Copied to
clipboard

## G = SL2(𝔽3).D6order 288 = 25·32

### 2nd non-split extension by SL2(𝔽3) of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C3×SL2(𝔽3) — SL2(𝔽3).D6
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — C6.6S4 — SL2(𝔽3).D6
 Lower central C3×SL2(𝔽3) — SL2(𝔽3).D6
 Upper central C1 — C2 — C22

Generators and relations for SL2(𝔽3).D6
G = < a,b,c,d,e | a4=c3=d6=1, b2=e2=a2, bab-1=a-1, cac-1=dad-1=b, eae-1=a-1b, cbc-1=dbd-1=ab, ebe-1=a2b, cd=dc, ece-1=c-1, ede-1=a2d-1 >

Subgroups: 632 in 104 conjugacy classes, 21 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, C32, Dic3, C12, D6, C2×C6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3⋊S3, C3×C6, C3⋊C8, SL2(𝔽3), Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×Q8, C3×Q8, C8.C22, C3⋊Dic3, C2×C3⋊S3, C62, C4.Dic3, Q82S3, C3⋊Q16, CSU2(𝔽3), GL2(𝔽3), C2×SL2(𝔽3), C4○D12, C6×Q8, C3×SL2(𝔽3), C327D4, Q8.11D6, Q8.D6, C6.5S4, C6.6S4, C6×SL2(𝔽3), SL2(𝔽3).D6
Quotients: C1, C2, C22, S3, D6, C3⋊S3, S4, C2×C3⋊S3, C2×S4, C3⋊S4, Q8.D6, C2×C3⋊S4, SL2(𝔽3).D6

Character table of SL2(𝔽3).D6

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 8A 8B 12A 12B size 1 1 2 36 2 8 8 8 6 6 36 2 2 2 8 8 8 8 8 8 8 8 8 36 36 12 12 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 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 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 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 -1 -1 1 1 linear of order 2 ρ5 2 2 -2 0 -1 -1 2 -1 -2 2 0 1 1 -1 1 1 -1 -2 -2 1 1 2 -1 0 0 1 -1 orthogonal lifted from D6 ρ6 2 2 2 0 -1 -1 2 -1 2 2 0 -1 -1 -1 -1 -1 -1 2 2 -1 -1 2 -1 0 0 -1 -1 orthogonal lifted from S3 ρ7 2 2 2 0 -1 -1 -1 2 2 2 0 -1 -1 -1 2 2 -1 -1 -1 -1 -1 -1 2 0 0 -1 -1 orthogonal lifted from S3 ρ8 2 2 -2 0 -1 2 -1 -1 -2 2 0 1 1 -1 1 1 2 1 1 -2 -2 -1 -1 0 0 1 -1 orthogonal lifted from D6 ρ9 2 2 -2 0 -1 -1 -1 2 -2 2 0 1 1 -1 -2 -2 -1 1 1 1 1 -1 2 0 0 1 -1 orthogonal lifted from D6 ρ10 2 2 2 0 2 -1 -1 -1 2 2 0 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 2 2 orthogonal lifted from S3 ρ11 2 2 -2 0 2 -1 -1 -1 -2 2 0 -2 -2 2 1 1 -1 1 1 1 1 -1 -1 0 0 -2 2 orthogonal lifted from D6 ρ12 2 2 2 0 -1 2 -1 -1 2 2 0 -1 -1 -1 -1 -1 2 -1 -1 2 2 -1 -1 0 0 -1 -1 orthogonal lifted from S3 ρ13 3 3 3 -1 3 0 0 0 -1 -1 -1 3 3 3 0 0 0 0 0 0 0 0 0 1 1 -1 -1 orthogonal lifted from S4 ρ14 3 3 3 1 3 0 0 0 -1 -1 1 3 3 3 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S4 ρ15 3 3 -3 -1 3 0 0 0 1 -1 1 -3 -3 3 0 0 0 0 0 0 0 0 0 1 -1 1 -1 orthogonal lifted from C2×S4 ρ16 3 3 -3 1 3 0 0 0 1 -1 -1 -3 -3 3 0 0 0 0 0 0 0 0 0 -1 1 1 -1 orthogonal lifted from C2×S4 ρ17 4 -4 0 0 4 -2 -2 -2 0 0 0 0 0 -4 0 0 2 0 0 0 0 2 2 0 0 0 0 symplectic lifted from Q8.D6, Schur index 2 ρ18 4 -4 0 0 -2 1 1 -2 0 0 0 -2√-3 2√-3 2 0 0 -1 √-3 -√-3 -√-3 √-3 -1 2 0 0 0 0 complex faithful ρ19 4 -4 0 0 -2 -2 1 1 0 0 0 2√-3 -2√-3 2 √-3 -√-3 2 √-3 -√-3 0 0 -1 -1 0 0 0 0 complex faithful ρ20 4 -4 0 0 4 1 1 1 0 0 0 0 0 -4 √-3 -√-3 -1 -√-3 √-3 -√-3 √-3 -1 -1 0 0 0 0 complex lifted from Q8.D6 ρ21 4 -4 0 0 -2 1 -2 1 0 0 0 -2√-3 2√-3 2 √-3 -√-3 -1 0 0 √-3 -√-3 2 -1 0 0 0 0 complex faithful ρ22 4 -4 0 0 -2 1 1 -2 0 0 0 2√-3 -2√-3 2 0 0 -1 -√-3 √-3 √-3 -√-3 -1 2 0 0 0 0 complex faithful ρ23 4 -4 0 0 -2 -2 1 1 0 0 0 -2√-3 2√-3 2 -√-3 √-3 2 -√-3 √-3 0 0 -1 -1 0 0 0 0 complex faithful ρ24 4 -4 0 0 4 1 1 1 0 0 0 0 0 -4 -√-3 √-3 -1 √-3 -√-3 √-3 -√-3 -1 -1 0 0 0 0 complex lifted from Q8.D6 ρ25 4 -4 0 0 -2 1 -2 1 0 0 0 2√-3 -2√-3 2 -√-3 √-3 -1 0 0 -√-3 √-3 2 -1 0 0 0 0 complex faithful ρ26 6 6 6 0 -3 0 0 0 -2 -2 0 -3 -3 -3 0 0 0 0 0 0 0 0 0 0 0 1 1 orthogonal lifted from C3⋊S4 ρ27 6 6 -6 0 -3 0 0 0 2 -2 0 3 3 -3 0 0 0 0 0 0 0 0 0 0 0 -1 1 orthogonal lifted from C2×C3⋊S4

Smallest permutation representation of SL2(𝔽3).D6
On 48 points
Generators in S48
(1 40 8 21)(2 45 9 15)(3 28 10 31)(4 37 11 24)(5 48 12 18)(6 25 7 34)(13 20 43 39)(14 26 44 35)(16 23 46 42)(17 29 47 32)(19 33 38 30)(22 36 41 27)
(1 44 8 14)(2 27 9 36)(3 42 10 23)(4 47 11 17)(5 30 12 33)(6 39 7 20)(13 25 43 34)(15 22 45 41)(16 28 46 31)(18 19 48 38)(21 35 40 26)(24 32 37 29)
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 36 24)(14 31 19)(15 32 20)(16 33 21)(17 34 22)(18 35 23)(25 41 47)(26 42 48)(27 37 43)(28 38 44)(29 39 45)(30 40 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)
(1 6 8 7)(2 12 9 5)(3 4 10 11)(13 21 43 40)(14 39 44 20)(15 19 45 38)(16 37 46 24)(17 23 47 42)(18 41 48 22)(25 26 34 35)(27 30 36 33)(28 32 31 29)

G:=sub<Sym(48)| (1,40,8,21)(2,45,9,15)(3,28,10,31)(4,37,11,24)(5,48,12,18)(6,25,7,34)(13,20,43,39)(14,26,44,35)(16,23,46,42)(17,29,47,32)(19,33,38,30)(22,36,41,27), (1,44,8,14)(2,27,9,36)(3,42,10,23)(4,47,11,17)(5,30,12,33)(6,39,7,20)(13,25,43,34)(15,22,45,41)(16,28,46,31)(18,19,48,38)(21,35,40,26)(24,32,37,29), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,36,24)(14,31,19)(15,32,20)(16,33,21)(17,34,22)(18,35,23)(25,41,47)(26,42,48)(27,37,43)(28,38,44)(29,39,45)(30,40,46), (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), (1,6,8,7)(2,12,9,5)(3,4,10,11)(13,21,43,40)(14,39,44,20)(15,19,45,38)(16,37,46,24)(17,23,47,42)(18,41,48,22)(25,26,34,35)(27,30,36,33)(28,32,31,29)>;

G:=Group( (1,40,8,21)(2,45,9,15)(3,28,10,31)(4,37,11,24)(5,48,12,18)(6,25,7,34)(13,20,43,39)(14,26,44,35)(16,23,46,42)(17,29,47,32)(19,33,38,30)(22,36,41,27), (1,44,8,14)(2,27,9,36)(3,42,10,23)(4,47,11,17)(5,30,12,33)(6,39,7,20)(13,25,43,34)(15,22,45,41)(16,28,46,31)(18,19,48,38)(21,35,40,26)(24,32,37,29), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,36,24)(14,31,19)(15,32,20)(16,33,21)(17,34,22)(18,35,23)(25,41,47)(26,42,48)(27,37,43)(28,38,44)(29,39,45)(30,40,46), (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), (1,6,8,7)(2,12,9,5)(3,4,10,11)(13,21,43,40)(14,39,44,20)(15,19,45,38)(16,37,46,24)(17,23,47,42)(18,41,48,22)(25,26,34,35)(27,30,36,33)(28,32,31,29) );

G=PermutationGroup([[(1,40,8,21),(2,45,9,15),(3,28,10,31),(4,37,11,24),(5,48,12,18),(6,25,7,34),(13,20,43,39),(14,26,44,35),(16,23,46,42),(17,29,47,32),(19,33,38,30),(22,36,41,27)], [(1,44,8,14),(2,27,9,36),(3,42,10,23),(4,47,11,17),(5,30,12,33),(6,39,7,20),(13,25,43,34),(15,22,45,41),(16,28,46,31),(18,19,48,38),(21,35,40,26),(24,32,37,29)], [(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,36,24),(14,31,19),(15,32,20),(16,33,21),(17,34,22),(18,35,23),(25,41,47),(26,42,48),(27,37,43),(28,38,44),(29,39,45),(30,40,46)], [(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)], [(1,6,8,7),(2,12,9,5),(3,4,10,11),(13,21,43,40),(14,39,44,20),(15,19,45,38),(16,37,46,24),(17,23,47,42),(18,41,48,22),(25,26,34,35),(27,30,36,33),(28,32,31,29)]])

Matrix representation of SL2(𝔽3).D6 in GL4(𝔽7) generated by

 0 0 6 4 3 0 6 3 4 4 2 6 6 1 4 5
,
 0 4 0 0 5 0 0 0 6 6 6 2 2 5 6 1
,
 4 5 6 6 3 3 3 6 3 3 2 1 5 2 1 3
,
 1 3 2 6 3 5 4 4 6 6 4 5 6 6 6 2
,
 5 1 3 3 5 6 5 3 2 2 5 3 4 6 0 5
G:=sub<GL(4,GF(7))| [0,3,4,6,0,0,4,1,6,6,2,4,4,3,6,5],[0,5,6,2,4,0,6,5,0,0,6,6,0,0,2,1],[4,3,3,5,5,3,3,2,6,3,2,1,6,6,1,3],[1,3,6,6,3,5,6,6,2,4,4,6,6,4,5,2],[5,5,2,4,1,6,2,6,3,5,5,0,3,3,3,5] >;

SL2(𝔽3).D6 in GAP, Magma, Sage, TeX

{\rm SL}_2({\mathbb F}_3).D_6
% in TeX

G:=Group("SL(2,3).D6");
// GroupNames label

G:=SmallGroup(288,912);
// by ID

G=gap.SmallGroup(288,912);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,2045,170,675,2524,1908,172,1517,1153,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^3=d^6=1,b^2=e^2=a^2,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=b,e*a*e^-1=a^-1*b,c*b*c^-1=d*b*d^-1=a*b,e*b*e^-1=a^2*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=a^2*d^-1>;
// generators/relations

Export

׿
×
𝔽