Copied to
clipboard

G = SL2(F3).D6order 288 = 25·32

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

non-abelian, soluble

Aliases: SL2(F3).8D6, (C6xQ8):2S3, C6.33(C2xS4), (C2xC6).16S4, C6.5S4:5C2, C6.6S4:5C2, (C3xQ8).15D6, C22.2(C3:S4), C3:3(Q8.D6), (C2xSL2(F3)):3S3, (C6xSL2(F3)):2C2, (C3xSL2(F3)).8C22, C2.7(C2xC3:S4), Q8.2(C2xC3:S3), (C2xQ8):2(C3:S3), SmallGroup(288,912)

Series: Derived Chief Lower central Upper central

C1C2Q8C3xSL2(F3) — SL2(F3).D6
C1C2Q8C3xQ8C3xSL2(F3)C6.6S4 — SL2(F3).D6
C3xSL2(F3) — SL2(F3).D6
C1C2C22

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, C2xC4, D4, Q8, Q8, C32, Dic3, C12, D6, C2xC6, C2xC6, M4(2), SD16, Q16, C2xQ8, C4oD4, C3:S3, C3xC6, C3:C8, SL2(F3), Dic6, C4xS3, D12, C3:D4, C2xC12, C3xQ8, C3xQ8, C8.C22, C3:Dic3, C2xC3:S3, C62, C4.Dic3, Q8:2S3, C3:Q16, CSU2(F3), GL2(F3), C2xSL2(F3), C4oD12, C6xQ8, C3xSL2(F3), C32:7D4, Q8.11D6, Q8.D6, C6.5S4, C6.6S4, C6xSL2(F3), SL2(F3).D6
Quotients: C1, C2, C22, S3, D6, C3:S3, S4, C2xC3:S3, C2xS4, C3:S4, Q8.D6, C2xC3:S4, SL2(F3).D6

Character table of SL2(F3).D6

 class 12A2B2C3A3B3C3D4A4B4C6A6B6C6D6E6F6G6H6I6J6K6L8A8B12A12B
 size 112362888663622288888888836361212
ρ1111111111111111111111111111    trivial
ρ211-1-11111-111-1-11-1-11-1-1-1-111-11-11    linear of order 2
ρ311-111111-11-1-1-11-1-11-1-1-1-1111-1-11    linear of order 2
ρ4111-1111111-1111111111111-1-111    linear of order 2
ρ522-20-1-12-1-22011-111-1-2-2112-1001-1    orthogonal lifted from D6
ρ62220-1-12-1220-1-1-1-1-1-122-1-12-100-1-1    orthogonal lifted from S3
ρ72220-1-1-12220-1-1-122-1-1-1-1-1-1200-1-1    orthogonal lifted from S3
ρ822-20-12-1-1-22011-111211-2-2-1-1001-1    orthogonal lifted from D6
ρ922-20-1-1-12-22011-1-2-2-11111-12001-1    orthogonal lifted from D6
ρ1022202-1-1-1220222-1-1-1-1-1-1-1-1-10022    orthogonal lifted from S3
ρ1122-202-1-1-1-220-2-2211-11111-1-100-22    orthogonal lifted from D6
ρ122220-12-1-1220-1-1-1-1-12-1-122-1-100-1-1    orthogonal lifted from S3
ρ13333-13000-1-1-133300000000011-1-1    orthogonal lifted from S4
ρ1433313000-1-11333000000000-1-1-1-1    orthogonal lifted from S4
ρ1533-3-130001-11-3-330000000001-11-1    orthogonal lifted from C2xS4
ρ1633-3130001-1-1-3-33000000000-111-1    orthogonal lifted from C2xS4
ρ174-4004-2-2-200000-40020000220000    symplectic lifted from Q8.D6, Schur index 2
ρ184-400-211-2000-2-32-3200-1-3--3--3-3-120000    complex faithful
ρ194-400-2-2110002-3-2-32-3--32-3--300-1-10000    complex faithful
ρ204-400411100000-4-3--3-1--3-3--3-3-1-10000    complex lifted from Q8.D6
ρ214-400-21-21000-2-32-32-3--3-100-3--32-10000    complex faithful
ρ224-400-211-20002-3-2-3200-1--3-3-3--3-120000    complex faithful
ρ234-400-2-211000-2-32-32--3-32--3-300-1-10000    complex faithful
ρ244-400411100000-4--3-3-1-3--3-3--3-1-10000    complex lifted from Q8.D6
ρ254-400-21-210002-3-2-32--3-3-100--3-32-10000    complex faithful
ρ266660-3000-2-20-3-3-30000000000011    orthogonal lifted from C3:S4
ρ2766-60-30002-2033-300000000000-11    orthogonal lifted from C2xC3: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

0064
3063
4426
6145
,
0400
5000
6662
2561
,
4566
3336
3321
5213
,
1326
3544
6645
6662
,
5133
5653
2253
4605
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

Character table of SL2(F3).D6 in TeX

׿
x
:
Z
F
o
wr
Q
<