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## G = C3×U2(𝔽3)  order 288 = 25·32

### Direct product of C3 and U2(𝔽3)

Aliases: C3×U2(𝔽3), C12.19S4, SL2(𝔽3)⋊2C12, C4.5(C3×S4), C4.A4.2C6, Q8.(C3×Dic3), C6.10(A4⋊C4), (C3×Q8).4Dic3, (C3×SL2(𝔽3))⋊5C4, C2.3(C3×A4⋊C4), (C3×C4.A4).5C2, (C3×C4○D4).7S3, C4○D4.1(C3×S3), SmallGroup(288,400)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C3×U2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C4.A4 — C3×C4.A4 — C3×U2(𝔽3)
 Lower central SL2(𝔽3) — C3×U2(𝔽3)
 Upper central C1 — C12

Generators and relations for C3×U2(𝔽3)
G = < a,b,c,d,e,f | a3=b4=e3=1, c2=d2=b2, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=b2c, ece-1=b2cd, fcf-1=cd, ede-1=c, fdf-1=b2d, fef-1=e-1 >

Smallest permutation representation of C3×U2(𝔽3)
On 72 points
Generators in S72
(1 50 26)(2 51 27)(3 52 28)(4 53 29)(5 54 30)(6 55 31)(7 56 32)(8 49 25)(9 57 33)(10 58 34)(11 59 35)(12 60 36)(13 61 37)(14 62 38)(15 63 39)(16 64 40)(17 65 41)(18 66 42)(19 67 43)(20 68 44)(21 69 45)(22 70 46)(23 71 47)(24 72 48)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)
(1 6 5 2)(3 8 7 4)(9 20 13 24)(10 16 14 12)(11 22 15 18)(17 19 21 23)(25 32 29 28)(26 31 30 27)(33 44 37 48)(34 40 38 36)(35 46 39 42)(41 43 45 47)(49 56 53 52)(50 55 54 51)(57 68 61 72)(58 64 62 60)(59 70 63 66)(65 67 69 71)
(1 7 5 3)(2 4 6 8)(9 18 13 22)(10 23 14 19)(11 20 15 24)(12 17 16 21)(25 27 29 31)(26 32 30 28)(33 42 37 46)(34 47 38 43)(35 44 39 48)(36 41 40 45)(49 51 53 55)(50 56 54 52)(57 66 61 70)(58 71 62 67)(59 68 63 72)(60 65 64 69)
(1 18 16)(2 9 19)(3 20 10)(4 11 21)(5 22 12)(6 13 23)(7 24 14)(8 15 17)(25 39 41)(26 42 40)(27 33 43)(28 44 34)(29 35 45)(30 46 36)(31 37 47)(32 48 38)(49 63 65)(50 66 64)(51 57 67)(52 68 58)(53 59 69)(54 70 60)(55 61 71)(56 72 62)
(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)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)

G:=sub<Sym(72)| (1,50,26)(2,51,27)(3,52,28)(4,53,29)(5,54,30)(6,55,31)(7,56,32)(8,49,25)(9,57,33)(10,58,34)(11,59,35)(12,60,36)(13,61,37)(14,62,38)(15,63,39)(16,64,40)(17,65,41)(18,66,42)(19,67,43)(20,68,44)(21,69,45)(22,70,46)(23,71,47)(24,72,48), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72), (1,6,5,2)(3,8,7,4)(9,20,13,24)(10,16,14,12)(11,22,15,18)(17,19,21,23)(25,32,29,28)(26,31,30,27)(33,44,37,48)(34,40,38,36)(35,46,39,42)(41,43,45,47)(49,56,53,52)(50,55,54,51)(57,68,61,72)(58,64,62,60)(59,70,63,66)(65,67,69,71), (1,7,5,3)(2,4,6,8)(9,18,13,22)(10,23,14,19)(11,20,15,24)(12,17,16,21)(25,27,29,31)(26,32,30,28)(33,42,37,46)(34,47,38,43)(35,44,39,48)(36,41,40,45)(49,51,53,55)(50,56,54,52)(57,66,61,70)(58,71,62,67)(59,68,63,72)(60,65,64,69), (1,18,16)(2,9,19)(3,20,10)(4,11,21)(5,22,12)(6,13,23)(7,24,14)(8,15,17)(25,39,41)(26,42,40)(27,33,43)(28,44,34)(29,35,45)(30,46,36)(31,37,47)(32,48,38)(49,63,65)(50,66,64)(51,57,67)(52,68,58)(53,59,69)(54,70,60)(55,61,71)(56,72,62), (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)>;

G:=Group( (1,50,26)(2,51,27)(3,52,28)(4,53,29)(5,54,30)(6,55,31)(7,56,32)(8,49,25)(9,57,33)(10,58,34)(11,59,35)(12,60,36)(13,61,37)(14,62,38)(15,63,39)(16,64,40)(17,65,41)(18,66,42)(19,67,43)(20,68,44)(21,69,45)(22,70,46)(23,71,47)(24,72,48), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72), (1,6,5,2)(3,8,7,4)(9,20,13,24)(10,16,14,12)(11,22,15,18)(17,19,21,23)(25,32,29,28)(26,31,30,27)(33,44,37,48)(34,40,38,36)(35,46,39,42)(41,43,45,47)(49,56,53,52)(50,55,54,51)(57,68,61,72)(58,64,62,60)(59,70,63,66)(65,67,69,71), (1,7,5,3)(2,4,6,8)(9,18,13,22)(10,23,14,19)(11,20,15,24)(12,17,16,21)(25,27,29,31)(26,32,30,28)(33,42,37,46)(34,47,38,43)(35,44,39,48)(36,41,40,45)(49,51,53,55)(50,56,54,52)(57,66,61,70)(58,71,62,67)(59,68,63,72)(60,65,64,69), (1,18,16)(2,9,19)(3,20,10)(4,11,21)(5,22,12)(6,13,23)(7,24,14)(8,15,17)(25,39,41)(26,42,40)(27,33,43)(28,44,34)(29,35,45)(30,46,36)(31,37,47)(32,48,38)(49,63,65)(50,66,64)(51,57,67)(52,68,58)(53,59,69)(54,70,60)(55,61,71)(56,72,62), (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72) );

G=PermutationGroup([[(1,50,26),(2,51,27),(3,52,28),(4,53,29),(5,54,30),(6,55,31),(7,56,32),(8,49,25),(9,57,33),(10,58,34),(11,59,35),(12,60,36),(13,61,37),(14,62,38),(15,63,39),(16,64,40),(17,65,41),(18,66,42),(19,67,43),(20,68,44),(21,69,45),(22,70,46),(23,71,47),(24,72,48)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72)], [(1,6,5,2),(3,8,7,4),(9,20,13,24),(10,16,14,12),(11,22,15,18),(17,19,21,23),(25,32,29,28),(26,31,30,27),(33,44,37,48),(34,40,38,36),(35,46,39,42),(41,43,45,47),(49,56,53,52),(50,55,54,51),(57,68,61,72),(58,64,62,60),(59,70,63,66),(65,67,69,71)], [(1,7,5,3),(2,4,6,8),(9,18,13,22),(10,23,14,19),(11,20,15,24),(12,17,16,21),(25,27,29,31),(26,32,30,28),(33,42,37,46),(34,47,38,43),(35,44,39,48),(36,41,40,45),(49,51,53,55),(50,56,54,52),(57,66,61,70),(58,71,62,67),(59,68,63,72),(60,65,64,69)], [(1,18,16),(2,9,19),(3,20,10),(4,11,21),(5,22,12),(6,13,23),(7,24,14),(8,15,17),(25,39,41),(26,42,40),(27,33,43),(28,44,34),(29,35,45),(30,46,36),(31,37,47),(32,48,38),(49,63,65),(50,66,64),(51,57,67),(52,68,58),(53,59,69),(54,70,60),(55,61,71),(56,72,62)], [(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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72)]])

48 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 4C ··· 4G 6A 6B 6C 6D 6E 6F 6G 8A 8B 12A 12B 12C 12D 12E ··· 12N 12O ··· 12T 24A 24B 24C 24D order 1 2 2 3 3 3 3 3 4 4 4 ··· 4 6 6 6 6 6 6 6 8 8 12 12 12 12 12 ··· 12 12 ··· 12 24 24 24 24 size 1 1 6 1 1 8 8 8 1 1 6 ··· 6 1 1 6 6 8 8 8 12 12 1 1 1 1 6 ··· 6 8 ··· 8 12 12 12 12

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 4 4 type + + + - + image C1 C2 C3 C4 C6 C12 S3 Dic3 C3×S3 C3×Dic3 U2(𝔽3) C3×U2(𝔽3) S4 A4⋊C4 C3×S4 C3×A4⋊C4 U2(𝔽3) C3×U2(𝔽3) kernel C3×U2(𝔽3) C3×C4.A4 U2(𝔽3) C3×SL2(𝔽3) C4.A4 SL2(𝔽3) C3×C4○D4 C3×Q8 C4○D4 Q8 C3 C1 C12 C6 C4 C2 C3 C1 # reps 1 1 2 2 2 4 1 1 2 2 4 8 2 2 4 4 2 4

Matrix representation of C3×U2(𝔽3) in GL2(𝔽13) generated by

 9 0 0 9
,
 5 0 0 5
,
 0 3 4 0
,
 5 0 0 8
,
 10 6 1 2
,
 0 3 6 0
G:=sub<GL(2,GF(13))| [9,0,0,9],[5,0,0,5],[0,4,3,0],[5,0,0,8],[10,1,6,2],[0,6,3,0] >;

C3×U2(𝔽3) in GAP, Magma, Sage, TeX

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

G:=Group("C3xU(2,3)");
// GroupNames label

G:=SmallGroup(288,400);
// by ID

G=gap.SmallGroup(288,400);
# by ID

G:=PCGroup([7,-2,-3,-2,-3,-2,2,-2,42,520,675,2524,655,172,1517,404,285,124]);
// Polycyclic

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

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