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## G = C3×CSU2(𝔽3)  order 144 = 24·32

### Direct product of C3 and CSU2(𝔽3)

Aliases: C3×CSU2(𝔽3), C6.8S4, SL2(𝔽3).C6, Q8.(C3×S3), C2.2(C3×S4), (C3×Q8).4S3, (C3×SL2(𝔽3)).2C2, SmallGroup(144,121)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C3×CSU2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3) — C3×CSU2(𝔽3)
 Lower central SL2(𝔽3) — C3×CSU2(𝔽3)
 Upper central C1 — C6

Generators and relations for C3×CSU2(𝔽3)
G = < a,b,c,d,e | a3=b4=d3=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece-1=b-1, dbd-1=bc, ebe-1=b2c, dcd-1=b, ede-1=d-1 >

Character table of C3×CSU2(𝔽3)

 class 1 2 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 8A 8B 12A 12B 12C 12D 24A 24B 24C 24D size 1 1 1 1 8 8 8 6 12 1 1 8 8 8 6 6 6 6 12 12 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 ζ32 ζ3 ζ3 ζ32 1 1 -1 ζ32 ζ3 1 ζ32 ζ3 -1 -1 ζ3 ζ32 ζ6 ζ65 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ4 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 ζ3 ζ32 1 ζ3 ζ32 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ5 1 1 ζ3 ζ32 ζ32 ζ3 1 1 -1 ζ3 ζ32 1 ζ3 ζ32 -1 -1 ζ32 ζ3 ζ65 ζ6 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ6 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 ζ32 ζ3 1 ζ32 ζ3 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ7 2 2 2 2 -1 -1 -1 2 0 2 2 -1 -1 -1 0 0 2 2 0 0 0 0 0 0 orthogonal lifted from S3 ρ8 2 -2 2 2 -1 -1 -1 0 0 -2 -2 1 1 1 -√2 √2 0 0 0 0 -√2 √2 √2 -√2 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ9 2 -2 2 2 -1 -1 -1 0 0 -2 -2 1 1 1 √2 -√2 0 0 0 0 √2 -√2 -√2 √2 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ10 2 2 -1-√-3 -1+√-3 ζ65 ζ6 -1 2 0 -1-√-3 -1+√-3 -1 ζ6 ζ65 0 0 -1+√-3 -1-√-3 0 0 0 0 0 0 complex lifted from C3×S3 ρ11 2 2 -1+√-3 -1-√-3 ζ6 ζ65 -1 2 0 -1+√-3 -1-√-3 -1 ζ65 ζ6 0 0 -1-√-3 -1+√-3 0 0 0 0 0 0 complex lifted from C3×S3 ρ12 2 -2 -1-√-3 -1+√-3 ζ65 ζ6 -1 0 0 1+√-3 1-√-3 1 ζ32 ζ3 -√2 √2 0 0 0 0 ζ83ζ32-ζ8ζ32 ζ87ζ32-ζ85ζ32 ζ87ζ3-ζ85ζ3 ζ83ζ3-ζ8ζ3 complex faithful ρ13 2 -2 -1-√-3 -1+√-3 ζ65 ζ6 -1 0 0 1+√-3 1-√-3 1 ζ32 ζ3 √2 -√2 0 0 0 0 ζ87ζ32-ζ85ζ32 ζ83ζ32-ζ8ζ32 ζ83ζ3-ζ8ζ3 ζ87ζ3-ζ85ζ3 complex faithful ρ14 2 -2 -1+√-3 -1-√-3 ζ6 ζ65 -1 0 0 1-√-3 1+√-3 1 ζ3 ζ32 -√2 √2 0 0 0 0 ζ83ζ3-ζ8ζ3 ζ87ζ3-ζ85ζ3 ζ87ζ32-ζ85ζ32 ζ83ζ32-ζ8ζ32 complex faithful ρ15 2 -2 -1+√-3 -1-√-3 ζ6 ζ65 -1 0 0 1-√-3 1+√-3 1 ζ3 ζ32 √2 -√2 0 0 0 0 ζ87ζ3-ζ85ζ3 ζ83ζ3-ζ8ζ3 ζ83ζ32-ζ8ζ32 ζ87ζ32-ζ85ζ32 complex faithful ρ16 3 3 3 3 0 0 0 -1 1 3 3 0 0 0 -1 -1 -1 -1 1 1 -1 -1 -1 -1 orthogonal lifted from S4 ρ17 3 3 3 3 0 0 0 -1 -1 3 3 0 0 0 1 1 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from S4 ρ18 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 -1 1 -3-3√-3/2 -3+3√-3/2 0 0 0 -1 -1 ζ65 ζ6 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 complex lifted from C3×S4 ρ19 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 1 1 ζ6 ζ65 ζ65 ζ6 ζ3 ζ3 ζ32 ζ32 complex lifted from C3×S4 ρ20 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 -1 1 -3+3√-3/2 -3-3√-3/2 0 0 0 -1 -1 ζ6 ζ65 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 complex lifted from C3×S4 ρ21 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 1 1 ζ65 ζ6 ζ6 ζ65 ζ32 ζ32 ζ3 ζ3 complex lifted from C3×S4 ρ22 4 -4 4 4 1 1 1 0 0 -4 -4 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ23 4 -4 -2-2√-3 -2+2√-3 ζ3 ζ32 1 0 0 2+2√-3 2-2√-3 -1 ζ6 ζ65 0 0 0 0 0 0 0 0 0 0 complex faithful ρ24 4 -4 -2+2√-3 -2-2√-3 ζ32 ζ3 1 0 0 2-2√-3 2+2√-3 -1 ζ65 ζ6 0 0 0 0 0 0 0 0 0 0 complex faithful

Smallest permutation representation of C3×CSU2(𝔽3)
On 48 points
Generators in S48
(1 13 25)(2 14 26)(3 15 27)(4 16 28)(5 34 39)(6 35 40)(7 36 37)(8 33 38)(9 17 21)(10 18 22)(11 19 23)(12 20 24)(29 44 48)(30 41 45)(31 42 46)(32 43 47)
(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)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)
(1 11 3 9)(2 10 4 12)(5 45 7 47)(6 48 8 46)(13 19 15 17)(14 18 16 20)(21 25 23 27)(22 28 24 26)(29 33 31 35)(30 36 32 34)(37 43 39 41)(38 42 40 44)
(1 25 13)(2 23 18)(3 27 15)(4 21 20)(5 32 40)(6 34 43)(7 30 38)(8 36 41)(9 24 16)(10 26 19)(11 22 14)(12 28 17)(29 44 48)(31 42 46)(33 37 45)(35 39 47)
(1 29 3 31)(2 35 4 33)(5 24 7 22)(6 28 8 26)(9 30 11 32)(10 34 12 36)(13 44 15 42)(14 40 16 38)(17 41 19 43)(18 39 20 37)(21 45 23 47)(25 48 27 46)

G:=sub<Sym(48)| (1,13,25)(2,14,26)(3,15,27)(4,16,28)(5,34,39)(6,35,40)(7,36,37)(8,33,38)(9,17,21)(10,18,22)(11,19,23)(12,20,24)(29,44,48)(30,41,45)(31,42,46)(32,43,47), (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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,11,3,9)(2,10,4,12)(5,45,7,47)(6,48,8,46)(13,19,15,17)(14,18,16,20)(21,25,23,27)(22,28,24,26)(29,33,31,35)(30,36,32,34)(37,43,39,41)(38,42,40,44), (1,25,13)(2,23,18)(3,27,15)(4,21,20)(5,32,40)(6,34,43)(7,30,38)(8,36,41)(9,24,16)(10,26,19)(11,22,14)(12,28,17)(29,44,48)(31,42,46)(33,37,45)(35,39,47), (1,29,3,31)(2,35,4,33)(5,24,7,22)(6,28,8,26)(9,30,11,32)(10,34,12,36)(13,44,15,42)(14,40,16,38)(17,41,19,43)(18,39,20,37)(21,45,23,47)(25,48,27,46)>;

G:=Group( (1,13,25)(2,14,26)(3,15,27)(4,16,28)(5,34,39)(6,35,40)(7,36,37)(8,33,38)(9,17,21)(10,18,22)(11,19,23)(12,20,24)(29,44,48)(30,41,45)(31,42,46)(32,43,47), (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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,11,3,9)(2,10,4,12)(5,45,7,47)(6,48,8,46)(13,19,15,17)(14,18,16,20)(21,25,23,27)(22,28,24,26)(29,33,31,35)(30,36,32,34)(37,43,39,41)(38,42,40,44), (1,25,13)(2,23,18)(3,27,15)(4,21,20)(5,32,40)(6,34,43)(7,30,38)(8,36,41)(9,24,16)(10,26,19)(11,22,14)(12,28,17)(29,44,48)(31,42,46)(33,37,45)(35,39,47), (1,29,3,31)(2,35,4,33)(5,24,7,22)(6,28,8,26)(9,30,11,32)(10,34,12,36)(13,44,15,42)(14,40,16,38)(17,41,19,43)(18,39,20,37)(21,45,23,47)(25,48,27,46) );

G=PermutationGroup([[(1,13,25),(2,14,26),(3,15,27),(4,16,28),(5,34,39),(6,35,40),(7,36,37),(8,33,38),(9,17,21),(10,18,22),(11,19,23),(12,20,24),(29,44,48),(30,41,45),(31,42,46),(32,43,47)], [(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),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48)], [(1,11,3,9),(2,10,4,12),(5,45,7,47),(6,48,8,46),(13,19,15,17),(14,18,16,20),(21,25,23,27),(22,28,24,26),(29,33,31,35),(30,36,32,34),(37,43,39,41),(38,42,40,44)], [(1,25,13),(2,23,18),(3,27,15),(4,21,20),(5,32,40),(6,34,43),(7,30,38),(8,36,41),(9,24,16),(10,26,19),(11,22,14),(12,28,17),(29,44,48),(31,42,46),(33,37,45),(35,39,47)], [(1,29,3,31),(2,35,4,33),(5,24,7,22),(6,28,8,26),(9,30,11,32),(10,34,12,36),(13,44,15,42),(14,40,16,38),(17,41,19,43),(18,39,20,37),(21,45,23,47),(25,48,27,46)]])

C3×CSU2(𝔽3) is a maximal subgroup of   CSU2(𝔽3)⋊S3  Dic3.5S4  D6.2S4  C322CSU2(𝔽3)
C3×CSU2(𝔽3) is a maximal quotient of   C32.CSU2(𝔽3)  C32⋊CSU2(𝔽3)

Matrix representation of C3×CSU2(𝔽3) in GL2(𝔽7) generated by

 2 0 0 2
,
 6 5 1 1
,
 5 6 5 2
,
 2 0 5 4
,
 2 5 6 5
G:=sub<GL(2,GF(7))| [2,0,0,2],[6,1,5,1],[5,5,6,2],[2,5,0,4],[2,6,5,5] >;

C3×CSU2(𝔽3) in GAP, Magma, Sage, TeX

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

G:=Group("C3xCSU(2,3)");
// GroupNames label

G:=SmallGroup(144,121);
// by ID

G=gap.SmallGroup(144,121);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,2,-2,432,218,867,447,117,544,286,202,88]);
// Polycyclic

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

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