direct product, non-abelian, soluble
Aliases: C6×CSU2(𝔽3), C2.5(C6×S4), C6.42(C2×S4), (C2×C6).19S4, Q8.1(S3×C6), (C6×Q8).8S3, C22.4(C3×S4), (C3×Q8).19D6, (C6×SL2(𝔽3)).5C2, SL2(𝔽3).1(C2×C6), (C2×SL2(𝔽3)).2C6, (C3×SL2(𝔽3)).13C22, (C2×Q8).2(C3×S3), SmallGroup(288,899)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3) — C3×CSU2(𝔽3) — C6×CSU2(𝔽3) |
| SL2(𝔽3) — C6×CSU2(𝔽3) |
Generators and relations for C6×CSU2(𝔽3)
G = < a,b,c,d,e | a6=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 >
Subgroups: 270 in 85 conjugacy classes, 24 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, Q8, Q8, C32, Dic3, C12, C2×C6, C2×C6, C2×C8, Q16, C2×Q8, C2×Q8, C3×C6, C24, SL2(𝔽3), SL2(𝔽3), C2×Dic3, C2×C12, C3×Q8, C3×Q8, C2×Q16, C3×Dic3, C62, C2×C24, C3×Q16, CSU2(𝔽3), C2×SL2(𝔽3), C2×SL2(𝔽3), C6×Q8, C6×Q8, C3×SL2(𝔽3), C6×Dic3, C6×Q16, C2×CSU2(𝔽3), C3×CSU2(𝔽3), C6×SL2(𝔽3), C6×CSU2(𝔽3)
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, S4, S3×C6, CSU2(𝔽3), C2×S4, C3×S4, C2×CSU2(𝔽3), C3×CSU2(𝔽3), C6×S4, C6×CSU2(𝔽3)
►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 76 85 83)(2 77 86 84)(3 78 87 79)(4 73 88 80)(5 74 89 81)(6 75 90 82)(7 17 58 38)(8 18 59 39)(9 13 60 40)(10 14 55 41)(11 15 56 42)(12 16 57 37)(19 49 62 44)(20 50 63 45)(21 51 64 46)(22 52 65 47)(23 53 66 48)(24 54 61 43)(25 67 92 35)(26 68 93 36)(27 69 94 31)(28 70 95 32)(29 71 96 33)(30 72 91 34)
(1 19 85 62)(2 20 86 63)(3 21 87 64)(4 22 88 65)(5 23 89 66)(6 24 90 61)(7 32 58 70)(8 33 59 71)(9 34 60 72)(10 35 55 67)(11 36 56 68)(12 31 57 69)(13 91 40 30)(14 92 41 25)(15 93 42 26)(16 94 37 27)(17 95 38 28)(18 96 39 29)(43 82 54 75)(44 83 49 76)(45 84 50 77)(46 79 51 78)(47 80 52 73)(48 81 53 74)
(1 5 3)(2 6 4)(7 72 15)(8 67 16)(9 68 17)(10 69 18)(11 70 13)(12 71 14)(19 48 78)(20 43 73)(21 44 74)(22 45 75)(23 46 76)(24 47 77)(25 27 29)(26 28 30)(31 39 55)(32 40 56)(33 41 57)(34 42 58)(35 37 59)(36 38 60)(49 81 64)(50 82 65)(51 83 66)(52 84 61)(53 79 62)(54 80 63)(85 89 87)(86 90 88)(91 93 95)(92 94 96)
(1 92 85 25)(2 93 86 26)(3 94 87 27)(4 95 88 28)(5 96 89 29)(6 91 90 30)(7 52 58 47)(8 53 59 48)(9 54 60 43)(10 49 55 44)(11 50 56 45)(12 51 57 46)(13 82 40 75)(14 83 41 76)(15 84 42 77)(16 79 37 78)(17 80 38 73)(18 81 39 74)(19 67 62 35)(20 68 63 36)(21 69 64 31)(22 70 65 32)(23 71 66 33)(24 72 61 34)
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), (1,76,85,83)(2,77,86,84)(3,78,87,79)(4,73,88,80)(5,74,89,81)(6,75,90,82)(7,17,58,38)(8,18,59,39)(9,13,60,40)(10,14,55,41)(11,15,56,42)(12,16,57,37)(19,49,62,44)(20,50,63,45)(21,51,64,46)(22,52,65,47)(23,53,66,48)(24,54,61,43)(25,67,92,35)(26,68,93,36)(27,69,94,31)(28,70,95,32)(29,71,96,33)(30,72,91,34), (1,19,85,62)(2,20,86,63)(3,21,87,64)(4,22,88,65)(5,23,89,66)(6,24,90,61)(7,32,58,70)(8,33,59,71)(9,34,60,72)(10,35,55,67)(11,36,56,68)(12,31,57,69)(13,91,40,30)(14,92,41,25)(15,93,42,26)(16,94,37,27)(17,95,38,28)(18,96,39,29)(43,82,54,75)(44,83,49,76)(45,84,50,77)(46,79,51,78)(47,80,52,73)(48,81,53,74), (1,5,3)(2,6,4)(7,72,15)(8,67,16)(9,68,17)(10,69,18)(11,70,13)(12,71,14)(19,48,78)(20,43,73)(21,44,74)(22,45,75)(23,46,76)(24,47,77)(25,27,29)(26,28,30)(31,39,55)(32,40,56)(33,41,57)(34,42,58)(35,37,59)(36,38,60)(49,81,64)(50,82,65)(51,83,66)(52,84,61)(53,79,62)(54,80,63)(85,89,87)(86,90,88)(91,93,95)(92,94,96), (1,92,85,25)(2,93,86,26)(3,94,87,27)(4,95,88,28)(5,96,89,29)(6,91,90,30)(7,52,58,47)(8,53,59,48)(9,54,60,43)(10,49,55,44)(11,50,56,45)(12,51,57,46)(13,82,40,75)(14,83,41,76)(15,84,42,77)(16,79,37,78)(17,80,38,73)(18,81,39,74)(19,67,62,35)(20,68,63,36)(21,69,64,31)(22,70,65,32)(23,71,66,33)(24,72,61,34)>;
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), (1,76,85,83)(2,77,86,84)(3,78,87,79)(4,73,88,80)(5,74,89,81)(6,75,90,82)(7,17,58,38)(8,18,59,39)(9,13,60,40)(10,14,55,41)(11,15,56,42)(12,16,57,37)(19,49,62,44)(20,50,63,45)(21,51,64,46)(22,52,65,47)(23,53,66,48)(24,54,61,43)(25,67,92,35)(26,68,93,36)(27,69,94,31)(28,70,95,32)(29,71,96,33)(30,72,91,34), (1,19,85,62)(2,20,86,63)(3,21,87,64)(4,22,88,65)(5,23,89,66)(6,24,90,61)(7,32,58,70)(8,33,59,71)(9,34,60,72)(10,35,55,67)(11,36,56,68)(12,31,57,69)(13,91,40,30)(14,92,41,25)(15,93,42,26)(16,94,37,27)(17,95,38,28)(18,96,39,29)(43,82,54,75)(44,83,49,76)(45,84,50,77)(46,79,51,78)(47,80,52,73)(48,81,53,74), (1,5,3)(2,6,4)(7,72,15)(8,67,16)(9,68,17)(10,69,18)(11,70,13)(12,71,14)(19,48,78)(20,43,73)(21,44,74)(22,45,75)(23,46,76)(24,47,77)(25,27,29)(26,28,30)(31,39,55)(32,40,56)(33,41,57)(34,42,58)(35,37,59)(36,38,60)(49,81,64)(50,82,65)(51,83,66)(52,84,61)(53,79,62)(54,80,63)(85,89,87)(86,90,88)(91,93,95)(92,94,96), (1,92,85,25)(2,93,86,26)(3,94,87,27)(4,95,88,28)(5,96,89,29)(6,91,90,30)(7,52,58,47)(8,53,59,48)(9,54,60,43)(10,49,55,44)(11,50,56,45)(12,51,57,46)(13,82,40,75)(14,83,41,76)(15,84,42,77)(16,79,37,78)(17,80,38,73)(18,81,39,74)(19,67,62,35)(20,68,63,36)(21,69,64,31)(22,70,65,32)(23,71,66,33)(24,72,61,34) );
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)], [(1,76,85,83),(2,77,86,84),(3,78,87,79),(4,73,88,80),(5,74,89,81),(6,75,90,82),(7,17,58,38),(8,18,59,39),(9,13,60,40),(10,14,55,41),(11,15,56,42),(12,16,57,37),(19,49,62,44),(20,50,63,45),(21,51,64,46),(22,52,65,47),(23,53,66,48),(24,54,61,43),(25,67,92,35),(26,68,93,36),(27,69,94,31),(28,70,95,32),(29,71,96,33),(30,72,91,34)], [(1,19,85,62),(2,20,86,63),(3,21,87,64),(4,22,88,65),(5,23,89,66),(6,24,90,61),(7,32,58,70),(8,33,59,71),(9,34,60,72),(10,35,55,67),(11,36,56,68),(12,31,57,69),(13,91,40,30),(14,92,41,25),(15,93,42,26),(16,94,37,27),(17,95,38,28),(18,96,39,29),(43,82,54,75),(44,83,49,76),(45,84,50,77),(46,79,51,78),(47,80,52,73),(48,81,53,74)], [(1,5,3),(2,6,4),(7,72,15),(8,67,16),(9,68,17),(10,69,18),(11,70,13),(12,71,14),(19,48,78),(20,43,73),(21,44,74),(22,45,75),(23,46,76),(24,47,77),(25,27,29),(26,28,30),(31,39,55),(32,40,56),(33,41,57),(34,42,58),(35,37,59),(36,38,60),(49,81,64),(50,82,65),(51,83,66),(52,84,61),(53,79,62),(54,80,63),(85,89,87),(86,90,88),(91,93,95),(92,94,96)], [(1,92,85,25),(2,93,86,26),(3,94,87,27),(4,95,88,28),(5,96,89,29),(6,91,90,30),(7,52,58,47),(8,53,59,48),(9,54,60,43),(10,49,55,44),(11,50,56,45),(12,51,57,46),(13,82,40,75),(14,83,41,76),(15,84,42,77),(16,79,37,78),(17,80,38,73),(18,81,39,74),(19,67,62,35),(20,68,63,36),(21,69,64,31),(22,70,65,32),(23,71,66,33),(24,72,61,34)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6O | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 6 | 6 | 12 | 12 | 1 | ··· | 1 | 8 | ··· | 8 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 6 | ··· | 6 |
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 | C2 | C3 | C6 | C6 | S3 | D6 | C3×S3 | S3×C6 | CSU2(𝔽3) | C3×CSU2(𝔽3) | S4 | C2×S4 | C3×S4 | C6×S4 | CSU2(𝔽3) | C3×CSU2(𝔽3) |
| kernel | C6×CSU2(𝔽3) | C3×CSU2(𝔽3) | C6×SL2(𝔽3) | C2×CSU2(𝔽3) | CSU2(𝔽3) | C2×SL2(𝔽3) | C6×Q8 | C3×Q8 | C2×Q8 | Q8 | C6 | C2 | C2×C6 | C6 | C22 | C2 | C6 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 2 | 4 | 8 | 2 | 2 | 4 | 4 | 2 | 4 |
Matrix representation of C6×CSU2(𝔽3) ►in GL3(𝔽73) generated by
| 9 | 0 | 0 |
| 0 | 65 | 0 |
| 0 | 0 | 65 |
| 1 | 0 | 0 |
| 0 | 9 | 65 |
| 0 | 65 | 64 |
| 1 | 0 | 0 |
| 0 | 0 | 72 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 1 | 64 |
| 72 | 0 | 0 |
| 0 | 2 | 39 |
| 0 | 71 | 71 |
G:=sub<GL(3,GF(73))| [9,0,0,0,65,0,0,0,65],[1,0,0,0,9,65,0,65,64],[1,0,0,0,0,1,0,72,0],[1,0,0,0,8,1,0,0,64],[72,0,0,0,2,71,0,39,71] >;
C6×CSU2(𝔽3) in GAP, Magma, Sage, TeX
C_6\times {\rm CSU}_2({\mathbb F}_3) % in TeX
G:=Group("C6xCSU(2,3)"); // GroupNames label
G:=SmallGroup(288,899);
// by ID
G=gap.SmallGroup(288,899);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1008,675,2524,655,172,1517,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=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