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## G = GL2(𝔽3)⋊C22order 192 = 26·3

### 3rd semidirect product of GL2(𝔽3) and C22 acting via C22/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — GL2(𝔽3)⋊C22
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — GL2(𝔽3) — C4.6S4 — GL2(𝔽3)⋊C22
 Lower central SL2(𝔽3) — GL2(𝔽3)⋊C22
 Upper central C1 — C4 — C2×C4

Generators and relations for GL2(𝔽3)⋊C22
G = < a,b,c,d,e,f | a4=c3=d2=e2=f2=1, b2=a2, bab-1=eae=dbd=a-1, cac-1=ab, dad=ebe=a2b, af=fa, cbc-1=a, bf=fb, dcd=c-1, ece=ac, cf=fc, ede=fdf=a2d, ef=fe >

Subgroups: 523 in 144 conjugacy classes, 27 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, D6, C2×C6, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, SL2(𝔽3), Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×M4(2), C4○D8, C8⋊C22, C8.C22, C2×C4○D4, C2×C4○D4, CSU2(𝔽3), GL2(𝔽3), C2×SL2(𝔽3), C4.A4, C4○D12, D8⋊C22, Q8.D6, C4.S4, C4.6S4, C4.3S4, C2×C4.A4, GL2(𝔽3)⋊C22
Quotients: C1, C2, C22, S3, C23, D6, S4, C22×S3, C2×S4, C22×S4, GL2(𝔽3)⋊C22

Character table of GL2(𝔽3)⋊C22

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D size 1 1 2 6 6 12 12 8 1 1 2 6 6 12 12 8 8 8 12 12 12 12 8 8 8 8 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 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 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 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 linear of order 2 ρ5 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 ρ6 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 ρ7 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 ρ8 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 ρ9 2 2 2 -2 -2 0 0 -1 -2 -2 -2 2 2 0 0 -1 -1 -1 0 0 0 0 1 1 1 1 orthogonal lifted from D6 ρ10 2 2 -2 2 -2 0 0 -1 2 2 -2 -2 2 0 0 -1 1 1 0 0 0 0 -1 -1 1 1 orthogonal lifted from D6 ρ11 2 2 -2 -2 2 0 0 -1 -2 -2 2 -2 2 0 0 -1 1 1 0 0 0 0 1 1 -1 -1 orthogonal lifted from D6 ρ12 2 2 2 2 2 0 0 -1 2 2 2 2 2 0 0 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ13 3 3 -3 -1 1 -1 1 0 3 3 -3 1 -1 -1 1 0 0 0 1 -1 1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ14 3 3 -3 1 -1 -1 -1 0 -3 -3 3 1 -1 1 1 0 0 0 1 -1 -1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ15 3 3 3 1 1 -1 1 0 -3 -3 -3 -1 -1 1 -1 0 0 0 1 1 -1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ16 3 3 3 -1 -1 -1 -1 0 3 3 3 -1 -1 -1 -1 0 0 0 1 1 1 1 0 0 0 0 orthogonal lifted from S4 ρ17 3 3 -3 1 -1 1 1 0 -3 -3 3 1 -1 -1 -1 0 0 0 -1 1 1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ18 3 3 -3 -1 1 1 -1 0 3 3 -3 1 -1 1 -1 0 0 0 -1 1 -1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ19 3 3 3 -1 -1 1 1 0 3 3 3 -1 -1 1 1 0 0 0 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S4 ρ20 3 3 3 1 1 1 -1 0 -3 -3 -3 -1 -1 -1 1 0 0 0 -1 -1 1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ21 4 -4 0 0 0 0 0 -2 4i -4i 0 0 0 0 0 2 0 0 0 0 0 0 2i -2i 0 0 complex faithful ρ22 4 -4 0 0 0 0 0 -2 -4i 4i 0 0 0 0 0 2 0 0 0 0 0 0 -2i 2i 0 0 complex faithful ρ23 4 -4 0 0 0 0 0 1 4i -4i 0 0 0 0 0 -1 -√-3 √-3 0 0 0 0 -i i √3 -√3 complex faithful ρ24 4 -4 0 0 0 0 0 1 4i -4i 0 0 0 0 0 -1 √-3 -√-3 0 0 0 0 -i i -√3 √3 complex faithful ρ25 4 -4 0 0 0 0 0 1 -4i 4i 0 0 0 0 0 -1 √-3 -√-3 0 0 0 0 i -i √3 -√3 complex faithful ρ26 4 -4 0 0 0 0 0 1 -4i 4i 0 0 0 0 0 -1 -√-3 √-3 0 0 0 0 i -i -√3 √3 complex faithful

Smallest permutation representation of GL2(𝔽3)⋊C22
On 32 points
Generators in S32
(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)
(1 11 3 9)(2 10 4 12)(5 31 7 29)(6 30 8 32)(13 18 15 20)(14 17 16 19)(21 27 23 25)(22 26 24 28)
(2 11 10)(4 9 12)(5 8 30)(6 32 7)(13 19 18)(15 17 20)(22 27 26)(24 25 28)
(1 31)(2 5)(3 29)(4 7)(6 12)(8 10)(9 32)(11 30)(13 25)(14 23)(15 27)(16 21)(17 22)(18 28)(19 24)(20 26)
(1 18)(2 17)(3 20)(4 19)(5 24)(6 23)(7 22)(8 21)(9 15)(10 14)(11 13)(12 16)(25 32)(26 31)(27 30)(28 29)
(5 7)(6 8)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)

G:=sub<Sym(32)| (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), (1,11,3,9)(2,10,4,12)(5,31,7,29)(6,30,8,32)(13,18,15,20)(14,17,16,19)(21,27,23,25)(22,26,24,28), (2,11,10)(4,9,12)(5,8,30)(6,32,7)(13,19,18)(15,17,20)(22,27,26)(24,25,28), (1,31)(2,5)(3,29)(4,7)(6,12)(8,10)(9,32)(11,30)(13,25)(14,23)(15,27)(16,21)(17,22)(18,28)(19,24)(20,26), (1,18)(2,17)(3,20)(4,19)(5,24)(6,23)(7,22)(8,21)(9,15)(10,14)(11,13)(12,16)(25,32)(26,31)(27,30)(28,29), (5,7)(6,8)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)>;

G:=Group( (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), (1,11,3,9)(2,10,4,12)(5,31,7,29)(6,30,8,32)(13,18,15,20)(14,17,16,19)(21,27,23,25)(22,26,24,28), (2,11,10)(4,9,12)(5,8,30)(6,32,7)(13,19,18)(15,17,20)(22,27,26)(24,25,28), (1,31)(2,5)(3,29)(4,7)(6,12)(8,10)(9,32)(11,30)(13,25)(14,23)(15,27)(16,21)(17,22)(18,28)(19,24)(20,26), (1,18)(2,17)(3,20)(4,19)(5,24)(6,23)(7,22)(8,21)(9,15)(10,14)(11,13)(12,16)(25,32)(26,31)(27,30)(28,29), (5,7)(6,8)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32) );

G=PermutationGroup([[(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)], [(1,11,3,9),(2,10,4,12),(5,31,7,29),(6,30,8,32),(13,18,15,20),(14,17,16,19),(21,27,23,25),(22,26,24,28)], [(2,11,10),(4,9,12),(5,8,30),(6,32,7),(13,19,18),(15,17,20),(22,27,26),(24,25,28)], [(1,31),(2,5),(3,29),(4,7),(6,12),(8,10),(9,32),(11,30),(13,25),(14,23),(15,27),(16,21),(17,22),(18,28),(19,24),(20,26)], [(1,18),(2,17),(3,20),(4,19),(5,24),(6,23),(7,22),(8,21),(9,15),(10,14),(11,13),(12,16),(25,32),(26,31),(27,30),(28,29)], [(5,7),(6,8),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32)]])

Matrix representation of GL2(𝔽3)⋊C22 in GL4(𝔽5) generated by

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

GL2(𝔽3)⋊C22 in GAP, Magma, Sage, TeX

{\rm GL}_2({\mathbb F}_3)\rtimes C_2^2
% in TeX

G:=Group("GL(2,3):C2^2");
// GroupNames label

G:=SmallGroup(192,1482);
// by ID

G=gap.SmallGroup(192,1482);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,680,2102,451,1684,655,172,1013,404,285,124]);
// Polycyclic

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

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