζ83-ζ8ζ32 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ32+ζ85ζ32+ζ85 | ζ87ζ32+ζ87-ζ85ζ32 | ζ87ζ3+ζ87-ζ85ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | complex faithful |
| ρ25 | 2 | -2 | 0 | -1 | -2i | 2i | 0 | 1 | √-3 | -√-3 | √-2 | -√-2 | √2 | -√2 | 0 | 0 | 0 | 0 | -√3 | ζ2 | √3 | ζ2 | ζ83ζ3+ζ8ζ3+ζ8 | ζ83ζ32+ζ83-ζ8ζ32 | ζ83ζ3+ζ83-ζ8ζ3 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ3+ζ87-ζ85ζ3 | ζ87ζ32+ζ87-ζ85ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | complex faithful |
| ρ26 | 2 | -2 | 0 | -1 | 2i | -2i | 0 | 1 | √-3 | -√-3 | √-2 | -√-2 | -√2 | √2 | 0 | 0 | 0 | 0 | √3 | ζ2 | -√3 | ζ2 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ3+ζ83-ζ8ζ3 | ζ83ζ32+ζ83-ζ8ζ32 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | ζ87ζ32+ζ87-ζ85ζ32 | ζ87ζ3+ζ87-ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | complex faithful |
| ρ27 | 2 | -2 | 0 | -1 | 2i | -2i | 0 | 1 | -√-3 | √-3 | √-2 | -√-2 | -√2 | √2 | 0 | 0 | 0 | 0 | -√3 | ζ2 | √3 | ζ2 | ζ83ζ32+ζ83+ζ8ζ32 | ζ83ζ32+ζ83-ζ8ζ32 | ζ83ζ3+ζ83-ζ8ζ3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87-ζ85ζ3 | ζ87ζ32+ζ87-ζ85ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | complex faithful |
| ρ28 | 2 | -2 | 0 | -1 | -2i | 2i | 0 | 1 | √-3 | -√-3 | -√-2 | √-2 | -√2 | √2 | 0 | 0 | 0 | 0 | -√3 | ζ2 | √3 | ζ2 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ87-ζ85ζ32 | ζ87ζ3+ζ87-ζ85ζ3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ3+ζ8ζ3+ζ8 | ζ83ζ3+ζ83-ζ8ζ3 | ζ83ζ32+ζ83-ζ8ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | complex faithful |
| ρ29 | 2 | -2 | 0 | -1 | -2i | 2i | 0 | 1 | -√-3 | √-3 | -√-2 | √-2 | -√2 | √2 | 0 | 0 | 0 | 0 | √3 | ζ2 | -√3 | ζ2 | ζ87ζ32+ζ85ζ32+ζ85 | ζ87ζ3+ζ87-ζ85ζ3 | ζ87ζ32+ζ87-ζ85ζ32 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ32+ζ83-ζ8ζ32 | ζ83ζ3+ζ83-ζ8ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | complex faithful |
| ρ30 | 2 | -2 | 0 | -1 | 2i | -2i | 0 | 1 | √-3 | -√-3 | -√-2 | √-2 | √2 | -√2 | 0 | 0 | 0 | 0 | √3 | ζ2 | -√3 | ζ2 | ζ87ζ3+ζ87+ζ85ζ3 | ζ87ζ3+ζ87-ζ85ζ3 | ζ87ζ32+ζ87-ζ85ζ32 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83-ζ8ζ32 | ζ83ζ3+ζ83-ζ8ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | complex faithful |
►On 48 points
Generators in S48
(1 18 10 15 7 24 4 21)(2 19 11 16 8 13 5 22)(3 20 12 17 9 14 6 23)(25 37 28 40 31 43 34 46)(26 38 29 41 32 44 35 47)(27 39 30 42 33 45 36 48)
(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)
(1 25 10 34 7 31 4 28)(2 30 11 27 8 36 5 33)(3 35 12 32 9 29 6 26)(13 45 22 42 19 39 16 48)(14 38 23 47 20 44 17 41)(15 43 24 40 21 37 18 46)
G:=sub<Sym(48)| (1,18,10,15,7,24,4,21)(2,19,11,16,8,13,5,22)(3,20,12,17,9,14,6,23)(25,37,28,40,31,43,34,46)(26,38,29,41,32,44,35,47)(27,39,30,42,33,45,36,48), (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), (1,25,10,34,7,31,4,28)(2,30,11,27,8,36,5,33)(3,35,12,32,9,29,6,26)(13,45,22,42,19,39,16,48)(14,38,23,47,20,44,17,41)(15,43,24,40,21,37,18,46)>;
G:=Group( (1,18,10,15,7,24,4,21)(2,19,11,16,8,13,5,22)(3,20,12,17,9,14,6,23)(25,37,28,40,31,43,34,46)(26,38,29,41,32,44,35,47)(27,39,30,42,33,45,36,48), (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), (1,25,10,34,7,31,4,28)(2,30,11,27,8,36,5,33)(3,35,12,32,9,29,6,26)(13,45,22,42,19,39,16,48)(14,38,23,47,20,44,17,41)(15,43,24,40,21,37,18,46) );
G=PermutationGroup([(1,18,10,15,7,24,4,21),(2,19,11,16,8,13,5,22),(3,20,12,17,9,14,6,23),(25,37,28,40,31,43,34,46),(26,38,29,41,32,44,35,47),(27,39,30,42,33,45,36,48)], [(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)], [(1,25,10,34,7,31,4,28),(2,30,11,27,8,36,5,33),(3,35,12,32,9,29,6,26),(13,45,22,42,19,39,16,48),(14,38,23,47,20,44,17,41),(15,43,24,40,21,37,18,46)])
Matrix representation of C8.Dic3 ►in GL2(𝔽73) generated by
| 63 | 0 |
| 28 | 51 |
| 49 | 0 |
| 24 | 70 |
| 72 | 63 |
| 12 | 1 |
G:=sub<GL(2,GF(73))| [63,28,0,51],[49,24,0,70],[72,12,63,1] >;
C8.Dic3 in GAP, Magma, Sage, TeX
C_8.{\rm Dic}_3 % in TeX
G:=Group("C8.Dic3"); // GroupNames label
G:=SmallGroup(96,26);
// by ID
G=gap.SmallGroup(96,26);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,55,86,579,69,2309]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^6=a^4,c^2=a^4*b^3,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations