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## G = C3×GL2(𝔽3)  order 144 = 24·32

### Direct product of C3 and GL2(𝔽3)

Aliases: C3×GL2(𝔽3), C6.9S4, SL2(𝔽3)⋊C6, Q8⋊(C3×S3), C2.3(C3×S4), (C3×Q8)⋊2S3, (C3×SL2(𝔽3))⋊4C2, SmallGroup(144,122)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C3×GL2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3) — C3×GL2(𝔽3)
 Lower central SL2(𝔽3) — C3×GL2(𝔽3)
 Upper central C1 — C6

Generators and relations for C3×GL2(𝔽3)
G = < a,b,c,d,e | a3=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece=b-1, dbd-1=bc, ebe=b2c, dcd-1=b, ede=d-1 >

Character table of C3×GL2(𝔽3)

 class 1 2A 2B 3A 3B 3C 3D 3E 4 6A 6B 6C 6D 6E 6F 6G 8A 8B 12A 12B 24A 24B 24C 24D size 1 1 12 1 1 8 8 8 6 1 1 8 8 8 12 12 6 6 6 6 6 6 6 6 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 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 linear of order 2 ρ3 1 1 -1 ζ3 ζ32 ζ3 1 ζ32 1 ζ3 ζ32 ζ3 1 ζ32 ζ6 ζ65 -1 -1 ζ32 ζ3 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ4 1 1 -1 ζ32 ζ3 ζ32 1 ζ3 1 ζ32 ζ3 ζ32 1 ζ3 ζ65 ζ6 -1 -1 ζ3 ζ32 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ5 1 1 1 ζ32 ζ3 ζ32 1 ζ3 1 ζ32 ζ3 ζ32 1 ζ3 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ6 1 1 1 ζ3 ζ32 ζ3 1 ζ32 1 ζ3 ζ32 ζ3 1 ζ32 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ7 2 2 0 2 2 -1 -1 -1 2 2 2 -1 -1 -1 0 0 0 0 2 2 0 0 0 0 orthogonal lifted from S3 ρ8 2 2 0 -1+√-3 -1-√-3 ζ65 -1 ζ6 2 -1+√-3 -1-√-3 ζ65 -1 ζ6 0 0 0 0 -1-√-3 -1+√-3 0 0 0 0 complex lifted from C3×S3 ρ9 2 2 0 -1-√-3 -1+√-3 ζ6 -1 ζ65 2 -1-√-3 -1+√-3 ζ6 -1 ζ65 0 0 0 0 -1+√-3 -1-√-3 0 0 0 0 complex lifted from C3×S3 ρ10 2 -2 0 2 2 -1 -1 -1 0 -2 -2 1 1 1 0 0 -√-2 √-2 0 0 -√-2 √-2 √-2 -√-2 complex lifted from GL2(𝔽3) ρ11 2 -2 0 2 2 -1 -1 -1 0 -2 -2 1 1 1 0 0 √-2 -√-2 0 0 √-2 -√-2 -√-2 √-2 complex lifted from GL2(𝔽3) ρ12 2 -2 0 -1-√-3 -1+√-3 ζ6 -1 ζ65 0 1+√-3 1-√-3 ζ32 1 ζ3 0 0 -√-2 √-2 0 0 ζ87ζ32+ζ85ζ32 ζ83ζ32+ζ8ζ32 ζ83ζ3+ζ8ζ3 ζ87ζ3+ζ85ζ3 complex faithful ρ13 2 -2 0 -1+√-3 -1-√-3 ζ65 -1 ζ6 0 1-√-3 1+√-3 ζ3 1 ζ32 0 0 -√-2 √-2 0 0 ζ87ζ3+ζ85ζ3 ζ83ζ3+ζ8ζ3 ζ83ζ32+ζ8ζ32 ζ87ζ32+ζ85ζ32 complex faithful ρ14 2 -2 0 -1+√-3 -1-√-3 ζ65 -1 ζ6 0 1-√-3 1+√-3 ζ3 1 ζ32 0 0 √-2 -√-2 0 0 ζ83ζ3+ζ8ζ3 ζ87ζ3+ζ85ζ3 ζ87ζ32+ζ85ζ32 ζ83ζ32+ζ8ζ32 complex faithful ρ15 2 -2 0 -1-√-3 -1+√-3 ζ6 -1 ζ65 0 1+√-3 1-√-3 ζ32 1 ζ3 0 0 √-2 -√-2 0 0 ζ83ζ32+ζ8ζ32 ζ87ζ32+ζ85ζ32 ζ87ζ3+ζ85ζ3 ζ83ζ3+ζ8ζ3 complex faithful ρ16 3 3 -1 3 3 0 0 0 -1 3 3 0 0 0 -1 -1 1 1 -1 -1 1 1 1 1 orthogonal lifted from S4 ρ17 3 3 1 3 3 0 0 0 -1 3 3 0 0 0 1 1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S4 ρ18 3 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 ζ6 ζ65 1 1 ζ6 ζ65 ζ3 ζ3 ζ32 ζ32 complex lifted from C3×S4 ρ19 3 3 1 -3-3√-3/2 -3+3√-3/2 0 0 0 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 ζ3 ζ32 -1 -1 ζ65 ζ6 ζ6 ζ6 ζ65 ζ65 complex lifted from C3×S4 ρ20 3 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 ζ65 ζ6 1 1 ζ65 ζ6 ζ32 ζ32 ζ3 ζ3 complex lifted from C3×S4 ρ21 3 3 1 -3+3√-3/2 -3-3√-3/2 0 0 0 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 ζ32 ζ3 -1 -1 ζ6 ζ65 ζ65 ζ65 ζ6 ζ6 complex lifted from C3×S4 ρ22 4 -4 0 4 4 1 1 1 0 -4 -4 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from GL2(𝔽3) ρ23 4 -4 0 -2+2√-3 -2-2√-3 ζ3 1 ζ32 0 2-2√-3 2+2√-3 ζ65 -1 ζ6 0 0 0 0 0 0 0 0 0 0 complex faithful ρ24 4 -4 0 -2-2√-3 -2+2√-3 ζ32 1 ζ3 0 2+2√-3 2-2√-3 ζ6 -1 ζ65 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C3×GL2(𝔽3)
On 24 points - transitive group 24T253
Generators in S24
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 19 11)(6 20 12)(7 17 9)(8 18 10)
(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 11 3 9)(2 10 4 12)(5 22 7 24)(6 21 8 23)(13 18 15 20)(14 17 16 19)
(2 11 10)(4 9 12)(5 8 21)(6 23 7)(13 19 18)(15 17 20)
(1 3)(2 11)(4 9)(5 21)(7 23)(13 19)(14 16)(15 17)(22 24)

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

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

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

G:=TransitiveGroup(24,253);

C3×GL2(𝔽3) is a maximal subgroup of   Dic3.4S4  GL2(𝔽3)⋊S3  D6.S4  C323GL2(𝔽3)
C3×GL2(𝔽3) is a maximal quotient of   C32.GL2(𝔽3)  C322GL2(𝔽3)

Matrix representation of C3×GL2(𝔽3) in GL2(𝔽19) generated by

 11 0 0 11
,
 6 1 1 13
,
 0 18 1 0
,
 2 3 4 16
,
 1 13 0 18
G:=sub<GL(2,GF(19))| [11,0,0,11],[6,1,1,13],[0,1,18,0],[2,4,3,16],[1,0,13,18] >;

C3×GL2(𝔽3) in GAP, Magma, Sage, TeX

C_3\times {\rm GL}_2({\mathbb F}_3)
% in TeX

G:=Group("C3xGL(2,3)");
// GroupNames label

G:=SmallGroup(144,122);
// by ID

G=gap.SmallGroup(144,122);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,2,-2,218,867,447,117,544,286,202,88]);
// Polycyclic

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

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