direct product, non-abelian, soluble
Aliases: C3×Q8.D6, GL2(𝔽3)⋊1C6, CSU2(𝔽3)⋊1C6, C2.7(C6×S4), (C2×C6).5S4, (C6×Q8)⋊4S3, C6.44(C2×S4), Q8.2(S3×C6), C22.2(C3×S4), (C3×Q8).20D6, (C2×SL2(𝔽3))⋊3C6, (C6×SL2(𝔽3))⋊7C2, (C3×GL2(𝔽3))⋊5C2, (C3×CSU2(𝔽3))⋊5C2, SL2(𝔽3).2(C2×C6), (C3×SL2(𝔽3)).14C22, (C2×Q8)⋊2(C3×S3), SmallGroup(288,901)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3) — C3×GL2(𝔽3) — C3×Q8.D6 |
| SL2(𝔽3) — C3×Q8.D6 |
Generators and relations for C3×Q8.D6
G = < a,b,c,d,e | a3=b4=d6=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, dbd-1=bc, dcd-1=b, ece-1=b-1c, ede-1=b2d-1 >
Subgroups: 310 in 85 conjugacy classes, 20 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, C32, Dic3, C12, D6, C2×C6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×S3, C3×C6, C24, SL2(𝔽3), SL2(𝔽3), C3⋊D4, C2×C12, C3×D4, C3×Q8, C3×Q8, C8.C22, C3×Dic3, S3×C6, C62, C3×M4(2), C3×SD16, C3×Q16, CSU2(𝔽3), GL2(𝔽3), C2×SL2(𝔽3), C2×SL2(𝔽3), C6×Q8, C3×C4○D4, C3×SL2(𝔽3), C3×C3⋊D4, C3×C8.C22, Q8.D6, C3×CSU2(𝔽3), C3×GL2(𝔽3), C6×SL2(𝔽3), C3×Q8.D6
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, S4, S3×C6, C2×S4, C3×S4, Q8.D6, C6×S4, C3×Q8.D6
►On 48 points
(1 8 6)(2 7 5)(3 12 10)(4 11 9)(13 21 33)(14 22 34)(15 23 35)(16 24 36)(17 19 31)(18 20 32)(25 45 38)(26 46 39)(27 47 40)(28 48 41)(29 43 42)(30 44 37)
(1 32 11 41)(2 35 12 38)(3 45 5 23)(4 48 6 20)(7 15 10 25)(8 18 9 28)(13 27 29 17)(14 16 30 26)(19 21 47 43)(22 24 44 46)(31 33 40 42)(34 36 37 39)
(1 36 11 39)(2 33 12 42)(3 43 5 21)(4 46 6 24)(7 13 10 29)(8 16 9 26)(14 28 30 18)(15 17 25 27)(19 45 47 23)(20 22 48 44)(31 38 40 35)(32 34 41 37)
(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)
(1 2 11 12)(3 6 5 4)(7 9 10 8)(13 14 29 30)(15 18 25 28)(16 27 26 17)(19 24 47 46)(20 45 48 23)(21 22 43 44)(31 36 40 39)(32 38 41 35)(33 34 42 37)
G:=sub<Sym(48)| (1,8,6)(2,7,5)(3,12,10)(4,11,9)(13,21,33)(14,22,34)(15,23,35)(16,24,36)(17,19,31)(18,20,32)(25,45,38)(26,46,39)(27,47,40)(28,48,41)(29,43,42)(30,44,37), (1,32,11,41)(2,35,12,38)(3,45,5,23)(4,48,6,20)(7,15,10,25)(8,18,9,28)(13,27,29,17)(14,16,30,26)(19,21,47,43)(22,24,44,46)(31,33,40,42)(34,36,37,39), (1,36,11,39)(2,33,12,42)(3,43,5,21)(4,46,6,24)(7,13,10,29)(8,16,9,26)(14,28,30,18)(15,17,25,27)(19,45,47,23)(20,22,48,44)(31,38,40,35)(32,34,41,37), (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), (1,2,11,12)(3,6,5,4)(7,9,10,8)(13,14,29,30)(15,18,25,28)(16,27,26,17)(19,24,47,46)(20,45,48,23)(21,22,43,44)(31,36,40,39)(32,38,41,35)(33,34,42,37)>;
G:=Group( (1,8,6)(2,7,5)(3,12,10)(4,11,9)(13,21,33)(14,22,34)(15,23,35)(16,24,36)(17,19,31)(18,20,32)(25,45,38)(26,46,39)(27,47,40)(28,48,41)(29,43,42)(30,44,37), (1,32,11,41)(2,35,12,38)(3,45,5,23)(4,48,6,20)(7,15,10,25)(8,18,9,28)(13,27,29,17)(14,16,30,26)(19,21,47,43)(22,24,44,46)(31,33,40,42)(34,36,37,39), (1,36,11,39)(2,33,12,42)(3,43,5,21)(4,46,6,24)(7,13,10,29)(8,16,9,26)(14,28,30,18)(15,17,25,27)(19,45,47,23)(20,22,48,44)(31,38,40,35)(32,34,41,37), (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), (1,2,11,12)(3,6,5,4)(7,9,10,8)(13,14,29,30)(15,18,25,28)(16,27,26,17)(19,24,47,46)(20,45,48,23)(21,22,43,44)(31,36,40,39)(32,38,41,35)(33,34,42,37) );
G=PermutationGroup([[(1,8,6),(2,7,5),(3,12,10),(4,11,9),(13,21,33),(14,22,34),(15,23,35),(16,24,36),(17,19,31),(18,20,32),(25,45,38),(26,46,39),(27,47,40),(28,48,41),(29,43,42),(30,44,37)], [(1,32,11,41),(2,35,12,38),(3,45,5,23),(4,48,6,20),(7,15,10,25),(8,18,9,28),(13,27,29,17),(14,16,30,26),(19,21,47,43),(22,24,44,46),(31,33,40,42),(34,36,37,39)], [(1,36,11,39),(2,33,12,42),(3,43,5,21),(4,46,6,24),(7,13,10,29),(8,16,9,26),(14,28,30,18),(15,17,25,27),(19,45,47,23),(20,22,48,44),(31,38,40,35),(32,34,41,37)], [(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)], [(1,2,11,12),(3,6,5,4),(7,9,10,8),(13,14,29,30),(15,18,25,28),(16,27,26,17),(19,24,47,46),(20,45,48,23),(21,22,43,44),(31,36,40,39),(32,38,41,35),(33,34,42,37)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | ··· | 6M | 6N | 6O | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 12 | 1 | 1 | 8 | 8 | 8 | 6 | 6 | 12 | 1 | 1 | 2 | 2 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D6 | C3×S3 | S3×C6 | S4 | C2×S4 | C3×S4 | C6×S4 | Q8.D6 | Q8.D6 | C3×Q8.D6 |
| kernel | C3×Q8.D6 | C3×CSU2(𝔽3) | C3×GL2(𝔽3) | C6×SL2(𝔽3) | Q8.D6 | CSU2(𝔽3) | GL2(𝔽3) | C2×SL2(𝔽3) | C6×Q8 | C3×Q8 | C2×Q8 | Q8 | C2×C6 | C6 | C22 | C2 | C3 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 6 |
Matrix representation of C3×Q8.D6 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 5 | 4 | 2 | 5 |
| 6 | 1 | 5 | 1 |
| 5 | 3 | 4 | 0 |
| 2 | 1 | 5 | 4 |
| 0 | 6 | 3 | 4 |
| 3 | 6 | 6 | 4 |
| 5 | 3 | 4 | 3 |
| 2 | 1 | 2 | 4 |
| 4 | 5 | 0 | 4 |
| 6 | 6 | 4 | 3 |
| 5 | 4 | 2 | 3 |
| 0 | 3 | 4 | 0 |
| 1 | 6 | 2 | 6 |
| 5 | 4 | 2 | 5 |
| 5 | 4 | 2 | 3 |
| 0 | 3 | 4 | 0 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[5,6,5,2,4,1,3,1,2,5,4,5,5,1,0,4],[0,3,5,2,6,6,3,1,3,6,4,2,4,4,3,4],[4,6,5,0,5,6,4,3,0,4,2,4,4,3,3,0],[1,5,5,0,6,4,4,3,2,2,2,4,6,5,3,0] >;
C3×Q8.D6 in GAP, Magma, Sage, TeX
C_3\times Q_8.D_6
% in TeX
G:=Group("C3xQ8.D6"); // GroupNames label
G:=SmallGroup(288,901);
// by ID
G=gap.SmallGroup(288,901);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,2045,675,2524,655,172,1517,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=d^6=1,c^2=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*b*e^-1=b^-1,d*b*d^-1=b*c,d*c*d^-1=b,e*c*e^-1=b^-1*c,e*d*e^-1=b^2*d^-1>;
// generators/relations