Aliases: Dic3.2S4, SL2(𝔽3)⋊1D6, GL2(𝔽3)⋊1S3, C2.7(S3×S4), C6.4(C2×S4), Q8.4(S32), (C3×Q8).4D6, C6.6S4⋊2C2, Q8⋊3S3⋊3S3, C3⋊1(C4.3S4), Dic3.A4⋊3C2, (C3×GL2(𝔽3))⋊1C2, (C3×SL2(𝔽3))⋊1C22, SmallGroup(288,847)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C3×SL2(𝔽3) — SL2(𝔽3)⋊D6 |
| C3×SL2(𝔽3) — SL2(𝔽3)⋊D6 |
Generators and relations for SL2(𝔽3)⋊D6
G = < a,b,c,d,e | a4=c3=d6=e2=1, b2=a2, bab-1=dbd-1=a-1, cac-1=eae=b, dad-1=a2b, cbc-1=ab, ebe=a, dcd-1=c-1, ece=ac-1, ede=d-1 >
Subgroups: 678 in 91 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2 [×3], C3, C3 [×2], C4 [×2], C22 [×5], S3 [×6], C6, C6 [×3], C8 [×2], C2×C4, D4 [×4], Q8, C23, C32, Dic3, C12 [×2], D6 [×7], C2×C6, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, SL2(𝔽3), SL2(𝔽3), C4×S3, D12 [×3], C3⋊D4, C3×D4, C3×Q8, C22×S3, C8⋊C22, C3×Dic3, S3×C6, C2×C3⋊S3, C8⋊S3, D24, D4⋊S3, Q8⋊2S3, C3×SD16, GL2(𝔽3), GL2(𝔽3) [×2], C4.A4, S3×D4, Q8⋊3S3, C3⋊D12, C3×SL2(𝔽3), Q8⋊3D6, C4.3S4, C3×GL2(𝔽3), C6.6S4, Dic3.A4, SL2(𝔽3)⋊D6
Quotients: C1, C2 [×3], C22, S3 [×2], D6 [×2], S4, S32, C2×S4, C4.3S4, S3×S4, SL2(𝔽3)⋊D6
Character table of SL2(𝔽3)⋊D6
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 6D | 8A | 8B | 12A | 12B | 12C | 24A | 24B | |
| size | 1 | 1 | 12 | 18 | 36 | 2 | 8 | 16 | 6 | 6 | 2 | 8 | 16 | 24 | 12 | 36 | 12 | 24 | 24 | 12 | 12 | |
| ρ1 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | -2 | 0 | 2 | -1 | -1 | 2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 2 | 1 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 2 | 0 | -1 | 2 | -1 | -1 | 2 | 0 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | 0 | 2 | 0 | 2 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | 2 | 0 | -1 | 2 | -1 | 1 | -2 | 0 | -1 | 0 | 0 | 1 | 1 | orthogonal lifted from D6 |
| ρ9 | 3 | 3 | -1 | 1 | 1 | 3 | 0 | 0 | -1 | -3 | 3 | 0 | 0 | -1 | 1 | -1 | -1 | 0 | 0 | 1 | 1 | orthogonal lifted from C2×S4 |
| ρ10 | 3 | 3 | -1 | -1 | -1 | 3 | 0 | 0 | -1 | 3 | 3 | 0 | 0 | -1 | 1 | 1 | -1 | 0 | 0 | 1 | 1 | orthogonal lifted from S4 |
| ρ11 | 3 | 3 | 1 | 1 | -1 | 3 | 0 | 0 | -1 | -3 | 3 | 0 | 0 | 1 | -1 | 1 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from C2×S4 |
| ρ12 | 3 | 3 | 1 | -1 | 1 | 3 | 0 | 0 | -1 | 3 | 3 | 0 | 0 | 1 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from S4 |
| ρ13 | 4 | 4 | 0 | 0 | 0 | -2 | -2 | 1 | 4 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ14 | 4 | -4 | 0 | 0 | 0 | 4 | -2 | -2 | 0 | 0 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.3S4 |
| ρ15 | 4 | -4 | 0 | 0 | 0 | 4 | 1 | 1 | 0 | 0 | -4 | -1 | -1 | 0 | 0 | 0 | 0 | -√3 | √3 | 0 | 0 | orthogonal lifted from C4.3S4 |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 4 | 1 | 1 | 0 | 0 | -4 | -1 | -1 | 0 | 0 | 0 | 0 | √3 | -√3 | 0 | 0 | orthogonal lifted from C4.3S4 |
| ρ17 | 4 | -4 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -√6 | √6 | orthogonal faithful |
| ρ18 | 4 | -4 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | √6 | -√6 | orthogonal faithful |
| ρ19 | 6 | 6 | 2 | 0 | 0 | -3 | 0 | 0 | -2 | 0 | -3 | 0 | 0 | -1 | -2 | 0 | 1 | 0 | 0 | 1 | 1 | orthogonal lifted from S3×S4 |
| ρ20 | 6 | 6 | -2 | 0 | 0 | -3 | 0 | 0 | -2 | 0 | -3 | 0 | 0 | 1 | 2 | 0 | 1 | 0 | 0 | -1 | -1 | orthogonal lifted from S3×S4 |
| ρ21 | 8 | -8 | 0 | 0 | 0 | -4 | 2 | -1 | 0 | 0 | 4 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 48 points
Generators in S48
(1 16 42 19)(2 7 37 25)(3 18 38 21)(4 9 39 27)(5 14 40 23)(6 11 41 29)(8 36 26 47)(10 32 28 43)(12 34 30 45)(13 48 22 31)(15 44 24 33)(17 46 20 35)
(1 30 42 12)(2 20 37 17)(3 26 38 8)(4 22 39 13)(5 28 40 10)(6 24 41 15)(7 46 25 35)(9 48 27 31)(11 44 29 33)(14 32 23 43)(16 34 19 45)(18 36 21 47)
(1 5 3)(2 4 6)(7 48 24)(8 19 43)(9 44 20)(10 21 45)(11 46 22)(12 23 47)(13 29 35)(14 36 30)(15 25 31)(16 32 26)(17 27 33)(18 34 28)(37 39 41)(38 42 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 41 42)(43 44 45 46 47 48)
(1 47)(2 46)(3 45)(4 44)(5 43)(6 48)(7 25)(8 30)(9 29)(10 28)(11 27)(12 26)(13 15)(16 18)(19 21)(22 24)(31 41)(32 40)(33 39)(34 38)(35 37)(36 42)
G:=sub<Sym(48)| (1,16,42,19)(2,7,37,25)(3,18,38,21)(4,9,39,27)(5,14,40,23)(6,11,41,29)(8,36,26,47)(10,32,28,43)(12,34,30,45)(13,48,22,31)(15,44,24,33)(17,46,20,35), (1,30,42,12)(2,20,37,17)(3,26,38,8)(4,22,39,13)(5,28,40,10)(6,24,41,15)(7,46,25,35)(9,48,27,31)(11,44,29,33)(14,32,23,43)(16,34,19,45)(18,36,21,47), (1,5,3)(2,4,6)(7,48,24)(8,19,43)(9,44,20)(10,21,45)(11,46,22)(12,23,47)(13,29,35)(14,36,30)(15,25,31)(16,32,26)(17,27,33)(18,34,28)(37,39,41)(38,42,40), (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)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,47)(2,46)(3,45)(4,44)(5,43)(6,48)(7,25)(8,30)(9,29)(10,28)(11,27)(12,26)(13,15)(16,18)(19,21)(22,24)(31,41)(32,40)(33,39)(34,38)(35,37)(36,42)>;
G:=Group( (1,16,42,19)(2,7,37,25)(3,18,38,21)(4,9,39,27)(5,14,40,23)(6,11,41,29)(8,36,26,47)(10,32,28,43)(12,34,30,45)(13,48,22,31)(15,44,24,33)(17,46,20,35), (1,30,42,12)(2,20,37,17)(3,26,38,8)(4,22,39,13)(5,28,40,10)(6,24,41,15)(7,46,25,35)(9,48,27,31)(11,44,29,33)(14,32,23,43)(16,34,19,45)(18,36,21,47), (1,5,3)(2,4,6)(7,48,24)(8,19,43)(9,44,20)(10,21,45)(11,46,22)(12,23,47)(13,29,35)(14,36,30)(15,25,31)(16,32,26)(17,27,33)(18,34,28)(37,39,41)(38,42,40), (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)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,47)(2,46)(3,45)(4,44)(5,43)(6,48)(7,25)(8,30)(9,29)(10,28)(11,27)(12,26)(13,15)(16,18)(19,21)(22,24)(31,41)(32,40)(33,39)(34,38)(35,37)(36,42) );
G=PermutationGroup([(1,16,42,19),(2,7,37,25),(3,18,38,21),(4,9,39,27),(5,14,40,23),(6,11,41,29),(8,36,26,47),(10,32,28,43),(12,34,30,45),(13,48,22,31),(15,44,24,33),(17,46,20,35)], [(1,30,42,12),(2,20,37,17),(3,26,38,8),(4,22,39,13),(5,28,40,10),(6,24,41,15),(7,46,25,35),(9,48,27,31),(11,44,29,33),(14,32,23,43),(16,34,19,45),(18,36,21,47)], [(1,5,3),(2,4,6),(7,48,24),(8,19,43),(9,44,20),(10,21,45),(11,46,22),(12,23,47),(13,29,35),(14,36,30),(15,25,31),(16,32,26),(17,27,33),(18,34,28),(37,39,41),(38,42,40)], [(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),(37,38,39,40,41,42),(43,44,45,46,47,48)], [(1,47),(2,46),(3,45),(4,44),(5,43),(6,48),(7,25),(8,30),(9,29),(10,28),(11,27),(12,26),(13,15),(16,18),(19,21),(22,24),(31,41),(32,40),(33,39),(34,38),(35,37),(36,42)])
Matrix representation of SL2(𝔽3)⋊D6 ►in GL4(𝔽5) generated by
| 3 | 0 | 3 | 1 |
| 1 | 4 | 2 | 2 |
| 3 | 3 | 2 | 2 |
| 1 | 1 | 0 | 1 |
| 2 | 2 | 0 | 2 |
| 4 | 3 | 4 | 1 |
| 1 | 3 | 3 | 3 |
| 1 | 0 | 1 | 2 |
| 2 | 4 | 1 | 2 |
| 1 | 1 | 4 | 4 |
| 4 | 0 | 2 | 2 |
| 0 | 2 | 2 | 3 |
| 1 | 2 | 4 | 2 |
| 0 | 4 | 0 | 3 |
| 3 | 2 | 0 | 3 |
| 2 | 3 | 1 | 0 |
| 0 | 1 | 1 | 4 |
| 0 | 0 | 3 | 0 |
| 0 | 2 | 0 | 0 |
| 4 | 2 | 3 | 0 |
G:=sub<GL(4,GF(5))| [3,1,3,1,0,4,3,1,3,2,2,0,1,2,2,1],[2,4,1,1,2,3,3,0,0,4,3,1,2,1,3,2],[2,1,4,0,4,1,0,2,1,4,2,2,2,4,2,3],[1,0,3,2,2,4,2,3,4,0,0,1,2,3,3,0],[0,0,0,4,1,0,2,2,1,3,0,3,4,0,0,0] >;
SL2(𝔽3)⋊D6 in GAP, Magma, Sage, TeX
{\rm SL}_2({\mathbb F}_3)\rtimes D_6 % in TeX
G:=Group("SL(2,3):D6"); // GroupNames label
G:=SmallGroup(288,847);
// by ID
G=gap.SmallGroup(288,847);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1008,2045,93,675,1271,1908,172,768,1153,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^3=d^6=e^2=1,b^2=a^2,b*a*b^-1=d*b*d^-1=a^-1,c*a*c^-1=e*a*e=b,d*a*d^-1=a^2*b,c*b*c^-1=a*b,e*b*e=a,d*c*d^-1=c^-1,e*c*e=a*c^-1,e*d*e=d^-1>;
// generators/relations