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## G = AGL2(𝔽3)  order 432 = 24·33

### Affine linear group on 𝔽32

Aliases: AGL2(𝔽3), PSU3(𝔽2)⋊S3, ASL2(𝔽3)⋊C2, C32⋊GL2(𝔽3), C3⋊S3.S4, Aut(C3⋊S3), Hol(C32), SmallGroup(432,734)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — PSU3(𝔽2) — ASL2(𝔽3) — AGL2(𝔽3)
 Chief series C1 — C32 — C3⋊S3 — PSU3(𝔽2) — ASL2(𝔽3) — AGL2(𝔽3)
 Lower central ASL2(𝔽3) — AGL2(𝔽3)
 Upper central C1

Generators and relations for AGL2(𝔽3)
G = < a,b,c,d,e,f | a3=b3=c4=e3=f2=1, d2=c2, dbd-1=ebe-1=ab=ba, cac-1=b-1, dad-1=a-1b, ae=ea, af=fa, cbc-1=a, fbf=a-1b-1, dcd-1=ede-1=c-1, ece-1=c-1d, fcf=cd, fdf=c2d, fef=e-1 >

9C2
36C2
4C3
12C3
24C3
27C4
54C22
12S3
12S3
12S3
24S3
36S3
36C6
36C6
4C32
8C32
9Q8
27C8
27D4
36D6
36D6
12C3×S3
12C3×S3
12C3×S3
24C3×S3
4He3
27SD16
6S32
12S32
3F9

Character table of AGL2(𝔽3)

 class 1 2A 2B 3A 3B 3C 4 6A 6B 8A 8B size 1 9 36 8 24 48 54 72 72 54 54 ρ1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 1 1 1 1 -1 -1 -1 linear of order 2 ρ3 2 2 0 2 -1 -1 2 -1 0 0 0 orthogonal lifted from S3 ρ4 2 -2 0 2 -1 -1 0 1 0 -√-2 √-2 complex lifted from GL2(𝔽3) ρ5 2 -2 0 2 -1 -1 0 1 0 √-2 -√-2 complex lifted from GL2(𝔽3) ρ6 3 3 1 3 0 0 -1 0 1 -1 -1 orthogonal lifted from S4 ρ7 3 3 -1 3 0 0 -1 0 -1 1 1 orthogonal lifted from S4 ρ8 4 -4 0 4 1 1 0 -1 0 0 0 orthogonal lifted from GL2(𝔽3) ρ9 8 0 2 -1 2 -1 0 0 -1 0 0 orthogonal faithful ρ10 8 0 -2 -1 2 -1 0 0 1 0 0 orthogonal faithful ρ11 16 0 0 -2 -2 1 0 0 0 0 0 orthogonal faithful

Permutation representations of AGL2(𝔽3)
On 9 points: primitive, doubly transitive - transitive group 9T26
Generators in S9
(1 9 7)(2 6 5)(3 8 4)
(1 6 8)(2 3 7)(4 9 5)
(2 3 4 5)(6 7 8 9)
(2 9 4 7)(3 8 5 6)
(2 5 6)(3 8 4)
(2 4)(3 6)(5 8)

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

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

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

G:=TransitiveGroup(9,26);

On 12 points - transitive group 12T157
Generators in S12
(1 10 12)(2 11 9)(4 8 6)
(1 12 10)(2 11 9)(3 5 7)
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 4)(2 3)(5 11 7 9)(6 10 8 12)
(1 4 2)(6 9 12)(8 11 10)
(1 4)(5 7)(6 12)(8 10)

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

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

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

G:=TransitiveGroup(12,157);

On 18 points - transitive group 18T157
Generators in S18
(1 17 15)(2 8 10)(3 6 7)(4 5 9)(11 12 18)(13 16 14)
(1 18 16)(2 9 7)(3 8 4)(5 6 10)(11 14 17)(12 13 15)
(3 4 5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
(3 10 5 8)(4 9 6 7)(11 18 13 16)(12 17 14 15)
(3 6 7)(4 9 5)(11 18 12)(13 16 14)
(1 2)(3 12)(4 16)(5 14)(6 18)(7 11)(8 17)(9 13)(10 15)

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

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

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

G:=TransitiveGroup(18,157);

On 24 points - transitive group 24T1325
Generators in S24
(1 14 16)(2 13 15)(3 24 22)(4 21 23)(6 11 9)(8 20 18)
(1 14 16)(2 15 13)(3 22 24)(4 21 23)(5 12 10)(7 17 19)
(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 5)(2 6)(3 8)(4 7)(9 13 11 15)(10 16 12 14)(17 21 19 23)(18 24 20 22)
(1 2 6)(3 8 4)(9 16 15)(11 14 13)(18 23 22)(20 21 24)
(1 4)(2 8)(3 6)(5 7)(9 22)(10 17)(11 24)(12 19)(13 20)(14 21)(15 18)(16 23)

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

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

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

G:=TransitiveGroup(24,1325);

On 24 points - transitive group 24T1326
Generators in S24
(2 15 20)(4 18 13)(5 21 10)(6 22 11)(7 12 23)(8 9 24)
(1 19 14)(3 16 17)(5 10 21)(6 22 11)(7 23 12)(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 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)
(2 22 21)(4 24 23)(5 20 6)(7 18 8)(9 12 13)(10 15 11)
(1 3)(2 21)(4 23)(5 20)(7 18)(10 15)(12 13)(14 16)(17 19)

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

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

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

G:=TransitiveGroup(24,1326);

On 24 points - transitive group 24T1327
Generators in S24
(2 15 20)(4 18 13)(5 21 10)(6 22 11)(7 12 23)(8 9 24)
(1 19 14)(3 16 17)(5 10 21)(6 22 11)(7 23 12)(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 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)
(2 22 21)(4 24 23)(5 20 6)(7 18 8)(9 12 13)(10 15 11)
(2 23)(4 21)(5 13)(6 9)(7 15)(8 11)(10 18)(12 20)(14 19)(16 17)(22 24)

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

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

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

G:=TransitiveGroup(24,1327);

On 24 points - transitive group 24T1334
Generators in S24
(2 15 20)(4 18 13)(5 21 10)(6 22 11)(7 12 23)(8 9 24)
(1 19 14)(3 16 17)(5 10 21)(6 22 11)(7 23 12)(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 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)
(1 14 19)(2 11 5)(3 16 17)(4 9 7)(6 21 15)(8 23 13)(10 20 22)(12 18 24)
(2 23)(4 21)(5 13)(6 9)(7 15)(8 11)(10 18)(12 20)(14 19)(16 17)(22 24)

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

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

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

G:=TransitiveGroup(24,1334);

On 27 points - transitive group 27T139
Generators in S27
(1 16 18)(2 23 21)(3 9 11)(4 8 5)(6 7 10)(12 13 17)(14 19 15)(20 25 26)(22 24 27)
(1 17 19)(2 20 22)(3 10 8)(4 11 7)(5 9 6)(12 15 16)(13 14 18)(21 26 27)(23 25 24)
(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 8 6 10)(5 11 7 9)(12 17 14 19)(13 16 15 18)(20 27 22 25)(21 26 23 24)
(1 2 3)(4 19 24)(5 14 22)(6 17 26)(7 12 20)(8 15 27)(9 16 23)(10 13 25)(11 18 21)
(1 2)(4 10)(5 7)(6 8)(12 22)(13 24)(14 20)(15 26)(16 23)(17 27)(18 21)(19 25)

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

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

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

G:=TransitiveGroup(27,139);

Polynomial with Galois group AGL2(𝔽3) over ℚ
actionf(x)Disc(f)
9T26x9-60x7-30x6+441x5+90x4-1116x3+180x2+972x-480220·315·54·114·134·7092
12T157x12-x11-53x10+53x9+847x8-482x7-5571x6+1485x5+16143x4-1208x3-17855x2-864x+2389212·116·1392·27774·629212·108496732

Matrix representation of AGL2(𝔽3) in GL8(ℤ)

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

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

AGL2(𝔽3) in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(432,734);
// by ID

G=gap.SmallGroup(432,734);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,-3,3,57,632,387,100,1179,262,185,80,14117,6060,1699,1034,201,14118,8245,1588,223,622]);
// Polycyclic

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

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