Aliases: Q8⋊D21, C14.2S4, C7⋊GL2(𝔽3), SL2(𝔽3)⋊D7, C2.3(C7⋊S4), (C7×Q8)⋊1S3, (C7×SL2(𝔽3))⋊1C2, SmallGroup(336,119)
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=d2=1, b2=a2, bab-1=a-1, cac-1=b, dad=a-1b, cbc-1=ab, dbd=a2b, dcd=c-1 >
Character table of Q8⋊D21
| class | 1 | 2A | 2B | 3 | 4 | 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 | 84 | 8 | 6 | 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 | 0 | -1 | 2 | -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 | 0 | 2 | 2 | 2 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | 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 | 0 | 2 | 2 | 2 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | 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 |
| ρ6 | 2 | 2 | 0 | 2 | 2 | 2 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | 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 |
| ρ7 | 2 | 2 | 0 | -1 | 2 | -1 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ3ζ75-ζ3ζ72-ζ72 | -ζ32ζ76+ζ32ζ7-ζ76 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ32ζ76-ζ32ζ7-ζ7 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ74+ζ3ζ73-ζ74 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ3ζ74+ζ3ζ73-ζ74 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ3ζ75-ζ3ζ72-ζ72 | -ζ32ζ76+ζ32ζ7-ζ76 | ζ32ζ76-ζ32ζ7-ζ7 | orthogonal lifted from D21 |
| ρ8 | 2 | 2 | 0 | -1 | 2 | -1 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ3ζ74+ζ3ζ73-ζ74 | ζ3ζ75-ζ3ζ72-ζ72 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ32ζ76-ζ32ζ7-ζ7 | -ζ32ζ76+ζ32ζ7-ζ76 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ32ζ76+ζ32ζ7-ζ76 | ζ32ζ76-ζ32ζ7-ζ7 | -ζ3ζ74+ζ3ζ73-ζ74 | ζ3ζ75-ζ3ζ72-ζ72 | -ζ3ζ75+ζ3ζ72-ζ75 | orthogonal lifted from D21 |
| ρ9 | 2 | 2 | 0 | -1 | 2 | -1 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ32ζ76-ζ32ζ7-ζ7 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ32ζ76+ζ32ζ7-ζ76 | -ζ3ζ74+ζ3ζ73-ζ74 | ζ3ζ75-ζ3ζ72-ζ72 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | -ζ32ζ76+ζ32ζ7-ζ76 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ3ζ75-ζ3ζ72-ζ72 | ζ32ζ76-ζ32ζ7-ζ7 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ74+ζ3ζ73-ζ74 | orthogonal lifted from D21 |
| ρ10 | 2 | 2 | 0 | -1 | 2 | -1 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ32ζ76+ζ32ζ7-ζ76 | -ζ3ζ74+ζ3ζ73-ζ74 | ζ32ζ76-ζ32ζ7-ζ7 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ3ζ75-ζ3ζ72-ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ32ζ76-ζ32ζ7-ζ7 | ζ3ζ75-ζ3ζ72-ζ72 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ32ζ76+ζ32ζ7-ζ76 | -ζ3ζ74+ζ3ζ73-ζ74 | -ζ32ζ74+ζ32ζ73-ζ74 | orthogonal lifted from D21 |
| ρ11 | 2 | 2 | 0 | -1 | 2 | -1 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ3ζ74+ζ3ζ73-ζ74 | ζ3ζ75-ζ3ζ72-ζ72 | -ζ32ζ76+ζ32ζ7-ζ76 | ζ32ζ76-ζ32ζ7-ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | -ζ3ζ74+ζ3ζ73-ζ74 | ζ32ζ76-ζ32ζ7-ζ7 | -ζ32ζ76+ζ32ζ7-ζ76 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ3ζ75-ζ3ζ72-ζ72 | orthogonal lifted from D21 |
| ρ12 | 2 | 2 | 0 | -1 | 2 | -1 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ32ζ76-ζ32ζ7-ζ7 | ζ3ζ75-ζ3ζ72-ζ72 | -ζ32ζ76+ζ32ζ7-ζ76 | -ζ3ζ74+ζ3ζ73-ζ74 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ3ζ75-ζ3ζ72-ζ72 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ74+ζ3ζ73-ζ74 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ32ζ76-ζ32ζ7-ζ7 | -ζ32ζ76+ζ32ζ7-ζ76 | orthogonal lifted from D21 |
| ρ13 | 2 | -2 | 0 | -1 | 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 | complex lifted from GL2(𝔽3) |
| ρ14 | 2 | -2 | 0 | -1 | 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 | complex lifted from GL2(𝔽3) |
| ρ15 | 3 | 3 | -1 | 0 | -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 | 1 | 0 | -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 | 0 | 1 | 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 | orthogonal lifted from GL2(𝔽3) |
| ρ18 | 4 | -4 | 0 | -2 | 0 | 2 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 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 | orthogonal faithful, Schur index 2 |
| ρ19 | 4 | -4 | 0 | -2 | 0 | 2 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 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 | orthogonal faithful, Schur index 2 |
| ρ20 | 4 | -4 | 0 | -2 | 0 | 2 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 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 | orthogonal faithful, Schur index 2 |
| ρ21 | 4 | -4 | 0 | 1 | 0 | -1 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 0 | 0 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | ζ3ζ75-ζ3ζ72+ζ75 | ζ3ζ76-ζ3ζ7+ζ76 | ζ32ζ75-ζ32ζ72+ζ75 | -ζ3ζ76+ζ3ζ7+ζ7 | -ζ32ζ74+ζ32ζ73+ζ73 | ζ32ζ74-ζ32ζ73+ζ74 | 0 | 0 | 0 | ζ3ζ75-ζ3ζ72-ζ72 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ74+ζ3ζ73-ζ74 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ32ζ76-ζ32ζ7-ζ7 | -ζ32ζ76+ζ32ζ7-ζ76 | orthogonal faithful |
| ρ22 | 4 | -4 | 0 | 1 | 0 | -1 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 0 | 0 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | ζ3ζ76-ζ3ζ7+ζ76 | ζ32ζ74-ζ32ζ73+ζ74 | -ζ3ζ76+ζ3ζ7+ζ7 | -ζ32ζ74+ζ32ζ73+ζ73 | ζ32ζ75-ζ32ζ72+ζ75 | ζ3ζ75-ζ3ζ72+ζ75 | 0 | 0 | 0 | -ζ32ζ76+ζ32ζ7-ζ76 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ3ζ75-ζ3ζ72-ζ72 | ζ32ζ76-ζ32ζ7-ζ7 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ74+ζ3ζ73-ζ74 | orthogonal faithful |
| ρ23 | 4 | -4 | 0 | 1 | 0 | -1 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 0 | 0 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | ζ32ζ74-ζ32ζ73+ζ74 | ζ3ζ75-ζ3ζ72+ζ75 | -ζ32ζ74+ζ32ζ73+ζ73 | ζ32ζ75-ζ32ζ72+ζ75 | -ζ3ζ76+ζ3ζ7+ζ7 | ζ3ζ76-ζ3ζ7+ζ76 | 0 | 0 | 0 | -ζ3ζ74+ζ3ζ73-ζ74 | ζ32ζ76-ζ32ζ7-ζ7 | -ζ32ζ76+ζ32ζ7-ζ76 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ3ζ75-ζ3ζ72-ζ72 | orthogonal faithful |
| ρ24 | 4 | -4 | 0 | 1 | 0 | -1 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 0 | 0 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | ζ32ζ75-ζ32ζ72+ζ75 | -ζ3ζ76+ζ3ζ7+ζ7 | ζ3ζ75-ζ3ζ72+ζ75 | ζ3ζ76-ζ3ζ7+ζ76 | ζ32ζ74-ζ32ζ73+ζ74 | -ζ32ζ74+ζ32ζ73+ζ73 | 0 | 0 | 0 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ3ζ74+ζ3ζ73-ζ74 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ3ζ75-ζ3ζ72-ζ72 | -ζ32ζ76+ζ32ζ7-ζ76 | ζ32ζ76-ζ32ζ7-ζ7 | orthogonal faithful |
| ρ25 | 4 | -4 | 0 | 1 | 0 | -1 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 0 | 0 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | -ζ32ζ74+ζ32ζ73+ζ73 | ζ32ζ75-ζ32ζ72+ζ75 | ζ32ζ74-ζ32ζ73+ζ74 | ζ3ζ75-ζ3ζ72+ζ75 | ζ3ζ76-ζ3ζ7+ζ76 | -ζ3ζ76+ζ3ζ7+ζ7 | 0 | 0 | 0 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ32ζ76+ζ32ζ7-ζ76 | ζ32ζ76-ζ32ζ7-ζ7 | -ζ3ζ74+ζ3ζ73-ζ74 | ζ3ζ75-ζ3ζ72-ζ72 | -ζ3ζ75+ζ3ζ72-ζ75 | orthogonal faithful |
| ρ26 | 4 | -4 | 0 | 1 | 0 | -1 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 0 | 0 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | -ζ3ζ76+ζ3ζ7+ζ7 | -ζ32ζ74+ζ32ζ73+ζ73 | ζ3ζ76-ζ3ζ7+ζ76 | ζ32ζ74-ζ32ζ73+ζ74 | ζ3ζ75-ζ3ζ72+ζ75 | ζ32ζ75-ζ32ζ72+ζ75 | 0 | 0 | 0 | ζ32ζ76-ζ32ζ7-ζ7 | ζ3ζ75-ζ3ζ72-ζ72 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ32ζ76+ζ32ζ7-ζ76 | -ζ3ζ74+ζ3ζ73-ζ74 | -ζ32ζ74+ζ32ζ73-ζ74 | orthogonal faithful |
| ρ27 | 6 | 6 | 0 | 0 | -2 | 0 | 3ζ76+3ζ7 | 3ζ75+3ζ72 | 3ζ74+3ζ73 | 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 |
| ρ28 | 6 | 6 | 0 | 0 | -2 | 0 | 3ζ75+3ζ72 | 3ζ74+3ζ73 | 3ζ76+3ζ7 | 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 | 0 | -2 | 0 | 3ζ74+3ζ73 | 3ζ76+3ζ7 | 3ζ75+3ζ72 | 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 |
►On 56 points
Generators in S56
(1 54 8 21)(2 48 9 15)(3 42 10 30)(4 36 11 24)(5 51 12 18)(6 45 13 33)(7 39 14 27)(16 23 49 56)(17 43 50 31)(19 26 52 38)(20 46 53 34)(22 29 55 41)(25 32 37 44)(28 35 40 47)
(1 47 8 35)(2 41 9 29)(3 56 10 23)(4 50 11 17)(5 44 12 32)(6 38 13 26)(7 53 14 20)(15 22 48 55)(16 42 49 30)(18 25 51 37)(19 45 52 33)(21 28 54 40)(24 31 36 43)(27 34 39 46)
(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)
(1 5)(2 4)(6 7)(8 12)(9 11)(13 14)(15 43)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 36)(23 56)(24 55)(25 54)(26 53)(27 52)(28 51)(29 50)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)
G:=sub<Sym(56)| (1,54,8,21)(2,48,9,15)(3,42,10,30)(4,36,11,24)(5,51,12,18)(6,45,13,33)(7,39,14,27)(16,23,49,56)(17,43,50,31)(19,26,52,38)(20,46,53,34)(22,29,55,41)(25,32,37,44)(28,35,40,47), (1,47,8,35)(2,41,9,29)(3,56,10,23)(4,50,11,17)(5,44,12,32)(6,38,13,26)(7,53,14,20)(15,22,48,55)(16,42,49,30)(18,25,51,37)(19,45,52,33)(21,28,54,40)(24,31,36,43)(27,34,39,46), (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), (1,5)(2,4)(6,7)(8,12)(9,11)(13,14)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)>;
G:=Group( (1,54,8,21)(2,48,9,15)(3,42,10,30)(4,36,11,24)(5,51,12,18)(6,45,13,33)(7,39,14,27)(16,23,49,56)(17,43,50,31)(19,26,52,38)(20,46,53,34)(22,29,55,41)(25,32,37,44)(28,35,40,47), (1,47,8,35)(2,41,9,29)(3,56,10,23)(4,50,11,17)(5,44,12,32)(6,38,13,26)(7,53,14,20)(15,22,48,55)(16,42,49,30)(18,25,51,37)(19,45,52,33)(21,28,54,40)(24,31,36,43)(27,34,39,46), (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), (1,5)(2,4)(6,7)(8,12)(9,11)(13,14)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44) );
G=PermutationGroup([[(1,54,8,21),(2,48,9,15),(3,42,10,30),(4,36,11,24),(5,51,12,18),(6,45,13,33),(7,39,14,27),(16,23,49,56),(17,43,50,31),(19,26,52,38),(20,46,53,34),(22,29,55,41),(25,32,37,44),(28,35,40,47)], [(1,47,8,35),(2,41,9,29),(3,56,10,23),(4,50,11,17),(5,44,12,32),(6,38,13,26),(7,53,14,20),(15,22,48,55),(16,42,49,30),(18,25,51,37),(19,45,52,33),(21,28,54,40),(24,31,36,43),(27,34,39,46)], [(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)], [(1,5),(2,4),(6,7),(8,12),(9,11),(13,14),(15,43),(16,42),(17,41),(18,40),(19,39),(20,38),(21,37),(22,36),(23,56),(24,55),(25,54),(26,53),(27,52),(28,51),(29,50),(30,49),(31,48),(32,47),(33,46),(34,45),(35,44)]])
Matrix representation of Q8⋊D21 ►in GL4(𝔽337) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 281 |
| 0 | 0 | 277 | 334 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 278 | 335 |
| 0 | 0 | 56 | 59 |
| 274 | 20 | 0 | 0 |
| 257 | 234 | 0 | 0 |
| 0 | 0 | 56 | 59 |
| 0 | 0 | 3 | 280 |
| 247 | 287 | 0 | 0 |
| 317 | 90 | 0 | 0 |
| 0 | 0 | 186 | 282 |
| 0 | 0 | 292 | 151 |
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,3,277,0,0,281,334],[1,0,0,0,0,1,0,0,0,0,278,56,0,0,335,59],[274,257,0,0,20,234,0,0,0,0,56,3,0,0,59,280],[247,317,0,0,287,90,0,0,0,0,186,292,0,0,282,151] >;
Q8⋊D21 in GAP, Magma, Sage, TeX
Q_8\rtimes D_{21} % in TeX
G:=Group("Q8:D21"); // GroupNames label
G:=SmallGroup(336,119);
// by ID
G=gap.SmallGroup(336,119);
# by ID
G:=PCGroup([6,-2,-3,-7,-2,2,-2,49,650,2019,3033,117,1264,1900,202,88]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^21=d^2=1,b^2=a^2,b*a*b^-1=a^-1,c*a*c^-1=b,d*a*d=a^-1*b,c*b*c^-1=a*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of Q8⋊D21 in TeX
Character table of Q8⋊D21 in TeX