Aliases: Q8⋊Dic3, C22.3S4, C2.GL2(𝔽3), C2.CSU2(𝔽3), SL2(𝔽3)⋊1C4, (C2×Q8).1S3, C2.2(A4⋊C4), (C2×SL2(𝔽3)).1C2, SmallGroup(96,66)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — Q8⋊Dic3 |
| SL2(𝔽3) — Q8⋊Dic3 |
Generators and relations for Q8⋊Dic3
G = < a,b,c,d | a4=c6=1, b2=a2, d2=c3, bab-1=dbd-1=a-1, cac-1=b, dad-1=a2b, cbc-1=ab, dcd-1=c-1 >
Character table of Q8⋊Dic3
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 8 | 6 | 6 | 12 | 12 | 8 | 8 | 8 | 6 | 6 | 6 | 6 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | i | -1 | 1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | i | -i | -1 | 1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ5 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ7 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √2 | -√2 | √2 | -√2 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ8 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√2 | √2 | -√2 | √2 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ9 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -√-2 | √-2 | √-2 | -√-2 | complex lifted from GL2(𝔽3) |
| ρ10 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | √-2 | -√-2 | -√-2 | √-2 | complex lifted from GL2(𝔽3) |
| ρ11 | 3 | 3 | 3 | 3 | 0 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ12 | 3 | 3 | 3 | 3 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from S4 |
| ρ13 | 3 | 3 | -3 | -3 | 0 | -1 | 1 | -i | i | 0 | 0 | 0 | -i | -i | i | i | complex lifted from A4⋊C4 |
| ρ14 | 3 | 3 | -3 | -3 | 0 | -1 | 1 | i | -i | 0 | 0 | 0 | i | i | -i | -i | complex lifted from A4⋊C4 |
| ρ15 | 4 | -4 | -4 | 4 | 1 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from GL2(𝔽3) |
| ρ16 | 4 | -4 | 4 | -4 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
►On 32 points
(1 20 7 14)(2 17 8 11)(3 26 6 31)(4 23 5 28)(9 13 15 19)(10 18 16 12)(21 25 32 30)(22 29 27 24)
(1 16 7 10)(2 19 8 13)(3 22 6 27)(4 25 5 30)(9 17 15 11)(12 20 18 14)(21 28 32 23)(24 31 29 26)
(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 5 2 6)(3 7 4 8)(9 29 12 32)(10 28 13 31)(11 27 14 30)(15 24 18 21)(16 23 19 26)(17 22 20 25)
G:=sub<Sym(32)| (1,20,7,14)(2,17,8,11)(3,26,6,31)(4,23,5,28)(9,13,15,19)(10,18,16,12)(21,25,32,30)(22,29,27,24), (1,16,7,10)(2,19,8,13)(3,22,6,27)(4,25,5,30)(9,17,15,11)(12,20,18,14)(21,28,32,23)(24,31,29,26), (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), (1,5,2,6)(3,7,4,8)(9,29,12,32)(10,28,13,31)(11,27,14,30)(15,24,18,21)(16,23,19,26)(17,22,20,25)>;
G:=Group( (1,20,7,14)(2,17,8,11)(3,26,6,31)(4,23,5,28)(9,13,15,19)(10,18,16,12)(21,25,32,30)(22,29,27,24), (1,16,7,10)(2,19,8,13)(3,22,6,27)(4,25,5,30)(9,17,15,11)(12,20,18,14)(21,28,32,23)(24,31,29,26), (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), (1,5,2,6)(3,7,4,8)(9,29,12,32)(10,28,13,31)(11,27,14,30)(15,24,18,21)(16,23,19,26)(17,22,20,25) );
G=PermutationGroup([[(1,20,7,14),(2,17,8,11),(3,26,6,31),(4,23,5,28),(9,13,15,19),(10,18,16,12),(21,25,32,30),(22,29,27,24)], [(1,16,7,10),(2,19,8,13),(3,22,6,27),(4,25,5,30),(9,17,15,11),(12,20,18,14),(21,28,32,23),(24,31,29,26)], [(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)], [(1,5,2,6),(3,7,4,8),(9,29,12,32),(10,28,13,31),(11,27,14,30),(15,24,18,21),(16,23,19,26),(17,22,20,25)]])
Q8⋊Dic3 is a maximal subgroup of
Q8⋊Dic6 C4×CSU2(𝔽3) CSU2(𝔽3)⋊C4 Q8.Dic6 SL2(𝔽3)⋊Q8 C4×GL2(𝔽3) GL2(𝔽3)⋊C4 C23.14S4 C23.15S4 C23.16S4 C4.A4⋊C4 SL2(𝔽3).D4 (C2×C4).S4 SL2(𝔽3)⋊D4 C6.GL2(𝔽3) C22.2S5 Q8⋊Dic15 D10.S4
Q8⋊Dic3 is a maximal quotient of
C2.U2(𝔽3) Q8⋊Dic9 C6.GL2(𝔽3) Q8⋊Dic15 D10.S4
Matrix representation of Q8⋊Dic3 ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 27 | 0 |
| 0 | 26 | 46 |
| 1 | 0 | 0 |
| 0 | 72 | 47 |
| 0 | 45 | 1 |
| 72 | 0 | 0 |
| 0 | 46 | 27 |
| 0 | 1 | 26 |
| 46 | 0 | 0 |
| 0 | 51 | 63 |
| 0 | 41 | 22 |
G:=sub<GL(3,GF(73))| [1,0,0,0,27,26,0,0,46],[1,0,0,0,72,45,0,47,1],[72,0,0,0,46,1,0,27,26],[46,0,0,0,51,41,0,63,22] >;
Q8⋊Dic3 in GAP, Magma, Sage, TeX
Q_8\rtimes {\rm Dic}_3 % in TeX
G:=Group("Q8:Dic3"); // GroupNames label
G:=SmallGroup(96,66);
// by ID
G=gap.SmallGroup(96,66);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,2,-2,12,146,579,447,117,364,286,202,88]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^6=1,b^2=a^2,d^2=c^3,b*a*b^-1=d*b*d^-1=a^-1,c*a*c^-1=b,d*a*d^-1=a^2*b,c*b*c^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of Q8⋊Dic3 in TeX
Character table of Q8⋊Dic3 in TeX