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## G = ASL2(𝔽3)  order 216 = 23·33

### Affine special linear group on 𝔽32

Aliases: ASL2(𝔽3), PU3(𝔽2), PSU3(𝔽2)⋊C3, C32⋊SL2(𝔽3), C3⋊S3.A4, Hessian group, SmallGroup(216,153)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — PSU3(𝔽2) — ASL2(𝔽3)
 Chief series C1 — C32 — C3⋊S3 — PSU3(𝔽2) — ASL2(𝔽3)
 Lower central PSU3(𝔽2) — ASL2(𝔽3)
 Upper central C1

Generators and relations for ASL2(𝔽3)
G = < a,b,c,d,e | a3=b3=c4=e3=1, d2=c2, eae-1=dbd-1=ab=ba, cac-1=b-1, dad-1=a-1b, cbc-1=a, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >

9C2
4C3
12C3
24C3
27C4
12S3
36C6
4C32
8C32
9Q8
12C3×S3
4He3

Character table of ASL2(𝔽3)

 class 1 2 3A 3B 3C 3D 3E 4 6A 6B size 1 9 8 12 12 24 24 54 36 36 ρ1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 ζ32 ζ3 ζ32 ζ3 1 ζ32 ζ3 linear of order 3 ρ3 1 1 1 ζ3 ζ32 ζ3 ζ32 1 ζ3 ζ32 linear of order 3 ρ4 2 -2 2 -1 -1 -1 -1 0 1 1 symplectic lifted from SL2(𝔽3), Schur index 2 ρ5 2 -2 2 ζ6 ζ65 ζ6 ζ65 0 ζ32 ζ3 complex lifted from SL2(𝔽3) ρ6 2 -2 2 ζ65 ζ6 ζ65 ζ6 0 ζ3 ζ32 complex lifted from SL2(𝔽3) ρ7 3 3 3 0 0 0 0 -1 0 0 orthogonal lifted from A4 ρ8 8 0 -1 2 2 -1 -1 0 0 0 orthogonal faithful ρ9 8 0 -1 -1-√-3 -1+√-3 ζ6 ζ65 0 0 0 complex faithful ρ10 8 0 -1 -1+√-3 -1-√-3 ζ65 ζ6 0 0 0 complex faithful

Permutation representations of ASL2(𝔽3)
On 9 points: primitive, doubly transitive - transitive group 9T23
Generators in S9
(1 8 6)(2 7 3)(4 5 9)
(1 9 7)(2 6 5)(3 8 4)
(2 3 4 5)(6 7 8 9)
(2 7 4 9)(3 6 5 8)
(2 6 5)(3 4 8)

G:=sub<Sym(9)| (1,8,6)(2,7,3)(4,5,9), (1,9,7)(2,6,5)(3,8,4), (2,3,4,5)(6,7,8,9), (2,7,4,9)(3,6,5,8), (2,6,5)(3,4,8)>;

G:=Group( (1,8,6)(2,7,3)(4,5,9), (1,9,7)(2,6,5)(3,8,4), (2,3,4,5)(6,7,8,9), (2,7,4,9)(3,6,5,8), (2,6,5)(3,4,8) );

G=PermutationGroup([(1,8,6),(2,7,3),(4,5,9)], [(1,9,7),(2,6,5),(3,8,4)], [(2,3,4,5),(6,7,8,9)], [(2,7,4,9),(3,6,5,8)], [(2,6,5),(3,4,8)])

G:=TransitiveGroup(9,23);

On 12 points - transitive group 12T122
Generators in S12
(1 9 11)(2 10 12)(3 8 6)
(1 11 9)(2 10 12)(4 5 7)
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 3)(2 4)(5 10 7 12)(6 9 8 11)
(1 4 2)(5 10 11)(7 12 9)

G:=sub<Sym(12)| (1,9,11)(2,10,12)(3,8,6), (1,11,9)(2,10,12)(4,5,7), (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,3)(2,4)(5,10,7,12)(6,9,8,11), (1,4,2)(5,10,11)(7,12,9)>;

G:=Group( (1,9,11)(2,10,12)(3,8,6), (1,11,9)(2,10,12)(4,5,7), (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,3)(2,4)(5,10,7,12)(6,9,8,11), (1,4,2)(5,10,11)(7,12,9) );

G=PermutationGroup([(1,9,11),(2,10,12),(3,8,6)], [(1,11,9),(2,10,12),(4,5,7)], [(1,2),(3,4),(5,6,7,8),(9,10,11,12)], [(1,3),(2,4),(5,10,7,12),(6,9,8,11)], [(1,4,2),(5,10,11),(7,12,9)])

G:=TransitiveGroup(12,122);

On 24 points - transitive group 24T562
Generators in S24
(1 14 19)(3 17 16)(5 21 10)(6 11 22)(7 12 23)(8 24 9)
(2 15 20)(4 18 13)(5 21 10)(6 22 11)(7 12 23)(8 9 24)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 22 3 24)(2 21 4 23)(5 18 7 20)(6 17 8 19)(9 14 11 16)(10 13 12 15)
(2 21 22)(4 23 24)(5 6 20)(7 8 18)(9 13 12)(10 11 15)

G:=sub<Sym(24)| (1,14,19)(3,17,16)(5,21,10)(6,11,22)(7,12,23)(8,24,9), (2,15,20)(4,18,13)(5,21,10)(6,22,11)(7,12,23)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,22,3,24)(2,21,4,23)(5,18,7,20)(6,17,8,19)(9,14,11,16)(10,13,12,15), (2,21,22)(4,23,24)(5,6,20)(7,8,18)(9,13,12)(10,11,15)>;

G:=Group( (1,14,19)(3,17,16)(5,21,10)(6,11,22)(7,12,23)(8,24,9), (2,15,20)(4,18,13)(5,21,10)(6,22,11)(7,12,23)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,22,3,24)(2,21,4,23)(5,18,7,20)(6,17,8,19)(9,14,11,16)(10,13,12,15), (2,21,22)(4,23,24)(5,6,20)(7,8,18)(9,13,12)(10,11,15) );

G=PermutationGroup([(1,14,19),(3,17,16),(5,21,10),(6,11,22),(7,12,23),(8,24,9)], [(2,15,20),(4,18,13),(5,21,10),(6,22,11),(7,12,23),(8,9,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,22,3,24),(2,21,4,23),(5,18,7,20),(6,17,8,19),(9,14,11,16),(10,13,12,15)], [(2,21,22),(4,23,24),(5,6,20),(7,8,18),(9,13,12),(10,11,15)])

G:=TransitiveGroup(24,562);

On 24 points - transitive group 24T569
Generators in S24
(1 14 19)(3 17 16)(5 21 10)(6 11 22)(7 12 23)(8 24 9)
(2 15 20)(4 18 13)(5 21 10)(6 22 11)(7 12 23)(8 9 24)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 22 3 24)(2 21 4 23)(5 18 7 20)(6 17 8 19)(9 14 11 16)(10 13 12 15)
(1 19 14)(2 5 11)(3 17 16)(4 7 9)(6 15 21)(8 13 23)(10 22 20)(12 24 18)

G:=sub<Sym(24)| (1,14,19)(3,17,16)(5,21,10)(6,11,22)(7,12,23)(8,24,9), (2,15,20)(4,18,13)(5,21,10)(6,22,11)(7,12,23)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,22,3,24)(2,21,4,23)(5,18,7,20)(6,17,8,19)(9,14,11,16)(10,13,12,15), (1,19,14)(2,5,11)(3,17,16)(4,7,9)(6,15,21)(8,13,23)(10,22,20)(12,24,18)>;

G:=Group( (1,14,19)(3,17,16)(5,21,10)(6,11,22)(7,12,23)(8,24,9), (2,15,20)(4,18,13)(5,21,10)(6,22,11)(7,12,23)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,22,3,24)(2,21,4,23)(5,18,7,20)(6,17,8,19)(9,14,11,16)(10,13,12,15), (1,19,14)(2,5,11)(3,17,16)(4,7,9)(6,15,21)(8,13,23)(10,22,20)(12,24,18) );

G=PermutationGroup([(1,14,19),(3,17,16),(5,21,10),(6,11,22),(7,12,23),(8,24,9)], [(2,15,20),(4,18,13),(5,21,10),(6,22,11),(7,12,23),(8,9,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,22,3,24),(2,21,4,23),(5,18,7,20),(6,17,8,19),(9,14,11,16),(10,13,12,15)], [(1,19,14),(2,5,11),(3,17,16),(4,7,9),(6,15,21),(8,13,23),(10,22,20),(12,24,18)])

G:=TransitiveGroup(24,569);

On 27 points - transitive group 27T82
Generators in S27
(1 19 17)(2 23 21)(3 8 10)(4 7 11)(5 6 9)(12 16 15)(13 18 14)(20 27 24)(22 26 25)
(1 16 18)(2 20 22)(3 9 11)(4 8 5)(6 7 10)(12 13 17)(14 19 15)(21 24 25)(23 27 26)
(4 5 6 7)(8 9 10 11)(12 13 14 15)(16 17 18 19)(20 21 22 23)(24 25 26 27)
(4 10 6 8)(5 9 7 11)(12 19 14 17)(13 18 15 16)(20 25 22 27)(21 24 23 26)
(1 3 2)(4 27 19)(5 23 14)(6 25 17)(7 21 12)(8 26 15)(9 20 16)(10 24 13)(11 22 18)

G:=sub<Sym(27)| (1,19,17)(2,23,21)(3,8,10)(4,7,11)(5,6,9)(12,16,15)(13,18,14)(20,27,24)(22,26,25), (1,16,18)(2,20,22)(3,9,11)(4,8,5)(6,7,10)(12,13,17)(14,19,15)(21,24,25)(23,27,26), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27), (4,10,6,8)(5,9,7,11)(12,19,14,17)(13,18,15,16)(20,25,22,27)(21,24,23,26), (1,3,2)(4,27,19)(5,23,14)(6,25,17)(7,21,12)(8,26,15)(9,20,16)(10,24,13)(11,22,18)>;

G:=Group( (1,19,17)(2,23,21)(3,8,10)(4,7,11)(5,6,9)(12,16,15)(13,18,14)(20,27,24)(22,26,25), (1,16,18)(2,20,22)(3,9,11)(4,8,5)(6,7,10)(12,13,17)(14,19,15)(21,24,25)(23,27,26), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27), (4,10,6,8)(5,9,7,11)(12,19,14,17)(13,18,15,16)(20,25,22,27)(21,24,23,26), (1,3,2)(4,27,19)(5,23,14)(6,25,17)(7,21,12)(8,26,15)(9,20,16)(10,24,13)(11,22,18) );

G=PermutationGroup([(1,19,17),(2,23,21),(3,8,10),(4,7,11),(5,6,9),(12,16,15),(13,18,14),(20,27,24),(22,26,25)], [(1,16,18),(2,20,22),(3,9,11),(4,8,5),(6,7,10),(12,13,17),(14,19,15),(21,24,25),(23,27,26)], [(4,5,6,7),(8,9,10,11),(12,13,14,15),(16,17,18,19),(20,21,22,23),(24,25,26,27)], [(4,10,6,8),(5,9,7,11),(12,19,14,17),(13,18,15,16),(20,25,22,27),(21,24,23,26)], [(1,3,2),(4,27,19),(5,23,14),(6,25,17),(7,21,12),(8,26,15),(9,20,16),(10,24,13),(11,22,18)])

G:=TransitiveGroup(27,82);

ASL2(𝔽3) is a maximal subgroup of   AGL2(𝔽3)

Polynomial with Galois group ASL2(𝔽3) over ℚ
actionf(x)Disc(f)
9T23x9+2x8-66x7-196x6+1022x5+3614x4-2711x3-14194x2-4931x+571428·36·52·72·716·2774·5872·19932
12T122x12+3x11-85x10-190x9+2069x8+3233x7-15840x6-16814x5+35725x4+31758x3-5433x2-1468x+8024·716·1092·2776·45134336168774164332

Matrix representation of ASL2(𝔽3) in GL8(ℤ)

 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0
,
 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1
,
 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0

G:=sub<GL(8,Integers())| [0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,1,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,1,0,0],[-1,0,0,0,1,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,1,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,0,0,1,0,0,-1],[1,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,1,0,0,-1,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;

ASL2(𝔽3) in GAP, Magma, Sage, TeX

{\rm ASL}_2({\mathbb F}_3)
% in TeX

G:=Group("ASL(2,3)");
// GroupNames label

G:=SmallGroup(216,153);
// by ID

G=gap.SmallGroup(216,153);
# by ID

G:=PCGroup([6,-3,-2,2,-2,-3,3,217,55,164,116,50,4324,1210,736,142,6053,1163,161,455]);
// Polycyclic

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

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