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G = C3×AΓL1(𝔽9)  order 432 = 24·33

Direct product of C3 and AΓL1(𝔽9)

direct product, non-abelian, soluble, monomial

Aliases: C3×AΓL1(𝔽9), F9⋊C6, C331SD16, PSU3(𝔽2)⋊2C6, S3≀C2.C6, (C3×F9)⋊3C2, C32⋊(C3×SD16), (C3×PSU3(𝔽2))⋊3C2, C3⋊S3.(C3×D4), C32⋊C4.(C2×C6), (C3×C3⋊S3).1D4, (C3×S3≀C2).2C2, (C3×C32⋊C4).6C22, SmallGroup(432,737)

Series: Derived Chief Lower central Upper central

C1C32C32⋊C4 — C3×AΓL1(𝔽9)
C1C32C3⋊S3C32⋊C4C3×C32⋊C4C3×F9 — C3×AΓL1(𝔽9)
C32C3⋊S3C32⋊C4 — C3×AΓL1(𝔽9)
C1C3

Generators and relations for C3×AΓL1(𝔽9)
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, ebe=b-1c, dcd-1=b, ce=ec, ede=d3 >

9C2
12C2
4C3
8C3
9C4
18C4
18C22
4S3
9C6
12C6
12C6
12S3
24C6
4C32
8C32
9C8
9Q8
9D4
9C12
12D6
18C12
18C2×C6
4C3×S3
4C3×S3
8C3×S3
12C3×S3
12C3×C6
9SD16
9C3×Q8
9C3×D4
9C24
2C32⋊C4
2S32
12S3×C6
4S3×C32
9C3×SD16
2C3×S32
2C3×C32⋊C4

Character table of C3×AΓL1(𝔽9)

 class 12A2B3A3B3C3D3E4A4B6A6B6C6D6E6F6G8A8B12A12B12C12D24A24B24C24D
 size 191211888183699121224242418181818363618181818
ρ1111111111111111111111111111    trivial
ρ211-1111111111-1-1-1-1-1-1-11111-1-1-1-1    linear of order 2
ρ3111111111-11111111-1-111-1-1-1-1-1-1    linear of order 2
ρ411-1111111-111-1-1-1-1-11111-1-11111    linear of order 2
ρ5111ζ3ζ321ζ32ζ31-1ζ32ζ3ζ32ζ31ζ3ζ32-1-1ζ3ζ32ζ65ζ6ζ65ζ6ζ6ζ65    linear of order 6
ρ6111ζ3ζ321ζ32ζ311ζ32ζ3ζ32ζ31ζ3ζ3211ζ3ζ32ζ3ζ32ζ3ζ32ζ32ζ3    linear of order 3
ρ7111ζ32ζ31ζ3ζ3211ζ3ζ32ζ3ζ321ζ32ζ311ζ32ζ3ζ32ζ3ζ32ζ3ζ3ζ32    linear of order 3
ρ811-1ζ32ζ31ζ3ζ321-1ζ3ζ32ζ65ζ6-1ζ6ζ6511ζ32ζ3ζ6ζ65ζ32ζ3ζ3ζ32    linear of order 6
ρ911-1ζ3ζ321ζ32ζ311ζ32ζ3ζ6ζ65-1ζ65ζ6-1-1ζ3ζ32ζ3ζ32ζ65ζ6ζ6ζ65    linear of order 6
ρ1011-1ζ32ζ31ζ3ζ3211ζ3ζ32ζ65ζ6-1ζ6ζ65-1-1ζ32ζ3ζ32ζ3ζ6ζ65ζ65ζ6    linear of order 6
ρ11111ζ32ζ31ζ3ζ321-1ζ3ζ32ζ3ζ321ζ32ζ3-1-1ζ32ζ3ζ6ζ65ζ6ζ65ζ65ζ6    linear of order 6
ρ1211-1ζ3ζ321ζ32ζ31-1ζ32ζ3ζ6ζ65-1ζ65ζ611ζ3ζ32ζ65ζ6ζ3ζ32ζ32ζ3    linear of order 6
ρ1322022222-20220000000-2-2000000    orthogonal lifted from D4
ρ14220-1--3-1+-32-1+-3-1--3-20-1+-3-1--300000001+-31--3000000    complex lifted from C3×D4
ρ15220-1+-3-1--32-1--3-1+-3-20-1--3-1+-300000001--31+-3000000    complex lifted from C3×D4
ρ162-202222200-2-200000--2-20000-2--2-2--2    complex lifted from SD16
ρ172-202222200-2-200000-2--20000--2-2--2-2    complex lifted from SD16
ρ182-20-1+-3-1--32-1--3-1+-3001+-31--300000-2--20000ζ87ζ385ζ3ζ83ζ328ζ32ζ87ζ3285ζ32ζ83ζ38ζ3    complex lifted from C3×SD16
ρ192-20-1--3-1+-32-1+-3-1--3001--31+-300000--2-20000ζ83ζ328ζ32ζ87ζ385ζ3ζ83ζ38ζ3ζ87ζ3285ζ32    complex lifted from C3×SD16
ρ202-20-1+-3-1--32-1--3-1+-3001+-31--300000--2-20000ζ83ζ38ζ3ζ87ζ3285ζ32ζ83ζ328ζ32ζ87ζ385ζ3    complex lifted from C3×SD16
ρ212-20-1--3-1+-32-1+-3-1--3001--31+-300000-2--20000ζ87ζ3285ζ32ζ83ζ38ζ3ζ87ζ385ζ3ζ83ζ328ζ32    complex lifted from C3×SD16
ρ2280-288-1-1-10000-2-21110000000000    orthogonal lifted from AΓL1(𝔽9)
ρ2380288-1-1-1000022-1-1-10000000000    orthogonal lifted from AΓL1(𝔽9)
ρ2480-2-4+4-3-4-4-3-1ζ6ζ6500001+-31--31ζ3ζ320000000000    complex faithful
ρ2580-2-4-4-3-4+4-3-1ζ65ζ600001--31+-31ζ32ζ30000000000    complex faithful
ρ26802-4+4-3-4-4-3-1ζ6ζ650000-1--3-1+-3-1ζ65ζ60000000000    complex faithful
ρ27802-4-4-3-4+4-3-1ζ65ζ60000-1+-3-1--3-1ζ6ζ650000000000    complex faithful

Permutation representations of C3×AΓL1(𝔽9)
On 24 points - transitive group 24T1330
Generators in S24
(1 9 21)(2 10 22)(3 11 23)(4 12 24)(5 13 17)(6 14 18)(7 15 19)(8 16 20)
(1 9 21)(3 11 23)(4 12 24)(5 17 13)(7 19 15)(8 20 16)
(1 21 9)(2 10 22)(4 12 24)(5 13 17)(6 18 14)(8 20 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(18 20)(19 23)(22 24)

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

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

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

G:=TransitiveGroup(24,1330);

On 27 points - transitive group 27T142
Generators in S27
(1 3 2)(4 19 23)(5 12 24)(6 13 25)(7 14 26)(8 15 27)(9 16 20)(10 17 21)(11 18 22)
(1 17 13)(2 10 6)(3 21 25)(4 5 7)(8 11 9)(12 14 19)(15 18 16)(20 27 22)(23 24 26)
(1 18 14)(2 11 7)(3 22 26)(4 10 9)(5 6 8)(12 13 15)(16 19 17)(20 23 21)(24 25 27)
(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 6)(5 9)(8 10)(12 16)(13 19)(15 17)(20 24)(21 27)(23 25)

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

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

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

G:=TransitiveGroup(27,142);

Matrix representation of C3×AΓL1(𝔽9) in GL8(𝔽73)

640000000
064000000
006400000
000640000
000064000
000006400
000000640
000000064
,
640000000
08000000
00800000
000640000
00008000
000006400
00000010
00000001
,
80000000
064000000
00100000
00010000
00008000
000006400
00000080
000000064
,
00001000
00000100
00000010
00000001
00100000
00010000
01000000
10000000
,
10000000
01000000
00010000
00100000
00000010
00000001
00001000
00000100

G:=sub<GL(8,GF(73))| [64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64],[64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64],[0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C3×AΓL1(𝔽9) in GAP, Magma, Sage, TeX

C_3\times {\rm AGammaL}_1({\mathbb F}_9)
% in TeX

G:=Group("C3xAGammaL(1,9)");
// GroupNames label

G:=SmallGroup(432,737);
// by ID

G=gap.SmallGroup(432,737);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,84,533,514,80,6053,7068,1202,201,16470,7069,1595,622]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,e*b*e=b^-1*c,d*c*d^-1=b,c*e=e*c,e*d*e=d^3>;
// generators/relations

Export

Subgroup lattice of C3×AΓL1(𝔽9) in TeX
Character table of C3×AΓL1(𝔽9) in TeX

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