metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.6D6, C3⋊3SD32, Q16⋊1S3, C6.10D8, C12.5D4, D24.2C2, C24.4C22, C3⋊C16⋊3C2, (C3×Q16)⋊1C2, C2.6(D4⋊S3), C4.3(C3⋊D4), SmallGroup(96,35)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.6D6
G = < a,b,c | a8=1, b6=a4, c2=a3, bab-1=a-1, ac=ca, cbc-1=a-1b5 >
Character table of C8.6D6
| class | 1 | 2A | 2B | 3 | 4A | 4B | 6 | 8A | 8B | 12A | 12B | 12C | 16A | 16B | 16C | 16D | 24A | 24B | |
| size | 1 | 1 | 24 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 8 | 8 | 6 | 6 | 6 | 6 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | -1 | 2 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 0 | -1 | 2 | -2 | -1 | 2 | 2 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | 0 | 2 | 2 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
| ρ8 | 2 | 2 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | -√2 | √2 | -√2 | √2 | 0 | 0 | orthogonal lifted from D8 |
| ρ9 | 2 | 2 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | √2 | -√2 | √2 | -√2 | 0 | 0 | orthogonal lifted from D8 |
| ρ10 | 2 | 2 | 0 | -1 | 2 | 0 | -1 | -2 | -2 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ11 | 2 | 2 | 0 | -1 | 2 | 0 | -1 | -2 | -2 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ12 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | √2 | -√2 | 0 | 0 | 0 | ζ1613+ζ1611 | ζ167+ζ16 | ζ165+ζ163 | ζ1615+ζ169 | -√2 | √2 | complex lifted from SD32 |
| ρ13 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | -√2 | √2 | 0 | 0 | 0 | ζ167+ζ16 | ζ165+ζ163 | ζ1615+ζ169 | ζ1613+ζ1611 | √2 | -√2 | complex lifted from SD32 |
| ρ14 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | -√2 | √2 | 0 | 0 | 0 | ζ1615+ζ169 | ζ1613+ζ1611 | ζ167+ζ16 | ζ165+ζ163 | √2 | -√2 | complex lifted from SD32 |
| ρ15 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | √2 | -√2 | 0 | 0 | 0 | ζ165+ζ163 | ζ1615+ζ169 | ζ1613+ζ1611 | ζ167+ζ16 | -√2 | √2 | complex lifted from SD32 |
| ρ16 | 4 | 4 | 0 | -2 | -4 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
| ρ17 | 4 | -4 | 0 | -2 | 0 | 0 | 2 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | orthogonal faithful, Schur index 2 |
| ρ18 | 4 | -4 | 0 | -2 | 0 | 0 | 2 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | orthogonal faithful, Schur index 2 |
►On 48 points
Generators in S48
(1 7 13 3 9 15 5 11)(2 8 14 4 10 16 6 12)(17 23 29 19 25 31 21 27)(18 24 30 20 26 32 22 28)(33 39 45 35 41 47 37 43)(34 40 46 36 42 48 38 44)
(1 46 21 22 45 2 9 38 29 30 37 10)(3 44 23 20 47 16 11 36 31 28 39 8)(4 7 40 27 32 35 12 15 48 19 24 43)(5 42 25 18 33 14 13 34 17 26 41 6)
(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)| (1,7,13,3,9,15,5,11)(2,8,14,4,10,16,6,12)(17,23,29,19,25,31,21,27)(18,24,30,20,26,32,22,28)(33,39,45,35,41,47,37,43)(34,40,46,36,42,48,38,44), (1,46,21,22,45,2,9,38,29,30,37,10)(3,44,23,20,47,16,11,36,31,28,39,8)(4,7,40,27,32,35,12,15,48,19,24,43)(5,42,25,18,33,14,13,34,17,26,41,6), (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( (1,7,13,3,9,15,5,11)(2,8,14,4,10,16,6,12)(17,23,29,19,25,31,21,27)(18,24,30,20,26,32,22,28)(33,39,45,35,41,47,37,43)(34,40,46,36,42,48,38,44), (1,46,21,22,45,2,9,38,29,30,37,10)(3,44,23,20,47,16,11,36,31,28,39,8)(4,7,40,27,32,35,12,15,48,19,24,43)(5,42,25,18,33,14,13,34,17,26,41,6), (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([[(1,7,13,3,9,15,5,11),(2,8,14,4,10,16,6,12),(17,23,29,19,25,31,21,27),(18,24,30,20,26,32,22,28),(33,39,45,35,41,47,37,43),(34,40,46,36,42,48,38,44)], [(1,46,21,22,45,2,9,38,29,30,37,10),(3,44,23,20,47,16,11,36,31,28,39,8),(4,7,40,27,32,35,12,15,48,19,24,43),(5,42,25,18,33,14,13,34,17,26,41,6)], [(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)]])
C8.6D6 is a maximal subgroup of
S3×SD32 D48⋊C2 Q32⋊S3 D48⋊5C2 C24.27C23 Q16⋊D6 Q16.D6 C9⋊SD32 D24.S3 C24.49D6 C32⋊10SD32 C15⋊SD32 C24.D10 C8.6D30
C8.6D6 is a maximal quotient of
C6.SD32 C6.D16 C6.5Q32 C9⋊SD32 D24.S3 C24.49D6 C32⋊10SD32 C15⋊SD32 C24.D10 C8.6D30
Matrix representation of C8.6D6 ►in GL4(𝔽7) generated by
| 6 | 3 | 0 | 2 |
| 1 | 6 | 1 | 1 |
| 2 | 2 | 2 | 3 |
| 3 | 4 | 2 | 6 |
| 2 | 5 | 1 | 5 |
| 0 | 0 | 2 | 4 |
| 2 | 1 | 1 | 5 |
| 1 | 2 | 1 | 4 |
| 0 | 1 | 0 | 2 |
| 2 | 6 | 6 | 3 |
| 2 | 6 | 5 | 2 |
| 4 | 2 | 4 | 3 |
G:=sub<GL(4,GF(7))| [6,1,2,3,3,6,2,4,0,1,2,2,2,1,3,6],[2,0,2,1,5,0,1,2,1,2,1,1,5,4,5,4],[0,2,2,4,1,6,6,2,0,6,5,4,2,3,2,3] >;
C8.6D6 in GAP, Magma, Sage, TeX
C_8._6D_6
% in TeX
G:=Group("C8.6D6"); // GroupNames label
G:=SmallGroup(96,35);
// by ID
G=gap.SmallGroup(96,35);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,73,103,218,116,122,579,297,69,2309]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^6=a^4,c^2=a^3,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^5>;
// generators/relations
Export
Subgroup lattice of C8.6D6 in TeX
Character table of C8.6D6 in TeX