direct product, non-abelian, soluble
Aliases: D5×SL2(𝔽3), D10.A4, (Q8×D5)⋊C3, (C5×Q8)⋊C6, Q8⋊(C3×D5), C2.3(D5×A4), C10.2(C2×A4), C5⋊(C2×SL2(𝔽3)), (C5×SL2(𝔽3))⋊3C2, SmallGroup(240,109)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — C10 — C5×Q8 — C5×SL2(𝔽3) — D5×SL2(𝔽3) |
| C5×Q8 — D5×SL2(𝔽3) |
Generators and relations for D5×SL2(𝔽3)
G = < a,b,c,d,e | a5=b2=c4=e3=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >
Character table of D5×SL2(𝔽3)
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 10A | 10B | 15A | 15B | 15C | 15D | 20A | 20B | 30A | 30B | 30C | 30D | |
| size | 1 | 1 | 5 | 5 | 4 | 4 | 6 | 30 | 2 | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 8 | 8 | 8 | 8 | 12 | 12 | 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 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ4 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 6 |
| ρ5 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ7 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | -1-√5/2 | -1+√5/2 | 2 | 2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ8 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | -1+√5/2 | -1-√5/2 | 2 | 2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ9 | 2 | -2 | -2 | 2 | -1 | -1 | 0 | 0 | 2 | 2 | 1 | 1 | 1 | 1 | -1 | -1 | -2 | -2 | -1 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ10 | 2 | -2 | 2 | -2 | -1 | -1 | 0 | 0 | 2 | 2 | 1 | 1 | -1 | -1 | 1 | 1 | -2 | -2 | -1 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ11 | 2 | -2 | -2 | 2 | ζ65 | ζ6 | 0 | 0 | 2 | 2 | ζ32 | ζ3 | ζ3 | ζ32 | ζ65 | ζ6 | -2 | -2 | ζ6 | ζ65 | ζ65 | ζ6 | 0 | 0 | ζ32 | ζ3 | ζ32 | ζ3 | complex lifted from SL2(𝔽3) |
| ρ12 | 2 | -2 | -2 | 2 | ζ6 | ζ65 | 0 | 0 | 2 | 2 | ζ3 | ζ32 | ζ32 | ζ3 | ζ6 | ζ65 | -2 | -2 | ζ65 | ζ6 | ζ6 | ζ65 | 0 | 0 | ζ3 | ζ32 | ζ3 | ζ32 | complex lifted from SL2(𝔽3) |
| ρ13 | 2 | -2 | 2 | -2 | ζ65 | ζ6 | 0 | 0 | 2 | 2 | ζ32 | ζ3 | ζ65 | ζ6 | ζ3 | ζ32 | -2 | -2 | ζ6 | ζ65 | ζ65 | ζ6 | 0 | 0 | ζ32 | ζ3 | ζ32 | ζ3 | complex lifted from SL2(𝔽3) |
| ρ14 | 2 | -2 | 2 | -2 | ζ6 | ζ65 | 0 | 0 | 2 | 2 | ζ3 | ζ32 | ζ6 | ζ65 | ζ32 | ζ3 | -2 | -2 | ζ65 | ζ6 | ζ6 | ζ65 | 0 | 0 | ζ3 | ζ32 | ζ3 | ζ32 | complex lifted from SL2(𝔽3) |
| ρ15 | 2 | 2 | 0 | 0 | -1-√-3 | -1+√-3 | 2 | 0 | -1+√5/2 | -1-√5/2 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | ζ3ζ54+ζ3ζ5 | ζ32ζ53+ζ32ζ52 | ζ32ζ54+ζ32ζ5 | ζ3ζ53+ζ3ζ52 | -1+√5/2 | -1-√5/2 | ζ3ζ53+ζ3ζ52 | ζ32ζ54+ζ32ζ5 | ζ3ζ54+ζ3ζ5 | ζ32ζ53+ζ32ζ52 | complex lifted from C3×D5 |
| ρ16 | 2 | 2 | 0 | 0 | -1+√-3 | -1-√-3 | 2 | 0 | -1-√5/2 | -1+√5/2 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ32ζ53+ζ32ζ52 | ζ3ζ54+ζ3ζ5 | ζ3ζ53+ζ3ζ52 | ζ32ζ54+ζ32ζ5 | -1-√5/2 | -1+√5/2 | ζ32ζ54+ζ32ζ5 | ζ3ζ53+ζ3ζ52 | ζ32ζ53+ζ32ζ52 | ζ3ζ54+ζ3ζ5 | complex lifted from C3×D5 |
| ρ17 | 2 | 2 | 0 | 0 | -1-√-3 | -1+√-3 | 2 | 0 | -1-√5/2 | -1+√5/2 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ3ζ53+ζ3ζ52 | ζ32ζ54+ζ32ζ5 | ζ32ζ53+ζ32ζ52 | ζ3ζ54+ζ3ζ5 | -1-√5/2 | -1+√5/2 | ζ3ζ54+ζ3ζ5 | ζ32ζ53+ζ32ζ52 | ζ3ζ53+ζ3ζ52 | ζ32ζ54+ζ32ζ5 | complex lifted from C3×D5 |
| ρ18 | 2 | 2 | 0 | 0 | -1+√-3 | -1-√-3 | 2 | 0 | -1+√5/2 | -1-√5/2 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | ζ32ζ54+ζ32ζ5 | ζ3ζ53+ζ3ζ52 | ζ3ζ54+ζ3ζ5 | ζ32ζ53+ζ32ζ52 | -1+√5/2 | -1-√5/2 | ζ32ζ53+ζ32ζ52 | ζ3ζ54+ζ3ζ5 | ζ32ζ54+ζ32ζ5 | ζ3ζ53+ζ3ζ52 | complex lifted from C3×D5 |
| ρ19 | 3 | 3 | 3 | 3 | 0 | 0 | -1 | -1 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ20 | 3 | 3 | -3 | -3 | 0 | 0 | -1 | 1 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ21 | 4 | -4 | 0 | 0 | -2 | -2 | 0 | 0 | -1-√5 | -1+√5 | 2 | 2 | 0 | 0 | 0 | 0 | 1-√5 | 1+√5 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | symplectic faithful, Schur index 2 |
| ρ22 | 4 | -4 | 0 | 0 | -2 | -2 | 0 | 0 | -1+√5 | -1-√5 | 2 | 2 | 0 | 0 | 0 | 0 | 1+√5 | 1-√5 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | symplectic faithful, Schur index 2 |
| ρ23 | 4 | -4 | 0 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | -1-√5 | -1+√5 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 1-√5 | 1+√5 | -ζ32ζ53-ζ32ζ52 | -ζ3ζ54-ζ3ζ5 | -ζ3ζ53-ζ3ζ52 | -ζ32ζ54-ζ32ζ5 | 0 | 0 | ζ32ζ54+ζ32ζ5 | ζ3ζ53+ζ3ζ52 | ζ32ζ53+ζ32ζ52 | ζ3ζ54+ζ3ζ5 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | -1+√5 | -1-√5 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 1+√5 | 1-√5 | -ζ32ζ54-ζ32ζ5 | -ζ3ζ53-ζ3ζ52 | -ζ3ζ54-ζ3ζ5 | -ζ32ζ53-ζ32ζ52 | 0 | 0 | ζ32ζ53+ζ32ζ52 | ζ3ζ54+ζ3ζ5 | ζ32ζ54+ζ32ζ5 | ζ3ζ53+ζ3ζ52 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | -1+√5 | -1-√5 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 1+√5 | 1-√5 | -ζ3ζ54-ζ3ζ5 | -ζ32ζ53-ζ32ζ52 | -ζ32ζ54-ζ32ζ5 | -ζ3ζ53-ζ3ζ52 | 0 | 0 | ζ3ζ53+ζ3ζ52 | ζ32ζ54+ζ32ζ5 | ζ3ζ54+ζ3ζ5 | ζ32ζ53+ζ32ζ52 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | -1-√5 | -1+√5 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 1-√5 | 1+√5 | -ζ3ζ53-ζ3ζ52 | -ζ32ζ54-ζ32ζ5 | -ζ32ζ53-ζ32ζ52 | -ζ3ζ54-ζ3ζ5 | 0 | 0 | ζ3ζ54+ζ3ζ5 | ζ32ζ53+ζ32ζ52 | ζ3ζ53+ζ3ζ52 | ζ32ζ54+ζ32ζ5 | complex faithful |
| ρ27 | 6 | 6 | 0 | 0 | 0 | 0 | -2 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | orthogonal lifted from D5×A4 |
| ρ28 | 6 | 6 | 0 | 0 | 0 | 0 | -2 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | orthogonal lifted from D5×A4 |
►On 40 points
Generators in S40
(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)
(1 12)(2 11)(3 15)(4 14)(5 13)(6 38)(7 37)(8 36)(9 40)(10 39)(16 23)(17 22)(18 21)(19 25)(20 24)(26 33)(27 32)(28 31)(29 35)(30 34)
(1 23 13 17)(2 24 14 18)(3 25 15 19)(4 21 11 20)(5 22 12 16)(6 30 40 31)(7 26 36 32)(8 27 37 33)(9 28 38 34)(10 29 39 35)
(1 8 13 37)(2 9 14 38)(3 10 15 39)(4 6 11 40)(5 7 12 36)(16 26 22 32)(17 27 23 33)(18 28 24 34)(19 29 25 35)(20 30 21 31)
(6 21 31)(7 22 32)(8 23 33)(9 24 34)(10 25 35)(16 26 36)(17 27 37)(18 28 38)(19 29 39)(20 30 40)
G:=sub<Sym(40)| (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), (1,12)(2,11)(3,15)(4,14)(5,13)(6,38)(7,37)(8,36)(9,40)(10,39)(16,23)(17,22)(18,21)(19,25)(20,24)(26,33)(27,32)(28,31)(29,35)(30,34), (1,23,13,17)(2,24,14,18)(3,25,15,19)(4,21,11,20)(5,22,12,16)(6,30,40,31)(7,26,36,32)(8,27,37,33)(9,28,38,34)(10,29,39,35), (1,8,13,37)(2,9,14,38)(3,10,15,39)(4,6,11,40)(5,7,12,36)(16,26,22,32)(17,27,23,33)(18,28,24,34)(19,29,25,35)(20,30,21,31), (6,21,31)(7,22,32)(8,23,33)(9,24,34)(10,25,35)(16,26,36)(17,27,37)(18,28,38)(19,29,39)(20,30,40)>;
G:=Group( (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), (1,12)(2,11)(3,15)(4,14)(5,13)(6,38)(7,37)(8,36)(9,40)(10,39)(16,23)(17,22)(18,21)(19,25)(20,24)(26,33)(27,32)(28,31)(29,35)(30,34), (1,23,13,17)(2,24,14,18)(3,25,15,19)(4,21,11,20)(5,22,12,16)(6,30,40,31)(7,26,36,32)(8,27,37,33)(9,28,38,34)(10,29,39,35), (1,8,13,37)(2,9,14,38)(3,10,15,39)(4,6,11,40)(5,7,12,36)(16,26,22,32)(17,27,23,33)(18,28,24,34)(19,29,25,35)(20,30,21,31), (6,21,31)(7,22,32)(8,23,33)(9,24,34)(10,25,35)(16,26,36)(17,27,37)(18,28,38)(19,29,39)(20,30,40) );
G=PermutationGroup([[(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)], [(1,12),(2,11),(3,15),(4,14),(5,13),(6,38),(7,37),(8,36),(9,40),(10,39),(16,23),(17,22),(18,21),(19,25),(20,24),(26,33),(27,32),(28,31),(29,35),(30,34)], [(1,23,13,17),(2,24,14,18),(3,25,15,19),(4,21,11,20),(5,22,12,16),(6,30,40,31),(7,26,36,32),(8,27,37,33),(9,28,38,34),(10,29,39,35)], [(1,8,13,37),(2,9,14,38),(3,10,15,39),(4,6,11,40),(5,7,12,36),(16,26,22,32),(17,27,23,33),(18,28,24,34),(19,29,25,35),(20,30,21,31)], [(6,21,31),(7,22,32),(8,23,33),(9,24,34),(10,25,35),(16,26,36),(17,27,37),(18,28,38),(19,29,39),(20,30,40)]])
D5×SL2(𝔽3) is a maximal subgroup of
D10.S4 D10.1S4 D10.2S4 SL2(𝔽3).11D10 D20.A4
Matrix representation of D5×SL2(𝔽3) ►in GL4(𝔽61) generated by
| 0 | 1 | 0 | 0 |
| 60 | 17 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 47 | 13 |
| 0 | 0 | 13 | 14 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 47 | 13 |
G:=sub<GL(4,GF(61))| [0,60,0,0,1,17,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,47,13,0,0,13,14],[1,0,0,0,0,1,0,0,0,0,1,47,0,0,0,13] >;
D5×SL2(𝔽3) in GAP, Magma, Sage, TeX
D_5\times {\rm SL}_2({\mathbb F}_3) % in TeX
G:=Group("D5xSL(2,3)"); // GroupNames label
G:=SmallGroup(240,109);
// by ID
G=gap.SmallGroup(240,109);
# by ID
G:=PCGroup([6,-2,-3,-2,2,-5,-2,170,374,81,543,261,2884]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^4=e^3=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,e*c*e^-1=d,e*d*e^-1=c*d>;
// generators/relations
Export
Subgroup lattice of D5×SL2(𝔽3) in TeX
Character table of D5×SL2(𝔽3) in TeX