Aliases: SL2(𝔽3)⋊3D4, (C2×C4).7S4, Q8⋊Dic3⋊2C2, (C2×Q8).19D6, C22.41(C2×S4), Q8.3(C3⋊D4), C2.7(C4.6S4), C2.5(C4.3S4), (C2×GL2(𝔽3))⋊6C2, C2.11(A4⋊D4), (C2×SL2(𝔽3)).19C22, (C2×C4.A4)⋊1C2, (C2×C4○D4)⋊1S3, SmallGroup(192,986)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C2×SL2(𝔽3) — SL2(𝔽3)⋊D4 |
| C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C2×GL2(𝔽3) — SL2(𝔽3)⋊D4 |
| SL2(𝔽3) — C2×SL2(𝔽3) — SL2(𝔽3)⋊D4 |
Generators and relations for SL2(𝔽3)⋊D4
G = < a,b,c,d,e | a4=c3=d4=e2=1, b2=a2, bab-1=ebe=a-1, cac-1=dad-1=b, eae=a2b, cbc-1=ab, dbd-1=a, dcd-1=ac-1, ece=c-1, ede=d-1 >
Subgroups: 397 in 83 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×Q8, C4○D4, SL2(𝔽3), C2×Dic3, C2×C12, C22×S3, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, D6⋊C4, GL2(𝔽3), C2×SL2(𝔽3), C4.A4, D4⋊D4, Q8⋊Dic3, C2×GL2(𝔽3), C2×C4.A4, SL2(𝔽3)⋊D4
Quotients: C1, C2, C22, S3, D4, D6, C3⋊D4, S4, C2×S4, C4.6S4, C4.3S4, A4⋊D4, SL2(𝔽3)⋊D4
Character table of SL2(𝔽3)⋊D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 12 | 24 | 8 | 2 | 2 | 6 | 6 | 24 | 8 | 8 | 8 | 12 | 12 | 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 | 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 | linear of order 2 |
| ρ3 | 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 |
| ρ4 | 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 |
| ρ5 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 2 | 2 | -2 | 0 | -1 | -2 | -2 | 2 | 2 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ8 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | -2 | 2 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | √-3 | √-3 | -√-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ9 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | -2 | 2 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -√-3 | -√-3 | √-3 | √-3 | complex lifted from C3⋊D4 |
| ρ10 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 1 | -1 | 1 | -√2 | -√-2 | √2 | √-2 | -i | i | -i | i | complex lifted from C4.6S4 |
| ρ11 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 1 | -1 | 1 | √2 | √-2 | -√2 | -√-2 | -i | i | -i | i | complex lifted from C4.6S4 |
| ρ12 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 1 | -1 | 1 | -√2 | √-2 | √2 | -√-2 | i | -i | i | -i | complex lifted from C4.6S4 |
| ρ13 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 1 | -1 | 1 | √2 | -√-2 | -√2 | √-2 | i | -i | i | -i | complex lifted from C4.6S4 |
| ρ14 | 3 | 3 | 3 | 3 | 1 | -1 | 0 | -3 | -3 | -1 | -1 | 1 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ15 | 3 | 3 | 3 | 3 | -1 | -1 | 0 | 3 | 3 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ16 | 3 | 3 | 3 | 3 | 1 | 1 | 0 | -3 | -3 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ17 | 3 | 3 | 3 | 3 | -1 | 1 | 0 | 3 | 3 | -1 | -1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ18 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.3S4 |
| ρ19 | 4 | -4 | -4 | 4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | -√3 | √3 | √3 | -√3 | orthogonal lifted from C4.3S4 |
| ρ20 | 4 | -4 | -4 | 4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | √3 | -√3 | -√3 | √3 | orthogonal lifted from C4.3S4 |
| ρ21 | 4 | -4 | 4 | -4 | 0 | 0 | 1 | 4i | -4i | 0 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | -i | i | -i | i | complex lifted from C4.6S4 |
| ρ22 | 4 | -4 | 4 | -4 | 0 | 0 | 1 | -4i | 4i | 0 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | i | -i | i | -i | complex lifted from C4.6S4 |
| ρ23 | 6 | 6 | -6 | -6 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4⋊D4 |
►On 32 points
Generators in S32
(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 3 7)(2 8 4 6)(9 15 11 13)(10 14 12 16)(17 23 19 21)(18 22 20 24)(25 31 27 29)(26 30 28 32)
(2 8 5)(4 6 7)(10 14 15)(12 16 13)(18 22 23)(20 24 21)(26 30 31)(28 32 29)
(1 22 9 32)(2 20 10 26)(3 24 11 30)(4 18 12 28)(5 23 15 29)(6 19 16 25)(7 21 13 31)(8 17 14 27)
(2 7)(4 5)(6 8)(10 13)(12 15)(14 16)(17 25)(18 29)(19 27)(20 31)(21 26)(22 32)(23 28)(24 30)
G:=sub<Sym(32)| (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), (1,5,3,7)(2,8,4,6)(9,15,11,13)(10,14,12,16)(17,23,19,21)(18,22,20,24)(25,31,27,29)(26,30,28,32), (2,8,5)(4,6,7)(10,14,15)(12,16,13)(18,22,23)(20,24,21)(26,30,31)(28,32,29), (1,22,9,32)(2,20,10,26)(3,24,11,30)(4,18,12,28)(5,23,15,29)(6,19,16,25)(7,21,13,31)(8,17,14,27), (2,7)(4,5)(6,8)(10,13)(12,15)(14,16)(17,25)(18,29)(19,27)(20,31)(21,26)(22,32)(23,28)(24,30)>;
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), (1,5,3,7)(2,8,4,6)(9,15,11,13)(10,14,12,16)(17,23,19,21)(18,22,20,24)(25,31,27,29)(26,30,28,32), (2,8,5)(4,6,7)(10,14,15)(12,16,13)(18,22,23)(20,24,21)(26,30,31)(28,32,29), (1,22,9,32)(2,20,10,26)(3,24,11,30)(4,18,12,28)(5,23,15,29)(6,19,16,25)(7,21,13,31)(8,17,14,27), (2,7)(4,5)(6,8)(10,13)(12,15)(14,16)(17,25)(18,29)(19,27)(20,31)(21,26)(22,32)(23,28)(24,30) );
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)], [(1,5,3,7),(2,8,4,6),(9,15,11,13),(10,14,12,16),(17,23,19,21),(18,22,20,24),(25,31,27,29),(26,30,28,32)], [(2,8,5),(4,6,7),(10,14,15),(12,16,13),(18,22,23),(20,24,21),(26,30,31),(28,32,29)], [(1,22,9,32),(2,20,10,26),(3,24,11,30),(4,18,12,28),(5,23,15,29),(6,19,16,25),(7,21,13,31),(8,17,14,27)], [(2,7),(4,5),(6,8),(10,13),(12,15),(14,16),(17,25),(18,29),(19,27),(20,31),(21,26),(22,32),(23,28),(24,30)]])
Matrix representation of SL2(𝔽3)⋊D4 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 53 | 40 |
| 0 | 0 | 21 | 20 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 41 | 52 |
| 0 | 0 | 21 | 32 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 72 | 0 |
| 0 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 29 | 61 |
| 0 | 0 | 58 | 44 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,53,21,0,0,40,20],[1,0,0,0,0,1,0,0,0,0,41,21,0,0,52,32],[1,0,0,0,0,1,0,0,0,0,72,72,0,0,1,0],[0,1,0,0,72,0,0,0,0,0,29,58,0,0,61,44],[1,0,0,0,0,72,0,0,0,0,72,0,0,0,1,1] >;
SL2(𝔽3)⋊D4 in GAP, Magma, Sage, TeX
{\rm SL}_2({\mathbb F}_3)\rtimes D_4 % in TeX
G:=Group("SL(2,3):D4"); // GroupNames label
G:=SmallGroup(192,986);
// by ID
G=gap.SmallGroup(192,986);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,85,680,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^3=d^4=e^2=1,b^2=a^2,b*a*b^-1=e*b*e=a^-1,c*a*c^-1=d*a*d^-1=b,e*a*e=a^2*b,c*b*c^-1=a*b,d*b*d^-1=a,d*c*d^-1=a*c^-1,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations
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