Aliases: Dic3.2S4, GL2(𝔽3)⋊1S3, SL2(𝔽3)⋊1D6, Q8.4S32, C2.7(S3×S4), C6.4(C2×S4), (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) — GL2(𝔽3)⋊S3 |
| C3×SL2(𝔽3) — GL2(𝔽3)⋊S3 |
Generators and relations for GL2(𝔽3)⋊S3
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, C3, C3, C4, C22, S3, C6, C6, C8, C2×C4, D4, Q8, C23, C32, Dic3, C12, D6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, SL2(𝔽3), SL2(𝔽3), C4×S3, D12, 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), C4.A4, S3×D4, Q8⋊3S3, C3⋊D12, C3×SL2(𝔽3), Q8⋊3D6, C4.3S4, C3×GL2(𝔽3), C6.6S4, Dic3.A4, GL2(𝔽3)⋊S3
Quotients: C1, C2, C22, S3, D6, S4, S32, C2×S4, C4.3S4, S3×S4, GL2(𝔽3)⋊S3
Character table of GL2(𝔽3)⋊S3
| 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 26 21)(2 7 27 44)(3 18 28 23)(4 9 29 46)(5 14 30 19)(6 11 25 48)(8 36 45 42)(10 32 47 38)(12 34 43 40)(13 37 24 31)(15 39 20 33)(17 41 22 35)
(1 43 26 12)(2 22 27 17)(3 45 28 8)(4 24 29 13)(5 47 30 10)(6 20 25 15)(7 41 44 35)(9 37 46 31)(11 39 48 33)(14 32 19 38)(16 34 21 40)(18 36 23 42)
(1 5 3)(2 4 6)(7 37 20)(8 21 38)(9 39 22)(10 23 40)(11 41 24)(12 19 42)(13 48 35)(14 36 43)(15 44 31)(16 32 45)(17 46 33)(18 34 47)(25 27 29)(26 30 28)
(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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 44)(8 43)(9 48)(10 47)(11 46)(12 45)(13 15)(16 18)(20 24)(21 23)(25 31)(26 36)(27 35)(28 34)(29 33)(30 32)
G:=sub<Sym(48)| (1,16,26,21)(2,7,27,44)(3,18,28,23)(4,9,29,46)(5,14,30,19)(6,11,25,48)(8,36,45,42)(10,32,47,38)(12,34,43,40)(13,37,24,31)(15,39,20,33)(17,41,22,35), (1,43,26,12)(2,22,27,17)(3,45,28,8)(4,24,29,13)(5,47,30,10)(6,20,25,15)(7,41,44,35)(9,37,46,31)(11,39,48,33)(14,32,19,38)(16,34,21,40)(18,36,23,42), (1,5,3)(2,4,6)(7,37,20)(8,21,38)(9,39,22)(10,23,40)(11,41,24)(12,19,42)(13,48,35)(14,36,43)(15,44,31)(16,32,45)(17,46,33)(18,34,47)(25,27,29)(26,30,28), (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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,44)(8,43)(9,48)(10,47)(11,46)(12,45)(13,15)(16,18)(20,24)(21,23)(25,31)(26,36)(27,35)(28,34)(29,33)(30,32)>;
G:=Group( (1,16,26,21)(2,7,27,44)(3,18,28,23)(4,9,29,46)(5,14,30,19)(6,11,25,48)(8,36,45,42)(10,32,47,38)(12,34,43,40)(13,37,24,31)(15,39,20,33)(17,41,22,35), (1,43,26,12)(2,22,27,17)(3,45,28,8)(4,24,29,13)(5,47,30,10)(6,20,25,15)(7,41,44,35)(9,37,46,31)(11,39,48,33)(14,32,19,38)(16,34,21,40)(18,36,23,42), (1,5,3)(2,4,6)(7,37,20)(8,21,38)(9,39,22)(10,23,40)(11,41,24)(12,19,42)(13,48,35)(14,36,43)(15,44,31)(16,32,45)(17,46,33)(18,34,47)(25,27,29)(26,30,28), (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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,44)(8,43)(9,48)(10,47)(11,46)(12,45)(13,15)(16,18)(20,24)(21,23)(25,31)(26,36)(27,35)(28,34)(29,33)(30,32) );
G=PermutationGroup([[(1,16,26,21),(2,7,27,44),(3,18,28,23),(4,9,29,46),(5,14,30,19),(6,11,25,48),(8,36,45,42),(10,32,47,38),(12,34,43,40),(13,37,24,31),(15,39,20,33),(17,41,22,35)], [(1,43,26,12),(2,22,27,17),(3,45,28,8),(4,24,29,13),(5,47,30,10),(6,20,25,15),(7,41,44,35),(9,37,46,31),(11,39,48,33),(14,32,19,38),(16,34,21,40),(18,36,23,42)], [(1,5,3),(2,4,6),(7,37,20),(8,21,38),(9,39,22),(10,23,40),(11,41,24),(12,19,42),(13,48,35),(14,36,43),(15,44,31),(16,32,45),(17,46,33),(18,34,47),(25,27,29),(26,30,28)], [(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)], [(1,42),(2,41),(3,40),(4,39),(5,38),(6,37),(7,44),(8,43),(9,48),(10,47),(11,46),(12,45),(13,15),(16,18),(20,24),(21,23),(25,31),(26,36),(27,35),(28,34),(29,33),(30,32)]])
Matrix representation of GL2(𝔽3)⋊S3 ►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] >;
GL2(𝔽3)⋊S3 in GAP, Magma, Sage, TeX
{\rm GL}_2({\mathbb F}_3)\rtimes S_3 % in TeX
G:=Group("GL(2,3):S3"); // 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
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