direct product, non-abelian, soluble
Aliases: C3×Q8⋊Dic3, C6.4GL2(𝔽3), SL2(𝔽3)⋊1C12, C6.4CSU2(𝔽3), Q8⋊(C3×Dic3), (C2×C6).18S4, (C6×Q8).7S3, C6.9(A4⋊C4), (C3×Q8)⋊2Dic3, C22.3(C3×S4), C2.(C3×GL2(𝔽3)), C2.(C3×CSU2(𝔽3)), (C3×SL2(𝔽3))⋊4C4, (C6×SL2(𝔽3)).4C2, (C2×SL2(𝔽3)).1C6, C2.2(C3×A4⋊C4), (C2×Q8).1(C3×S3), SmallGroup(288,399)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C6×SL2(𝔽3) — C3×Q8⋊Dic3 |
| SL2(𝔽3) — C3×Q8⋊Dic3 |
Generators and relations for C3×Q8⋊Dic3
G = < a,b,c,d,e | a3=b4=d6=1, c2=b2, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece-1=b-1, dbd-1=c, ebe-1=b2c, dcd-1=bc, ede-1=d-1 >
Subgroups: 218 in 63 conjugacy classes, 20 normal (all characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, Q8, Q8, C32, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C2×Q8, C3×C6, C24, SL2(𝔽3), SL2(𝔽3), C2×Dic3, C2×C12, C3×Q8, C3×Q8, Q8⋊C4, C3×Dic3, C62, C3×C4⋊C4, C2×C24, C2×SL2(𝔽3), C2×SL2(𝔽3), C6×Q8, C3×SL2(𝔽3), C6×Dic3, C3×Q8⋊C4, Q8⋊Dic3, C6×SL2(𝔽3), C3×Q8⋊Dic3
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, C3×S3, S4, C3×Dic3, CSU2(𝔽3), GL2(𝔽3), A4⋊C4, C3×S4, Q8⋊Dic3, C3×CSU2(𝔽3), C3×GL2(𝔽3), C3×A4⋊C4, C3×Q8⋊Dic3
►On 96 points
Generators in S96
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 35 25)(14 36 26)(15 31 27)(16 32 28)(17 33 29)(18 34 30)(19 43 73)(20 44 74)(21 45 75)(22 46 76)(23 47 77)(24 48 78)(37 85 96)(38 86 91)(39 87 92)(40 88 93)(41 89 94)(42 90 95)(49 69 83)(50 70 84)(51 71 79)(52 72 80)(53 67 81)(54 68 82)(55 59 57)(56 60 58)(61 63 65)(62 64 66)
(1 77 57 49)(2 20 58 72)(3 47 59 83)(4 74 60 52)(5 23 55 69)(6 44 56 80)(7 92 66 31)(8 90 61 18)(9 39 62 27)(10 95 63 34)(11 87 64 15)(12 42 65 30)(13 33 85 94)(14 40 86 28)(16 36 88 91)(17 37 89 25)(19 81 71 45)(21 73 67 51)(22 84 68 48)(24 76 70 54)(26 93 38 32)(29 96 41 35)(43 53 79 75)(46 50 82 78)
(1 19 57 71)(2 46 58 82)(3 73 59 51)(4 22 60 68)(5 43 55 79)(6 76 56 54)(7 89 66 17)(8 38 61 26)(9 94 62 33)(10 86 63 14)(11 41 64 29)(12 91 65 36)(13 39 85 27)(15 35 87 96)(16 42 88 30)(18 32 90 93)(20 78 72 50)(21 83 67 47)(23 75 69 53)(24 80 70 44)(25 92 37 31)(28 95 40 34)(45 49 81 77)(48 52 84 74)
(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 64 4 61)(2 63 5 66)(3 62 6 65)(7 58 10 55)(8 57 11 60)(9 56 12 59)(13 70 16 67)(14 69 17 72)(15 68 18 71)(19 87 22 90)(20 86 23 89)(21 85 24 88)(25 50 28 53)(26 49 29 52)(27 54 30 51)(31 82 34 79)(32 81 35 84)(33 80 36 83)(37 78 40 75)(38 77 41 74)(39 76 42 73)(43 92 46 95)(44 91 47 94)(45 96 48 93)
G:=sub<Sym(96)| (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,35,25)(14,36,26)(15,31,27)(16,32,28)(17,33,29)(18,34,30)(19,43,73)(20,44,74)(21,45,75)(22,46,76)(23,47,77)(24,48,78)(37,85,96)(38,86,91)(39,87,92)(40,88,93)(41,89,94)(42,90,95)(49,69,83)(50,70,84)(51,71,79)(52,72,80)(53,67,81)(54,68,82)(55,59,57)(56,60,58)(61,63,65)(62,64,66), (1,77,57,49)(2,20,58,72)(3,47,59,83)(4,74,60,52)(5,23,55,69)(6,44,56,80)(7,92,66,31)(8,90,61,18)(9,39,62,27)(10,95,63,34)(11,87,64,15)(12,42,65,30)(13,33,85,94)(14,40,86,28)(16,36,88,91)(17,37,89,25)(19,81,71,45)(21,73,67,51)(22,84,68,48)(24,76,70,54)(26,93,38,32)(29,96,41,35)(43,53,79,75)(46,50,82,78), (1,19,57,71)(2,46,58,82)(3,73,59,51)(4,22,60,68)(5,43,55,79)(6,76,56,54)(7,89,66,17)(8,38,61,26)(9,94,62,33)(10,86,63,14)(11,41,64,29)(12,91,65,36)(13,39,85,27)(15,35,87,96)(16,42,88,30)(18,32,90,93)(20,78,72,50)(21,83,67,47)(23,75,69,53)(24,80,70,44)(25,92,37,31)(28,95,40,34)(45,49,81,77)(48,52,84,74), (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), (1,64,4,61)(2,63,5,66)(3,62,6,65)(7,58,10,55)(8,57,11,60)(9,56,12,59)(13,70,16,67)(14,69,17,72)(15,68,18,71)(19,87,22,90)(20,86,23,89)(21,85,24,88)(25,50,28,53)(26,49,29,52)(27,54,30,51)(31,82,34,79)(32,81,35,84)(33,80,36,83)(37,78,40,75)(38,77,41,74)(39,76,42,73)(43,92,46,95)(44,91,47,94)(45,96,48,93)>;
G:=Group( (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,35,25)(14,36,26)(15,31,27)(16,32,28)(17,33,29)(18,34,30)(19,43,73)(20,44,74)(21,45,75)(22,46,76)(23,47,77)(24,48,78)(37,85,96)(38,86,91)(39,87,92)(40,88,93)(41,89,94)(42,90,95)(49,69,83)(50,70,84)(51,71,79)(52,72,80)(53,67,81)(54,68,82)(55,59,57)(56,60,58)(61,63,65)(62,64,66), (1,77,57,49)(2,20,58,72)(3,47,59,83)(4,74,60,52)(5,23,55,69)(6,44,56,80)(7,92,66,31)(8,90,61,18)(9,39,62,27)(10,95,63,34)(11,87,64,15)(12,42,65,30)(13,33,85,94)(14,40,86,28)(16,36,88,91)(17,37,89,25)(19,81,71,45)(21,73,67,51)(22,84,68,48)(24,76,70,54)(26,93,38,32)(29,96,41,35)(43,53,79,75)(46,50,82,78), (1,19,57,71)(2,46,58,82)(3,73,59,51)(4,22,60,68)(5,43,55,79)(6,76,56,54)(7,89,66,17)(8,38,61,26)(9,94,62,33)(10,86,63,14)(11,41,64,29)(12,91,65,36)(13,39,85,27)(15,35,87,96)(16,42,88,30)(18,32,90,93)(20,78,72,50)(21,83,67,47)(23,75,69,53)(24,80,70,44)(25,92,37,31)(28,95,40,34)(45,49,81,77)(48,52,84,74), (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), (1,64,4,61)(2,63,5,66)(3,62,6,65)(7,58,10,55)(8,57,11,60)(9,56,12,59)(13,70,16,67)(14,69,17,72)(15,68,18,71)(19,87,22,90)(20,86,23,89)(21,85,24,88)(25,50,28,53)(26,49,29,52)(27,54,30,51)(31,82,34,79)(32,81,35,84)(33,80,36,83)(37,78,40,75)(38,77,41,74)(39,76,42,73)(43,92,46,95)(44,91,47,94)(45,96,48,93) );
G=PermutationGroup([[(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,35,25),(14,36,26),(15,31,27),(16,32,28),(17,33,29),(18,34,30),(19,43,73),(20,44,74),(21,45,75),(22,46,76),(23,47,77),(24,48,78),(37,85,96),(38,86,91),(39,87,92),(40,88,93),(41,89,94),(42,90,95),(49,69,83),(50,70,84),(51,71,79),(52,72,80),(53,67,81),(54,68,82),(55,59,57),(56,60,58),(61,63,65),(62,64,66)], [(1,77,57,49),(2,20,58,72),(3,47,59,83),(4,74,60,52),(5,23,55,69),(6,44,56,80),(7,92,66,31),(8,90,61,18),(9,39,62,27),(10,95,63,34),(11,87,64,15),(12,42,65,30),(13,33,85,94),(14,40,86,28),(16,36,88,91),(17,37,89,25),(19,81,71,45),(21,73,67,51),(22,84,68,48),(24,76,70,54),(26,93,38,32),(29,96,41,35),(43,53,79,75),(46,50,82,78)], [(1,19,57,71),(2,46,58,82),(3,73,59,51),(4,22,60,68),(5,43,55,79),(6,76,56,54),(7,89,66,17),(8,38,61,26),(9,94,62,33),(10,86,63,14),(11,41,64,29),(12,91,65,36),(13,39,85,27),(15,35,87,96),(16,42,88,30),(18,32,90,93),(20,78,72,50),(21,83,67,47),(23,75,69,53),(24,80,70,44),(25,92,37,31),(28,95,40,34),(45,49,81,77),(48,52,84,74)], [(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)], [(1,64,4,61),(2,63,5,66),(3,62,6,65),(7,58,10,55),(8,57,11,60),(9,56,12,59),(13,70,16,67),(14,69,17,72),(15,68,18,71),(19,87,22,90),(20,86,23,89),(21,85,24,88),(25,50,28,53),(26,49,29,52),(27,54,30,51),(31,82,34,79),(32,81,35,84),(33,80,36,83),(37,78,40,75),(38,77,41,74),(39,76,42,73),(43,92,46,95),(44,91,47,94),(45,96,48,93)]])
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 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | - | + | - | + | ||||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | S3 | Dic3 | C3×S3 | C3×Dic3 | CSU2(𝔽3) | GL2(𝔽3) | C3×CSU2(𝔽3) | C3×GL2(𝔽3) | S4 | A4⋊C4 | C3×S4 | C3×A4⋊C4 | CSU2(𝔽3) | GL2(𝔽3) | C3×CSU2(𝔽3) | C3×GL2(𝔽3) |
| kernel | C3×Q8⋊Dic3 | C6×SL2(𝔽3) | Q8⋊Dic3 | C3×SL2(𝔽3) | C2×SL2(𝔽3) | SL2(𝔽3) | C6×Q8 | C3×Q8 | C2×Q8 | Q8 | C6 | C6 | C2 | C2 | C2×C6 | C6 | C22 | C2 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C3×Q8⋊Dic3 ►in GL3(𝔽73) generated by
| 64 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 1 | 0 | 0 |
| 0 | 65 | 9 |
| 0 | 9 | 8 |
| 1 | 0 | 0 |
| 0 | 0 | 72 |
| 0 | 1 | 0 |
| 72 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 72 | 64 |
| 46 | 0 | 0 |
| 0 | 48 | 13 |
| 0 | 25 | 25 |
G:=sub<GL(3,GF(73))| [64,0,0,0,8,0,0,0,8],[1,0,0,0,65,9,0,9,8],[1,0,0,0,0,1,0,72,0],[72,0,0,0,8,72,0,0,64],[46,0,0,0,48,25,0,13,25] >;
C3×Q8⋊Dic3 in GAP, Magma, Sage, TeX
C_3\times Q_8\rtimes {\rm Dic}_3 % in TeX
G:=Group("C3xQ8:Dic3"); // GroupNames label
G:=SmallGroup(288,399);
// by ID
G=gap.SmallGroup(288,399);
# by ID
G:=PCGroup([7,-2,-3,-2,-3,-2,2,-2,42,675,2524,655,172,1517,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=d^6=1,c^2=b^2,e^2=d^3,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=c,e*b*e^-1=b^2*c,d*c*d^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations