metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.5Q8, C8.1Dic6, C12.2SD16, C3⋊C16⋊1C4, C3⋊1(C8.Q8), C8.28(C4×S3), (C2×C8).40D6, C12.1(C4⋊C4), C4.Q8.1S3, C24.32(C2×C4), (C2×C12).87D4, C6.2(C4.Q8), (C2×C6).29SD16, C24.C4.4C2, C12.C8.2C2, C4.1(Dic3⋊C4), (C2×C24).46C22, C4.6(Q8⋊2S3), C22.5(D4.S3), C2.3(C12.Q8), (C3×C4.Q8).1C2, (C2×C4).15(C3⋊D4), SmallGroup(192,46)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.Dic6
G = < a,b,c | a8=b12=1, c2=ab6, bab-1=a3, cac-1=a5, cbc-1=a-1b-1 >
Character table of C8.Dic6
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 12C | 12D | 12E | 12F | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 4 | 24 | 24 | 4 | 4 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 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 | 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 | -i | i | -1 | -1 | 1 | 1 | 1 | -1 | i | -i | -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 | i | -i | -1 | -1 | 1 | 1 | 1 | -1 | -i | i | -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 | -i | i | -1 | -1 | 1 | 1 | 1 | -1 | -i | i | -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 | i | -i | -1 | -1 | 1 | 1 | 1 | -1 | i | -i | -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 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 2 | -1 | 2 | 2 | -2 | -2 | -1 | -1 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | symplectic lifted from Q8, Schur index 2 |
| ρ13 | 2 | 2 | -2 | -1 | -2 | 2 | 0 | 0 | 1 | 1 | -1 | -2 | -2 | 2 | 0 | 0 | 1 | -1 | √3 | √3 | -√3 | -√3 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | symplectic lifted from Dic6, Schur index 2 |
| ρ14 | 2 | 2 | -2 | -1 | -2 | 2 | 0 | 0 | 1 | 1 | -1 | -2 | -2 | 2 | 0 | 0 | 1 | -1 | -√3 | -√3 | √3 | √3 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | symplectic lifted from Dic6, Schur index 2 |
| ρ15 | 2 | 2 | -2 | -1 | -2 | 2 | -2i | 2i | 1 | 1 | -1 | 2 | 2 | -2 | 0 | 0 | 1 | -1 | i | -i | -i | i | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | complex lifted from C4×S3 |
| ρ16 | 2 | 2 | -2 | -1 | -2 | 2 | 2i | -2i | 1 | 1 | -1 | 2 | 2 | -2 | 0 | 0 | 1 | -1 | -i | i | i | -i | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | complex lifted from C4×S3 |
| ρ17 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | -2 | 0 | 0 | -1 | -1 | -√-3 | √-3 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | -2 | 0 | 0 | -1 | -1 | √-3 | -√-3 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ19 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | √-2 | -√-2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ21 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ22 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | -√-2 | √-2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ23 | 4 | 4 | -4 | -2 | 4 | -4 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊2S3 |
| ρ24 | 4 | 4 | 4 | -2 | -4 | -4 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4.S3, Schur index 2 |
| ρ25 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | -2√-2 | 2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-2 | 2√-2 | 0 | 0 | complex lifted from C8.Q8 |
| ρ26 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 2√-2 | -2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-2 | -2√-2 | 0 | 0 | complex lifted from C8.Q8 |
| ρ27 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | -2√-3 | 2√-3 | 2 | -2√-2 | 2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | √6 | -√6 | complex faithful |
| ρ28 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | -2√-3 | 2√-3 | 2 | 2√-2 | -2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | -√6 | √6 | complex faithful |
| ρ29 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 2√-3 | -2√-3 | 2 | 2√-2 | -2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | √6 | -√6 | complex faithful |
| ρ30 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 2√-3 | -2√-3 | 2 | -2√-2 | 2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | -√6 | √6 | complex faithful |
►On 48 points
Generators in S48
(1 3 5 7 9 11 13 15)(2 12 6 16 10 4 14 8)(17 27 21 31 25 19 29 23)(18 20 22 24 26 28 30 32)(33 43 37 47 41 35 45 39)(34 36 38 40 42 44 46 48)
(1 32 40)(2 35 25 4 41 27 10 43 17 12 33 19)(3 22 42 7 18 46)(5 28 44 13 20 36)(6 47 29 16 45 23 14 39 21 8 37 31)(9 24 48)(11 30 34 15 26 38)
(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)
G:=sub<Sym(48)| (1,3,5,7,9,11,13,15)(2,12,6,16,10,4,14,8)(17,27,21,31,25,19,29,23)(18,20,22,24,26,28,30,32)(33,43,37,47,41,35,45,39)(34,36,38,40,42,44,46,48), (1,32,40)(2,35,25,4,41,27,10,43,17,12,33,19)(3,22,42,7,18,46)(5,28,44,13,20,36)(6,47,29,16,45,23,14,39,21,8,37,31)(9,24,48)(11,30,34,15,26,38), (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)>;
G:=Group( (1,3,5,7,9,11,13,15)(2,12,6,16,10,4,14,8)(17,27,21,31,25,19,29,23)(18,20,22,24,26,28,30,32)(33,43,37,47,41,35,45,39)(34,36,38,40,42,44,46,48), (1,32,40)(2,35,25,4,41,27,10,43,17,12,33,19)(3,22,42,7,18,46)(5,28,44,13,20,36)(6,47,29,16,45,23,14,39,21,8,37,31)(9,24,48)(11,30,34,15,26,38), (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) );
G=PermutationGroup([[(1,3,5,7,9,11,13,15),(2,12,6,16,10,4,14,8),(17,27,21,31,25,19,29,23),(18,20,22,24,26,28,30,32),(33,43,37,47,41,35,45,39),(34,36,38,40,42,44,46,48)], [(1,32,40),(2,35,25,4,41,27,10,43,17,12,33,19),(3,22,42,7,18,46),(5,28,44,13,20,36),(6,47,29,16,45,23,14,39,21,8,37,31),(9,24,48),(11,30,34,15,26,38)], [(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)]])
Matrix representation of C8.Dic6 ►in GL6(𝔽97)
| 96 | 0 | 0 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 57 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 57 | 40 |
| 0 | 0 | 0 | 0 | 57 | 57 |
| 0 | 22 | 0 | 0 | 0 | 0 |
| 75 | 75 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 96 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 57 |
| 0 | 0 | 0 | 0 | 57 | 57 |
| 82 | 55 | 0 | 0 | 0 | 0 |
| 70 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 57 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
G:=sub<GL(6,GF(97))| [96,0,0,0,0,0,0,96,0,0,0,0,0,0,40,40,0,0,0,0,57,40,0,0,0,0,0,0,57,57,0,0,0,0,40,57],[0,75,0,0,0,0,22,75,0,0,0,0,0,0,1,0,0,0,0,0,0,96,0,0,0,0,0,0,40,57,0,0,0,0,57,57],[82,70,0,0,0,0,55,15,0,0,0,0,0,0,0,0,40,40,0,0,0,0,57,40,0,0,1,0,0,0,0,0,0,1,0,0] >;
C8.Dic6 in GAP, Magma, Sage, TeX
C_8.{\rm Dic}_6 % in TeX
G:=Group("C8.Dic6"); // GroupNames label
G:=SmallGroup(192,46);
// by ID
G=gap.SmallGroup(192,46);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,365,36,758,346,80,851,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^8=b^12=1,c^2=a*b^6,b*a*b^-1=a^3,c*a*c^-1=a^5,c*b*c^-1=a^-1*b^-1>;
// generators/relations
Export
Subgroup lattice of C8.Dic6 in TeX
Character table of C8.Dic6 in TeX