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## G = CSU2(𝔽3)⋊S3order 288 = 25·32

### 1st semidirect product of CSU2(𝔽3) and S3 acting via S3/C3=C2

Aliases: Dic3.1S4, CSU2(𝔽3)⋊1S3, SL2(𝔽3).1D6, Q8.1S32, C2.4(S3×S4), C6.1(C2×S4), Q83S3.S3, (C3×Q8).1D6, C6.5S41C2, Dic3.A4.C2, C31(C4.S4), (C3×CSU2(𝔽3))⋊3C2, (C3×SL2(𝔽3)).1C22, SmallGroup(288,844)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C3×SL2(𝔽3) — CSU2(𝔽3)⋊S3
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — Dic3.A4 — CSU2(𝔽3)⋊S3
 Lower central C3×SL2(𝔽3) — CSU2(𝔽3)⋊S3
 Upper central C1 — C2

Generators and relations for CSU2(𝔽3)⋊S3
G = < a,b,c,d,e,f | a4=c3=e3=f2=1, b2=d2=a2, bab-1=faf=dbd-1=a-1, cac-1=ab, dad-1=fbf=a2b, ae=ea, cbc-1=a, be=eb, dcd-1=c-1, ce=ec, fcf=ac, de=ed, df=fd, fef=e-1 >

Subgroups: 414 in 77 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, C32, Dic3, Dic3, C12, D6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×C6, C3⋊C8, C24, SL2(𝔽3), SL2(𝔽3), Dic6, C4×S3, D12, C3×Q8, C3×Q8, C8.C22, C3×Dic3, C3⋊Dic3, C8⋊S3, C24⋊C2, Q82S3, C3⋊Q16, C3×Q16, CSU2(𝔽3), CSU2(𝔽3), C4.A4, S3×Q8, Q83S3, C322Q8, C3×SL2(𝔽3), Q16⋊S3, C4.S4, C3×CSU2(𝔽3), C6.5S4, Dic3.A4, CSU2(𝔽3)⋊S3
Quotients: C1, C2, C22, S3, D6, S4, S32, C2×S4, C4.S4, S3×S4, CSU2(𝔽3)⋊S3

Character table of CSU2(𝔽3)⋊S3

 class 1 2A 2B 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 8A 8B 12A 12B 12C 12D 24A 24B size 1 1 18 2 8 16 6 6 12 36 2 8 16 12 36 12 24 24 24 12 12 ρ1 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 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 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 linear of order 2 ρ5 2 2 2 2 -1 -1 2 2 0 0 2 -1 -1 0 0 2 -1 0 -1 0 0 orthogonal lifted from S3 ρ6 2 2 0 -1 2 -1 2 0 2 0 -1 2 -1 2 0 -1 0 -1 0 -1 -1 orthogonal lifted from S3 ρ7 2 2 -2 2 -1 -1 2 -2 0 0 2 -1 -1 0 0 2 1 0 1 0 0 orthogonal lifted from D6 ρ8 2 2 0 -1 2 -1 2 0 -2 0 -1 2 -1 -2 0 -1 0 1 0 1 1 orthogonal lifted from D6 ρ9 3 3 1 3 0 0 -1 -3 -1 1 3 0 0 1 -1 -1 0 -1 0 1 1 orthogonal lifted from C2×S4 ρ10 3 3 -1 3 0 0 -1 3 1 1 3 0 0 -1 -1 -1 0 1 0 -1 -1 orthogonal lifted from S4 ρ11 3 3 -1 3 0 0 -1 3 -1 -1 3 0 0 1 1 -1 0 -1 0 1 1 orthogonal lifted from S4 ρ12 3 3 1 3 0 0 -1 -3 1 -1 3 0 0 -1 1 -1 0 1 0 -1 -1 orthogonal lifted from C2×S4 ρ13 4 4 0 -2 -2 1 4 0 0 0 -2 -2 1 0 0 -2 0 0 0 0 0 orthogonal lifted from S32 ρ14 4 -4 0 4 -2 -2 0 0 0 0 -4 2 2 0 0 0 0 0 0 0 0 symplectic lifted from C4.S4, Schur index 2 ρ15 4 -4 0 4 1 1 0 0 0 0 -4 -1 -1 0 0 0 -√3 0 √3 0 0 symplectic lifted from C4.S4, Schur index 2 ρ16 4 -4 0 4 1 1 0 0 0 0 -4 -1 -1 0 0 0 √3 0 -√3 0 0 symplectic lifted from C4.S4, Schur index 2 ρ17 4 -4 0 -2 -2 1 0 0 0 0 2 2 -1 0 0 0 0 0 0 √-6 -√-6 complex faithful ρ18 4 -4 0 -2 -2 1 0 0 0 0 2 2 -1 0 0 0 0 0 0 -√-6 √-6 complex faithful ρ19 6 6 0 -3 0 0 -2 0 -2 0 -3 0 0 2 0 1 0 1 0 -1 -1 orthogonal lifted from S3×S4 ρ20 6 6 0 -3 0 0 -2 0 2 0 -3 0 0 -2 0 1 0 -1 0 1 1 orthogonal lifted from S3×S4 ρ21 8 -8 0 -4 2 -1 0 0 0 0 4 -2 1 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of CSU2(𝔽3)⋊S3
On 96 points
Generators in S96
(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)(73 74 75 76)(77 78 79 80)(81 82 83 84)(85 86 87 88)(89 90 91 92)(93 94 95 96)
(1 11 3 9)(2 10 4 12)(5 95 7 93)(6 94 8 96)(13 18 15 20)(14 17 16 19)(21 26 23 28)(22 25 24 27)(29 35 31 33)(30 34 32 36)(37 43 39 41)(38 42 40 44)(45 51 47 49)(46 50 48 52)(53 58 55 60)(54 57 56 59)(61 66 63 68)(62 65 64 67)(69 74 71 76)(70 73 72 75)(77 83 79 81)(78 82 80 84)(85 91 87 89)(86 90 88 92)
(1 16 24)(2 19 26)(3 14 22)(4 17 28)(5 90 78)(6 88 81)(7 92 80)(8 86 83)(9 20 23)(10 13 27)(11 18 21)(12 15 25)(29 45 37)(30 51 42)(31 47 39)(32 49 44)(33 52 40)(34 46 43)(35 50 38)(36 48 41)(53 67 74)(54 62 70)(55 65 76)(56 64 72)(57 68 71)(58 61 75)(59 66 69)(60 63 73)(77 93 85)(79 95 87)(82 94 91)(84 96 89)
(1 29 3 31)(2 33 4 35)(5 71 7 69)(6 74 8 76)(9 30 11 32)(10 36 12 34)(13 41 15 43)(14 39 16 37)(17 38 19 40)(18 44 20 42)(21 49 23 51)(22 47 24 45)(25 46 27 48)(26 52 28 50)(53 83 55 81)(54 77 56 79)(57 80 59 78)(58 82 60 84)(61 91 63 89)(62 85 64 87)(65 88 67 86)(66 90 68 92)(70 93 72 95)(73 96 75 94)
(1 16 24)(2 13 21)(3 14 22)(4 15 23)(5 91 83)(6 92 84)(7 89 81)(8 90 82)(9 17 25)(10 18 26)(11 19 27)(12 20 28)(29 37 45)(30 38 46)(31 39 47)(32 40 48)(33 41 49)(34 42 50)(35 43 51)(36 44 52)(53 69 61)(54 70 62)(55 71 63)(56 72 64)(57 73 65)(58 74 66)(59 75 67)(60 76 68)(77 93 85)(78 94 86)(79 95 87)(80 96 88)
(1 58)(2 57)(3 60)(4 59)(5 48)(6 47)(7 46)(8 45)(9 55)(10 54)(11 53)(12 56)(13 65)(14 68)(15 67)(16 66)(17 63)(18 62)(19 61)(20 64)(21 73)(22 76)(23 75)(24 74)(25 71)(26 70)(27 69)(28 72)(29 82)(30 81)(31 84)(32 83)(33 80)(34 79)(35 78)(36 77)(37 90)(38 89)(39 92)(40 91)(41 88)(42 87)(43 86)(44 85)(49 96)(50 95)(51 94)(52 93)

G:=sub<Sym(96)| (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)(73,74,75,76)(77,78,79,80)(81,82,83,84)(85,86,87,88)(89,90,91,92)(93,94,95,96), (1,11,3,9)(2,10,4,12)(5,95,7,93)(6,94,8,96)(13,18,15,20)(14,17,16,19)(21,26,23,28)(22,25,24,27)(29,35,31,33)(30,34,32,36)(37,43,39,41)(38,42,40,44)(45,51,47,49)(46,50,48,52)(53,58,55,60)(54,57,56,59)(61,66,63,68)(62,65,64,67)(69,74,71,76)(70,73,72,75)(77,83,79,81)(78,82,80,84)(85,91,87,89)(86,90,88,92), (1,16,24)(2,19,26)(3,14,22)(4,17,28)(5,90,78)(6,88,81)(7,92,80)(8,86,83)(9,20,23)(10,13,27)(11,18,21)(12,15,25)(29,45,37)(30,51,42)(31,47,39)(32,49,44)(33,52,40)(34,46,43)(35,50,38)(36,48,41)(53,67,74)(54,62,70)(55,65,76)(56,64,72)(57,68,71)(58,61,75)(59,66,69)(60,63,73)(77,93,85)(79,95,87)(82,94,91)(84,96,89), (1,29,3,31)(2,33,4,35)(5,71,7,69)(6,74,8,76)(9,30,11,32)(10,36,12,34)(13,41,15,43)(14,39,16,37)(17,38,19,40)(18,44,20,42)(21,49,23,51)(22,47,24,45)(25,46,27,48)(26,52,28,50)(53,83,55,81)(54,77,56,79)(57,80,59,78)(58,82,60,84)(61,91,63,89)(62,85,64,87)(65,88,67,86)(66,90,68,92)(70,93,72,95)(73,96,75,94), (1,16,24)(2,13,21)(3,14,22)(4,15,23)(5,91,83)(6,92,84)(7,89,81)(8,90,82)(9,17,25)(10,18,26)(11,19,27)(12,20,28)(29,37,45)(30,38,46)(31,39,47)(32,40,48)(33,41,49)(34,42,50)(35,43,51)(36,44,52)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(57,73,65)(58,74,66)(59,75,67)(60,76,68)(77,93,85)(78,94,86)(79,95,87)(80,96,88), (1,58)(2,57)(3,60)(4,59)(5,48)(6,47)(7,46)(8,45)(9,55)(10,54)(11,53)(12,56)(13,65)(14,68)(15,67)(16,66)(17,63)(18,62)(19,61)(20,64)(21,73)(22,76)(23,75)(24,74)(25,71)(26,70)(27,69)(28,72)(29,82)(30,81)(31,84)(32,83)(33,80)(34,79)(35,78)(36,77)(37,90)(38,89)(39,92)(40,91)(41,88)(42,87)(43,86)(44,85)(49,96)(50,95)(51,94)(52,93)>;

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)(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)(73,74,75,76)(77,78,79,80)(81,82,83,84)(85,86,87,88)(89,90,91,92)(93,94,95,96), (1,11,3,9)(2,10,4,12)(5,95,7,93)(6,94,8,96)(13,18,15,20)(14,17,16,19)(21,26,23,28)(22,25,24,27)(29,35,31,33)(30,34,32,36)(37,43,39,41)(38,42,40,44)(45,51,47,49)(46,50,48,52)(53,58,55,60)(54,57,56,59)(61,66,63,68)(62,65,64,67)(69,74,71,76)(70,73,72,75)(77,83,79,81)(78,82,80,84)(85,91,87,89)(86,90,88,92), (1,16,24)(2,19,26)(3,14,22)(4,17,28)(5,90,78)(6,88,81)(7,92,80)(8,86,83)(9,20,23)(10,13,27)(11,18,21)(12,15,25)(29,45,37)(30,51,42)(31,47,39)(32,49,44)(33,52,40)(34,46,43)(35,50,38)(36,48,41)(53,67,74)(54,62,70)(55,65,76)(56,64,72)(57,68,71)(58,61,75)(59,66,69)(60,63,73)(77,93,85)(79,95,87)(82,94,91)(84,96,89), (1,29,3,31)(2,33,4,35)(5,71,7,69)(6,74,8,76)(9,30,11,32)(10,36,12,34)(13,41,15,43)(14,39,16,37)(17,38,19,40)(18,44,20,42)(21,49,23,51)(22,47,24,45)(25,46,27,48)(26,52,28,50)(53,83,55,81)(54,77,56,79)(57,80,59,78)(58,82,60,84)(61,91,63,89)(62,85,64,87)(65,88,67,86)(66,90,68,92)(70,93,72,95)(73,96,75,94), (1,16,24)(2,13,21)(3,14,22)(4,15,23)(5,91,83)(6,92,84)(7,89,81)(8,90,82)(9,17,25)(10,18,26)(11,19,27)(12,20,28)(29,37,45)(30,38,46)(31,39,47)(32,40,48)(33,41,49)(34,42,50)(35,43,51)(36,44,52)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(57,73,65)(58,74,66)(59,75,67)(60,76,68)(77,93,85)(78,94,86)(79,95,87)(80,96,88), (1,58)(2,57)(3,60)(4,59)(5,48)(6,47)(7,46)(8,45)(9,55)(10,54)(11,53)(12,56)(13,65)(14,68)(15,67)(16,66)(17,63)(18,62)(19,61)(20,64)(21,73)(22,76)(23,75)(24,74)(25,71)(26,70)(27,69)(28,72)(29,82)(30,81)(31,84)(32,83)(33,80)(34,79)(35,78)(36,77)(37,90)(38,89)(39,92)(40,91)(41,88)(42,87)(43,86)(44,85)(49,96)(50,95)(51,94)(52,93) );

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),(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),(73,74,75,76),(77,78,79,80),(81,82,83,84),(85,86,87,88),(89,90,91,92),(93,94,95,96)], [(1,11,3,9),(2,10,4,12),(5,95,7,93),(6,94,8,96),(13,18,15,20),(14,17,16,19),(21,26,23,28),(22,25,24,27),(29,35,31,33),(30,34,32,36),(37,43,39,41),(38,42,40,44),(45,51,47,49),(46,50,48,52),(53,58,55,60),(54,57,56,59),(61,66,63,68),(62,65,64,67),(69,74,71,76),(70,73,72,75),(77,83,79,81),(78,82,80,84),(85,91,87,89),(86,90,88,92)], [(1,16,24),(2,19,26),(3,14,22),(4,17,28),(5,90,78),(6,88,81),(7,92,80),(8,86,83),(9,20,23),(10,13,27),(11,18,21),(12,15,25),(29,45,37),(30,51,42),(31,47,39),(32,49,44),(33,52,40),(34,46,43),(35,50,38),(36,48,41),(53,67,74),(54,62,70),(55,65,76),(56,64,72),(57,68,71),(58,61,75),(59,66,69),(60,63,73),(77,93,85),(79,95,87),(82,94,91),(84,96,89)], [(1,29,3,31),(2,33,4,35),(5,71,7,69),(6,74,8,76),(9,30,11,32),(10,36,12,34),(13,41,15,43),(14,39,16,37),(17,38,19,40),(18,44,20,42),(21,49,23,51),(22,47,24,45),(25,46,27,48),(26,52,28,50),(53,83,55,81),(54,77,56,79),(57,80,59,78),(58,82,60,84),(61,91,63,89),(62,85,64,87),(65,88,67,86),(66,90,68,92),(70,93,72,95),(73,96,75,94)], [(1,16,24),(2,13,21),(3,14,22),(4,15,23),(5,91,83),(6,92,84),(7,89,81),(8,90,82),(9,17,25),(10,18,26),(11,19,27),(12,20,28),(29,37,45),(30,38,46),(31,39,47),(32,40,48),(33,41,49),(34,42,50),(35,43,51),(36,44,52),(53,69,61),(54,70,62),(55,71,63),(56,72,64),(57,73,65),(58,74,66),(59,75,67),(60,76,68),(77,93,85),(78,94,86),(79,95,87),(80,96,88)], [(1,58),(2,57),(3,60),(4,59),(5,48),(6,47),(7,46),(8,45),(9,55),(10,54),(11,53),(12,56),(13,65),(14,68),(15,67),(16,66),(17,63),(18,62),(19,61),(20,64),(21,73),(22,76),(23,75),(24,74),(25,71),(26,70),(27,69),(28,72),(29,82),(30,81),(31,84),(32,83),(33,80),(34,79),(35,78),(36,77),(37,90),(38,89),(39,92),(40,91),(41,88),(42,87),(43,86),(44,85),(49,96),(50,95),(51,94),(52,93)]])

Matrix representation of CSU2(𝔽3)⋊S3 in GL4(𝔽5) generated by

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

CSU2(𝔽3)⋊S3 in GAP, Magma, Sage, TeX

{\rm CSU}_2({\mathbb F}_3)\rtimes S_3
% in TeX

G:=Group("CSU(2,3):S3");
// GroupNames label

G:=SmallGroup(288,844);
// by ID

G=gap.SmallGroup(288,844);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1008,2045,1016,93,675,1271,1908,172,768,1153,285,124]);
// Polycyclic

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

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