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G = C3⋊U2(𝔽3)  order 288 = 25·32

The semidirect product of C3 and U2(𝔽3) acting via U2(𝔽3)/C4.A4=C2

Aliases: C3⋊U2(𝔽3), C12.13S4, SL2(𝔽3)⋊2Dic3, C4.5(C3⋊S4), C4.A4.2S3, C6.7(A4⋊C4), Q8.(C3⋊Dic3), (C3×Q8).3Dic3, C2.3(C6.7S4), (C3×SL2(𝔽3))⋊2C4, (C3×C4○D4).5S3, (C3×C4.A4).1C2, C4○D4.1(C3⋊S3), SmallGroup(288,404)

Series: Derived Chief Lower central Upper central

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

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, faf-1=a-1, 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 >

Subgroups: 284 in 62 conjugacy classes, 19 normal (13 characteristic)
C1, C2, C2, C3, C3 [×3], C4, C4 [×3], C22, C6, C6 [×4], C8, C2×C4 [×2], D4, Q8, C32, Dic3 [×2], C12, C12 [×4], C2×C6, C42, M4(2), C4○D4, C3×C6, C3⋊C8 [×4], SL2(𝔽3) [×3], C2×Dic3, C2×C12, C3×D4, C3×Q8, C4≀C2, C3×C12, C4.Dic3, C4×Dic3, C4.A4 [×3], C3×C4○D4, C324C8, C3×SL2(𝔽3), Q83Dic3, U2(𝔽3) [×3], C3×C4.A4, C3⋊U2(𝔽3)
Quotients: C1, C2, C4, S3 [×4], Dic3 [×4], C3⋊S3, S4, C3⋊Dic3, A4⋊C4, C3⋊S4, U2(𝔽3), C6.7S4, C3⋊U2(𝔽3)

Character table of C3⋊U2(𝔽3)

 class 1 2A 2B 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G 6A 6B 6C 6D 6E 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H 12I size 1 1 6 2 8 8 8 1 1 6 18 18 18 18 2 8 8 8 12 36 36 2 2 8 8 8 8 8 8 12 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 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 1 1 1 1 linear of order 2 ρ3 1 1 -1 1 1 1 1 -1 -1 1 i -i i -i 1 1 1 1 -1 -i i -1 -1 -1 -1 -1 -1 -1 -1 1 linear of order 4 ρ4 1 1 -1 1 1 1 1 -1 -1 1 -i i -i i 1 1 1 1 -1 i -i -1 -1 -1 -1 -1 -1 -1 -1 1 linear of order 4 ρ5 2 2 2 -1 2 -1 -1 2 2 2 0 0 0 0 -1 -1 -1 2 -1 0 0 -1 -1 2 -1 -1 -1 -1 2 -1 orthogonal lifted from S3 ρ6 2 2 2 -1 -1 2 -1 2 2 2 0 0 0 0 -1 -1 2 -1 -1 0 0 -1 -1 -1 -1 2 2 -1 -1 -1 orthogonal lifted from S3 ρ7 2 2 2 2 -1 -1 -1 2 2 2 0 0 0 0 2 -1 -1 -1 2 0 0 2 2 -1 -1 -1 -1 -1 -1 2 orthogonal lifted from S3 ρ8 2 2 2 -1 -1 -1 2 2 2 2 0 0 0 0 -1 2 -1 -1 -1 0 0 -1 -1 -1 2 -1 -1 2 -1 -1 orthogonal lifted from S3 ρ9 2 2 -2 2 -1 -1 -1 -2 -2 2 0 0 0 0 2 -1 -1 -1 -2 0 0 -2 -2 1 1 1 1 1 1 2 symplectic lifted from Dic3, Schur index 2 ρ10 2 2 -2 -1 -1 2 -1 -2 -2 2 0 0 0 0 -1 -1 2 -1 1 0 0 1 1 1 1 -2 -2 1 1 -1 symplectic lifted from Dic3, Schur index 2 ρ11 2 2 -2 -1 -1 -1 2 -2 -2 2 0 0 0 0 -1 2 -1 -1 1 0 0 1 1 1 -2 1 1 -2 1 -1 symplectic lifted from Dic3, Schur index 2 ρ12 2 2 -2 -1 2 -1 -1 -2 -2 2 0 0 0 0 -1 -1 -1 2 1 0 0 1 1 -2 1 1 1 1 -2 -1 symplectic lifted from Dic3, Schur index 2 ρ13 2 -2 0 2 -1 -1 -1 2i -2i 0 1+i 1-i -1-i -1+i -2 1 1 1 0 0 0 2i -2i -i i -i i -i i 0 complex lifted from U2(𝔽3) ρ14 2 -2 0 2 -1 -1 -1 -2i 2i 0 1-i 1+i -1+i -1-i -2 1 1 1 0 0 0 -2i 2i i -i i -i i -i 0 complex lifted from U2(𝔽3) ρ15 2 -2 0 2 -1 -1 -1 2i -2i 0 -1-i -1+i 1+i 1-i -2 1 1 1 0 0 0 2i -2i -i i -i i -i i 0 complex lifted from U2(𝔽3) ρ16 2 -2 0 2 -1 -1 -1 -2i 2i 0 -1+i -1-i 1-i 1+i -2 1 1 1 0 0 0 -2i 2i i -i i -i i -i 0 complex lifted from U2(𝔽3) ρ17 3 3 -1 3 0 0 0 3 3 -1 1 1 1 1 3 0 0 0 -1 -1 -1 3 3 0 0 0 0 0 0 -1 orthogonal lifted from S4 ρ18 3 3 -1 3 0 0 0 3 3 -1 -1 -1 -1 -1 3 0 0 0 -1 1 1 3 3 0 0 0 0 0 0 -1 orthogonal lifted from S4 ρ19 3 3 1 3 0 0 0 -3 -3 -1 i -i i -i 3 0 0 0 1 i -i -3 -3 0 0 0 0 0 0 -1 complex lifted from A4⋊C4 ρ20 3 3 1 3 0 0 0 -3 -3 -1 -i i -i i 3 0 0 0 1 -i i -3 -3 0 0 0 0 0 0 -1 complex lifted from A4⋊C4 ρ21 4 -4 0 -2 -2 1 1 4i -4i 0 0 0 0 0 2 -1 -1 2 0 0 0 -2i 2i -2i -i i -i i 2i 0 complex faithful ρ22 4 -4 0 4 1 1 1 -4i 4i 0 0 0 0 0 -4 -1 -1 -1 0 0 0 -4i 4i -i i -i i -i i 0 complex lifted from U2(𝔽3) ρ23 4 -4 0 -2 1 1 -2 4i -4i 0 0 0 0 0 2 2 -1 -1 0 0 0 -2i 2i i 2i i -i -2i -i 0 complex faithful ρ24 4 -4 0 4 1 1 1 4i -4i 0 0 0 0 0 -4 -1 -1 -1 0 0 0 4i -4i i -i i -i i -i 0 complex lifted from U2(𝔽3) ρ25 4 -4 0 -2 1 1 -2 -4i 4i 0 0 0 0 0 2 2 -1 -1 0 0 0 2i -2i -i -2i -i i 2i i 0 complex faithful ρ26 4 -4 0 -2 -2 1 1 -4i 4i 0 0 0 0 0 2 -1 -1 2 0 0 0 2i -2i 2i i -i i -i -2i 0 complex faithful ρ27 4 -4 0 -2 1 -2 1 -4i 4i 0 0 0 0 0 2 -1 2 -1 0 0 0 2i -2i -i i 2i -2i -i i 0 complex faithful ρ28 4 -4 0 -2 1 -2 1 4i -4i 0 0 0 0 0 2 -1 2 -1 0 0 0 -2i 2i i -i -2i 2i i -i 0 complex faithful ρ29 6 6 -2 -3 0 0 0 6 6 -2 0 0 0 0 -3 0 0 0 1 0 0 -3 -3 0 0 0 0 0 0 1 orthogonal lifted from C3⋊S4 ρ30 6 6 2 -3 0 0 0 -6 -6 -2 0 0 0 0 -3 0 0 0 -1 0 0 3 3 0 0 0 0 0 0 1 symplectic lifted from C6.7S4, Schur index 2

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

Matrix representation of C3⋊U2(𝔽3) in GL4(𝔽5) generated by

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

C3⋊U2(𝔽3) in GAP, Magma, Sage, TeX

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

G:=Group("C3:U(2,3)");
// GroupNames label

G:=SmallGroup(288,404);
// by ID

G=gap.SmallGroup(288,404);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,14,1016,170,675,2524,1908,172,1517,1153,285,124]);
// 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,f*a*f^-1=a^-1,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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