Aliases: Q8.D21, C14.1S4, C7⋊CSU2(𝔽3), SL2(𝔽3).D7, C2.2(C7⋊S4), (C7×Q8).1S3, (C7×SL2(𝔽3)).1C2, SmallGroup(336,118)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C7×SL2(𝔽3) — Q8.D21 |
| C1 — C2 — Q8 — C7×Q8 — C7×SL2(𝔽3) — Q8.D21 |
| C7×SL2(𝔽3) — Q8.D21 |
Generators and relations for Q8.D21
G = < a,b,c,d | a4=c21=1, b2=d2=a2, bab-1=a-1, cac-1=b, dad-1=a-1b, cbc-1=ab, dbd-1=a2b, dcd-1=c-1 >
Character table of Q8.D21
| class | 1 | 2 | 3 | 4A | 4B | 6 | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 21A | 21B | 21C | 21D | 21E | 21F | 28A | 28B | 28C | 42A | 42B | 42C | 42D | 42E | 42F | |
| size | 1 | 1 | 8 | 6 | 84 | 8 | 2 | 2 | 2 | 42 | 42 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 8 | 8 | 8 | 8 | 8 | 8 | |
| 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 | 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 | linear of order 2 |
| ρ3 | 2 | 2 | -1 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ4 | 2 | 2 | 2 | 2 | 0 | 2 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ5 | 2 | 2 | 2 | 2 | 0 | 2 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ6 | 2 | 2 | 2 | 2 | 0 | 2 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ7 | 2 | 2 | -1 | 2 | 0 | -1 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ32ζ75+ζ32ζ72-ζ75 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ3ζ76+ζ3ζ7-ζ76 | ζ3ζ76-ζ3ζ7-ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ3ζ76+ζ3ζ7-ζ76 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ32ζ75+ζ32ζ72-ζ75 | -ζ3ζ75+ζ3ζ72-ζ75 | orthogonal lifted from D21 |
| ρ8 | 2 | 2 | -1 | 2 | 0 | -1 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ3ζ76+ζ3ζ7-ζ76 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ3ζ76-ζ3ζ7-ζ7 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ32ζ75+ζ32ζ72-ζ75 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ32ζ75+ζ32ζ72-ζ75 | -ζ3ζ76+ζ3ζ7-ζ76 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ32ζ74-ζ32ζ73-ζ73 | orthogonal lifted from D21 |
| ρ9 | 2 | 2 | -1 | 2 | 0 | -1 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ3ζ76+ζ3ζ7-ζ76 | -ζ32ζ75+ζ32ζ72-ζ75 | ζ3ζ76-ζ3ζ7-ζ7 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | -ζ32ζ75+ζ32ζ72-ζ75 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ3ζ76+ζ3ζ7-ζ76 | ζ3ζ76-ζ3ζ7-ζ7 | orthogonal lifted from D21 |
| ρ10 | 2 | 2 | -1 | 2 | 0 | -1 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ32ζ75+ζ32ζ72-ζ75 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ3ζ76+ζ3ζ7-ζ76 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ3ζ76+ζ3ζ7-ζ76 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ32ζ75+ζ32ζ72-ζ75 | orthogonal lifted from D21 |
| ρ11 | 2 | 2 | -1 | 2 | 0 | -1 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ3ζ76-ζ3ζ7-ζ7 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ3ζ76+ζ3ζ7-ζ76 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ32ζ75+ζ32ζ72-ζ75 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | -ζ3ζ76+ζ3ζ7-ζ76 | -ζ32ζ75+ζ32ζ72-ζ75 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ3ζ76-ζ3ζ7-ζ7 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ32ζ74+ζ32ζ73-ζ74 | orthogonal lifted from D21 |
| ρ12 | 2 | 2 | -1 | 2 | 0 | -1 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ32ζ75+ζ32ζ72-ζ75 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ3ζ76+ζ3ζ7-ζ76 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ32ζ74-ζ32ζ73-ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ32ζ75+ζ32ζ72-ζ75 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ3ζ76+ζ3ζ7-ζ76 | orthogonal lifted from D21 |
| ρ13 | 2 | -2 | -1 | 0 | 0 | 1 | 2 | 2 | 2 | √2 | -√2 | -2 | -2 | -2 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ14 | 2 | -2 | -1 | 0 | 0 | 1 | 2 | 2 | 2 | -√2 | √2 | -2 | -2 | -2 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ15 | 3 | 3 | 0 | -1 | -1 | 0 | 3 | 3 | 3 | 1 | 1 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ16 | 3 | 3 | 0 | -1 | 1 | 0 | 3 | 3 | 3 | -1 | -1 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ17 | 4 | -4 | 1 | 0 | 0 | -1 | 4 | 4 | 4 | 0 | 0 | -4 | -4 | -4 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ18 | 4 | -4 | -2 | 0 | 0 | 2 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 0 | 0 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ75-ζ72 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | symplectic faithful, Schur index 2 |
| ρ19 | 4 | -4 | -2 | 0 | 0 | 2 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 0 | 0 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ76-ζ7 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | symplectic faithful, Schur index 2 |
| ρ20 | 4 | -4 | -2 | 0 | 0 | 2 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 0 | 0 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ74-ζ73 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | symplectic faithful, Schur index 2 |
| ρ21 | 4 | -4 | 1 | 0 | 0 | -1 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 0 | 0 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | -ζ3ζ75+ζ3ζ72+ζ72 | -ζ3ζ76+ζ3ζ7+ζ7 | -ζ32ζ75+ζ32ζ72+ζ72 | ζ3ζ76-ζ3ζ7+ζ76 | ζ32ζ74-ζ32ζ73+ζ74 | -ζ32ζ74+ζ32ζ73+ζ73 | 0 | 0 | 0 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ32ζ75+ζ32ζ72-ζ75 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ3ζ76+ζ3ζ7-ζ76 | symplectic faithful, Schur index 2 |
| ρ22 | 4 | -4 | 1 | 0 | 0 | -1 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 0 | 0 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | ζ32ζ74-ζ32ζ73+ζ74 | -ζ32ζ75+ζ32ζ72+ζ72 | -ζ32ζ74+ζ32ζ73+ζ73 | -ζ3ζ75+ζ3ζ72+ζ72 | -ζ3ζ76+ζ3ζ7+ζ7 | ζ3ζ76-ζ3ζ7+ζ76 | 0 | 0 | 0 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ3ζ76+ζ3ζ7-ζ76 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ32ζ75+ζ32ζ72-ζ75 | symplectic faithful, Schur index 2 |
| ρ23 | 4 | -4 | 1 | 0 | 0 | -1 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 0 | 0 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | ζ3ζ76-ζ3ζ7+ζ76 | ζ32ζ74-ζ32ζ73+ζ74 | -ζ3ζ76+ζ3ζ7+ζ7 | -ζ32ζ74+ζ32ζ73+ζ73 | -ζ3ζ75+ζ3ζ72+ζ72 | -ζ32ζ75+ζ32ζ72+ζ72 | 0 | 0 | 0 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ32ζ75+ζ32ζ72-ζ75 | -ζ3ζ76+ζ3ζ7-ζ76 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ32ζ74-ζ32ζ73-ζ73 | symplectic faithful, Schur index 2 |
| ρ24 | 4 | -4 | 1 | 0 | 0 | -1 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 0 | 0 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | -ζ32ζ75+ζ32ζ72+ζ72 | ζ3ζ76-ζ3ζ7+ζ76 | -ζ3ζ75+ζ3ζ72+ζ72 | -ζ3ζ76+ζ3ζ7+ζ7 | -ζ32ζ74+ζ32ζ73+ζ73 | ζ32ζ74-ζ32ζ73+ζ74 | 0 | 0 | 0 | -ζ32ζ75+ζ32ζ72-ζ75 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ3ζ76+ζ3ζ7-ζ76 | ζ3ζ76-ζ3ζ7-ζ7 | symplectic faithful, Schur index 2 |
| ρ25 | 4 | -4 | 1 | 0 | 0 | -1 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 0 | 0 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | -ζ32ζ74+ζ32ζ73+ζ73 | -ζ3ζ75+ζ3ζ72+ζ72 | ζ32ζ74-ζ32ζ73+ζ74 | -ζ32ζ75+ζ32ζ72+ζ72 | ζ3ζ76-ζ3ζ7+ζ76 | -ζ3ζ76+ζ3ζ7+ζ7 | 0 | 0 | 0 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ3ζ76+ζ3ζ7-ζ76 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ32ζ75+ζ32ζ72-ζ75 | -ζ3ζ75+ζ3ζ72-ζ75 | symplectic faithful, Schur index 2 |
| ρ26 | 4 | -4 | 1 | 0 | 0 | -1 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 0 | 0 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | -ζ3ζ76+ζ3ζ7+ζ7 | -ζ32ζ74+ζ32ζ73+ζ73 | ζ3ζ76-ζ3ζ7+ζ76 | ζ32ζ74-ζ32ζ73+ζ74 | -ζ32ζ75+ζ32ζ72+ζ72 | -ζ3ζ75+ζ3ζ72+ζ72 | 0 | 0 | 0 | -ζ3ζ76+ζ3ζ7-ζ76 | -ζ32ζ75+ζ32ζ72-ζ75 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ3ζ76-ζ3ζ7-ζ7 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ32ζ74+ζ32ζ73-ζ74 | symplectic faithful, Schur index 2 |
| ρ27 | 6 | 6 | 0 | -2 | 0 | 0 | 3ζ75+3ζ72 | 3ζ76+3ζ7 | 3ζ74+3ζ73 | 0 | 0 | 3ζ76+3ζ7 | 3ζ75+3ζ72 | 3ζ74+3ζ73 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C7⋊S4 |
| ρ28 | 6 | 6 | 0 | -2 | 0 | 0 | 3ζ76+3ζ7 | 3ζ74+3ζ73 | 3ζ75+3ζ72 | 0 | 0 | 3ζ74+3ζ73 | 3ζ76+3ζ7 | 3ζ75+3ζ72 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C7⋊S4 |
| ρ29 | 6 | 6 | 0 | -2 | 0 | 0 | 3ζ74+3ζ73 | 3ζ75+3ζ72 | 3ζ76+3ζ7 | 0 | 0 | 3ζ75+3ζ72 | 3ζ74+3ζ73 | 3ζ76+3ζ7 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C7⋊S4 |
►On 112 points
Generators in S112
(1 47 9 64)(2 41 10 58)(3 35 11 52)(4 29 12 67)(5 44 13 61)(6 38 14 55)(7 32 8 70)(15 93 28 90)(16 108 22 84)(17 102 23 78)(18 96 24 72)(19 111 25 87)(20 105 26 81)(21 99 27 75)(30 37 68 54)(31 62 69 45)(33 40 50 57)(34 65 51 48)(36 43 53 60)(39 46 56 63)(42 49 59 66)(71 109 95 85)(73 80 97 104)(74 112 98 88)(76 83 100 107)(77 94 101 91)(79 86 103 110)(82 89 106 92)
(1 40 9 57)(2 34 10 51)(3 49 11 66)(4 43 12 60)(5 37 13 54)(6 31 14 69)(7 46 8 63)(15 107 28 83)(16 101 22 77)(17 95 23 71)(18 110 24 86)(19 104 25 80)(20 98 26 74)(21 92 27 89)(29 36 67 53)(30 61 68 44)(32 39 70 56)(33 64 50 47)(35 42 52 59)(38 45 55 62)(41 48 58 65)(72 79 96 103)(73 111 97 87)(75 82 99 106)(76 93 100 90)(78 85 102 109)(81 88 105 112)(84 91 108 94)
(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 49)(50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91)(92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 26 9 20)(2 25 10 19)(3 24 11 18)(4 23 12 17)(5 22 13 16)(6 28 14 15)(7 27 8 21)(29 109 67 85)(30 108 68 84)(31 107 69 83)(32 106 70 82)(33 105 50 81)(34 104 51 80)(35 103 52 79)(36 102 53 78)(37 101 54 77)(38 100 55 76)(39 99 56 75)(40 98 57 74)(41 97 58 73)(42 96 59 72)(43 95 60 71)(44 94 61 91)(45 93 62 90)(46 92 63 89)(47 112 64 88)(48 111 65 87)(49 110 66 86)
G:=sub<Sym(112)| (1,47,9,64)(2,41,10,58)(3,35,11,52)(4,29,12,67)(5,44,13,61)(6,38,14,55)(7,32,8,70)(15,93,28,90)(16,108,22,84)(17,102,23,78)(18,96,24,72)(19,111,25,87)(20,105,26,81)(21,99,27,75)(30,37,68,54)(31,62,69,45)(33,40,50,57)(34,65,51,48)(36,43,53,60)(39,46,56,63)(42,49,59,66)(71,109,95,85)(73,80,97,104)(74,112,98,88)(76,83,100,107)(77,94,101,91)(79,86,103,110)(82,89,106,92), (1,40,9,57)(2,34,10,51)(3,49,11,66)(4,43,12,60)(5,37,13,54)(6,31,14,69)(7,46,8,63)(15,107,28,83)(16,101,22,77)(17,95,23,71)(18,110,24,86)(19,104,25,80)(20,98,26,74)(21,92,27,89)(29,36,67,53)(30,61,68,44)(32,39,70,56)(33,64,50,47)(35,42,52,59)(38,45,55,62)(41,48,58,65)(72,79,96,103)(73,111,97,87)(75,82,99,106)(76,93,100,90)(78,85,102,109)(81,88,105,112)(84,91,108,94), (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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49)(50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,26,9,20)(2,25,10,19)(3,24,11,18)(4,23,12,17)(5,22,13,16)(6,28,14,15)(7,27,8,21)(29,109,67,85)(30,108,68,84)(31,107,69,83)(32,106,70,82)(33,105,50,81)(34,104,51,80)(35,103,52,79)(36,102,53,78)(37,101,54,77)(38,100,55,76)(39,99,56,75)(40,98,57,74)(41,97,58,73)(42,96,59,72)(43,95,60,71)(44,94,61,91)(45,93,62,90)(46,92,63,89)(47,112,64,88)(48,111,65,87)(49,110,66,86)>;
G:=Group( (1,47,9,64)(2,41,10,58)(3,35,11,52)(4,29,12,67)(5,44,13,61)(6,38,14,55)(7,32,8,70)(15,93,28,90)(16,108,22,84)(17,102,23,78)(18,96,24,72)(19,111,25,87)(20,105,26,81)(21,99,27,75)(30,37,68,54)(31,62,69,45)(33,40,50,57)(34,65,51,48)(36,43,53,60)(39,46,56,63)(42,49,59,66)(71,109,95,85)(73,80,97,104)(74,112,98,88)(76,83,100,107)(77,94,101,91)(79,86,103,110)(82,89,106,92), (1,40,9,57)(2,34,10,51)(3,49,11,66)(4,43,12,60)(5,37,13,54)(6,31,14,69)(7,46,8,63)(15,107,28,83)(16,101,22,77)(17,95,23,71)(18,110,24,86)(19,104,25,80)(20,98,26,74)(21,92,27,89)(29,36,67,53)(30,61,68,44)(32,39,70,56)(33,64,50,47)(35,42,52,59)(38,45,55,62)(41,48,58,65)(72,79,96,103)(73,111,97,87)(75,82,99,106)(76,93,100,90)(78,85,102,109)(81,88,105,112)(84,91,108,94), (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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49)(50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,26,9,20)(2,25,10,19)(3,24,11,18)(4,23,12,17)(5,22,13,16)(6,28,14,15)(7,27,8,21)(29,109,67,85)(30,108,68,84)(31,107,69,83)(32,106,70,82)(33,105,50,81)(34,104,51,80)(35,103,52,79)(36,102,53,78)(37,101,54,77)(38,100,55,76)(39,99,56,75)(40,98,57,74)(41,97,58,73)(42,96,59,72)(43,95,60,71)(44,94,61,91)(45,93,62,90)(46,92,63,89)(47,112,64,88)(48,111,65,87)(49,110,66,86) );
G=PermutationGroup([[(1,47,9,64),(2,41,10,58),(3,35,11,52),(4,29,12,67),(5,44,13,61),(6,38,14,55),(7,32,8,70),(15,93,28,90),(16,108,22,84),(17,102,23,78),(18,96,24,72),(19,111,25,87),(20,105,26,81),(21,99,27,75),(30,37,68,54),(31,62,69,45),(33,40,50,57),(34,65,51,48),(36,43,53,60),(39,46,56,63),(42,49,59,66),(71,109,95,85),(73,80,97,104),(74,112,98,88),(76,83,100,107),(77,94,101,91),(79,86,103,110),(82,89,106,92)], [(1,40,9,57),(2,34,10,51),(3,49,11,66),(4,43,12,60),(5,37,13,54),(6,31,14,69),(7,46,8,63),(15,107,28,83),(16,101,22,77),(17,95,23,71),(18,110,24,86),(19,104,25,80),(20,98,26,74),(21,92,27,89),(29,36,67,53),(30,61,68,44),(32,39,70,56),(33,64,50,47),(35,42,52,59),(38,45,55,62),(41,48,58,65),(72,79,96,103),(73,111,97,87),(75,82,99,106),(76,93,100,90),(78,85,102,109),(81,88,105,112),(84,91,108,94)], [(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),(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49),(50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91),(92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,26,9,20),(2,25,10,19),(3,24,11,18),(4,23,12,17),(5,22,13,16),(6,28,14,15),(7,27,8,21),(29,109,67,85),(30,108,68,84),(31,107,69,83),(32,106,70,82),(33,105,50,81),(34,104,51,80),(35,103,52,79),(36,102,53,78),(37,101,54,77),(38,100,55,76),(39,99,56,75),(40,98,57,74),(41,97,58,73),(42,96,59,72),(43,95,60,71),(44,94,61,91),(45,93,62,90),(46,92,63,89),(47,112,64,88),(48,111,65,87),(49,110,66,86)]])
Matrix representation of Q8.D21 ►in GL4(𝔽337) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 312 | 35 |
| 0 | 0 | 11 | 25 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 326 |
| 0 | 0 | 26 | 301 |
| 63 | 234 | 0 | 0 |
| 103 | 297 | 0 | 0 |
| 0 | 0 | 336 | 1 |
| 0 | 0 | 336 | 0 |
| 123 | 8 | 0 | 0 |
| 131 | 214 | 0 | 0 |
| 0 | 0 | 142 | 119 |
| 0 | 0 | 261 | 195 |
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,312,11,0,0,35,25],[1,0,0,0,0,1,0,0,0,0,36,26,0,0,326,301],[63,103,0,0,234,297,0,0,0,0,336,336,0,0,1,0],[123,131,0,0,8,214,0,0,0,0,142,261,0,0,119,195] >;
Q8.D21 in GAP, Magma, Sage, TeX
Q_8.D_{21} % in TeX
G:=Group("Q8.D21"); // GroupNames label
G:=SmallGroup(336,118);
// by ID
G=gap.SmallGroup(336,118);
# by ID
G:=PCGroup([6,-2,-3,-7,-2,2,-2,1008,49,650,2019,3033,117,1264,1900,202,88]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^21=1,b^2=d^2=a^2,b*a*b^-1=a^-1,c*a*c^-1=b,d*a*d^-1=a^-1*b,c*b*c^-1=a*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of Q8.D21 in TeX
Character table of Q8.D21 in TeX