Aliases: Q8⋊C9, C6.1A4, C3.SL2(𝔽3), (C3×Q8).C3, C2.(C3.A4), SmallGroup(72,3)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — Q8⋊C9 |
Generators and relations for Q8⋊C9
G = < a,b,c | a4=c9=1, b2=a2, bab-1=a-1, cac-1=b, cbc-1=ab >
Character table of Q8⋊C9
| class | 1 | 2 | 3A | 3B | 4 | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 18A | 18B | 18C | 18D | 18E | 18F | |
| size | 1 | 1 | 1 | 1 | 6 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ4 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ97 | ζ92 | ζ98 | ζ95 | ζ94 | ζ9 | ζ3 | ζ32 | ζ97 | ζ94 | ζ92 | ζ9 | ζ95 | ζ98 | linear of order 9 |
| ρ5 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ92 | ζ97 | ζ9 | ζ94 | ζ95 | ζ98 | ζ32 | ζ3 | ζ92 | ζ95 | ζ97 | ζ98 | ζ94 | ζ9 | linear of order 9 |
| ρ6 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ95 | ζ94 | ζ97 | ζ9 | ζ98 | ζ92 | ζ32 | ζ3 | ζ95 | ζ98 | ζ94 | ζ92 | ζ9 | ζ97 | linear of order 9 |
| ρ7 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ9 | ζ98 | ζ95 | ζ92 | ζ97 | ζ94 | ζ3 | ζ32 | ζ9 | ζ97 | ζ98 | ζ94 | ζ92 | ζ95 | linear of order 9 |
| ρ8 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ98 | ζ9 | ζ94 | ζ97 | ζ92 | ζ95 | ζ32 | ζ3 | ζ98 | ζ92 | ζ9 | ζ95 | ζ97 | ζ94 | linear of order 9 |
| ρ9 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ94 | ζ95 | ζ92 | ζ98 | ζ9 | ζ97 | ζ3 | ζ32 | ζ94 | ζ9 | ζ95 | ζ97 | ζ98 | ζ92 | linear of order 9 |
| ρ10 | 2 | -2 | 2 | 2 | 0 | -2 | -2 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ11 | 2 | -2 | 2 | 2 | 0 | -2 | -2 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | 0 | 0 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | complex lifted from SL2(𝔽3) |
| ρ12 | 2 | -2 | 2 | 2 | 0 | -2 | -2 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | 0 | 0 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | complex lifted from SL2(𝔽3) |
| ρ13 | 2 | -2 | -1+√-3 | -1-√-3 | 0 | 1+√-3 | 1-√-3 | -ζ92 | -ζ97 | -ζ9 | -ζ94 | -ζ95 | -ζ98 | 0 | 0 | ζ92 | ζ95 | ζ97 | ζ98 | ζ94 | ζ9 | complex faithful |
| ρ14 | 2 | -2 | -1-√-3 | -1+√-3 | 0 | 1-√-3 | 1+√-3 | -ζ94 | -ζ95 | -ζ92 | -ζ98 | -ζ9 | -ζ97 | 0 | 0 | ζ94 | ζ9 | ζ95 | ζ97 | ζ98 | ζ92 | complex faithful |
| ρ15 | 2 | -2 | -1+√-3 | -1-√-3 | 0 | 1+√-3 | 1-√-3 | -ζ95 | -ζ94 | -ζ97 | -ζ9 | -ζ98 | -ζ92 | 0 | 0 | ζ95 | ζ98 | ζ94 | ζ92 | ζ9 | ζ97 | complex faithful |
| ρ16 | 2 | -2 | -1+√-3 | -1-√-3 | 0 | 1+√-3 | 1-√-3 | -ζ98 | -ζ9 | -ζ94 | -ζ97 | -ζ92 | -ζ95 | 0 | 0 | ζ98 | ζ92 | ζ9 | ζ95 | ζ97 | ζ94 | complex faithful |
| ρ17 | 2 | -2 | -1-√-3 | -1+√-3 | 0 | 1-√-3 | 1+√-3 | -ζ9 | -ζ98 | -ζ95 | -ζ92 | -ζ97 | -ζ94 | 0 | 0 | ζ9 | ζ97 | ζ98 | ζ94 | ζ92 | ζ95 | complex faithful |
| ρ18 | 2 | -2 | -1-√-3 | -1+√-3 | 0 | 1-√-3 | 1+√-3 | -ζ97 | -ζ92 | -ζ98 | -ζ95 | -ζ94 | -ζ9 | 0 | 0 | ζ97 | ζ94 | ζ92 | ζ9 | ζ95 | ζ98 | complex faithful |
| ρ19 | 3 | 3 | 3 | 3 | -1 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ20 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3.A4 |
| ρ21 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3.A4 |
►Regular action on 72 points
Generators in S72
(1 50 22 28)(2 56 23 44)(3 15 24 66)(4 53 25 31)(5 59 26 38)(6 18 27 69)(7 47 19 34)(8 62 20 41)(9 12 21 72)(10 61 70 40)(11 35 71 48)(13 55 64 43)(14 29 65 51)(16 58 67 37)(17 32 68 54)(30 57 52 45)(33 60 46 39)(36 63 49 42)
(1 55 22 43)(2 14 23 65)(3 52 24 30)(4 58 25 37)(5 17 26 68)(6 46 27 33)(7 61 19 40)(8 11 20 71)(9 49 21 36)(10 34 70 47)(12 63 72 42)(13 28 64 50)(15 57 66 45)(16 31 67 53)(18 60 69 39)(29 56 51 44)(32 59 54 38)(35 62 48 41)
(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)| (1,50,22,28)(2,56,23,44)(3,15,24,66)(4,53,25,31)(5,59,26,38)(6,18,27,69)(7,47,19,34)(8,62,20,41)(9,12,21,72)(10,61,70,40)(11,35,71,48)(13,55,64,43)(14,29,65,51)(16,58,67,37)(17,32,68,54)(30,57,52,45)(33,60,46,39)(36,63,49,42), (1,55,22,43)(2,14,23,65)(3,52,24,30)(4,58,25,37)(5,17,26,68)(6,46,27,33)(7,61,19,40)(8,11,20,71)(9,49,21,36)(10,34,70,47)(12,63,72,42)(13,28,64,50)(15,57,66,45)(16,31,67,53)(18,60,69,39)(29,56,51,44)(32,59,54,38)(35,62,48,41), (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( (1,50,22,28)(2,56,23,44)(3,15,24,66)(4,53,25,31)(5,59,26,38)(6,18,27,69)(7,47,19,34)(8,62,20,41)(9,12,21,72)(10,61,70,40)(11,35,71,48)(13,55,64,43)(14,29,65,51)(16,58,67,37)(17,32,68,54)(30,57,52,45)(33,60,46,39)(36,63,49,42), (1,55,22,43)(2,14,23,65)(3,52,24,30)(4,58,25,37)(5,17,26,68)(6,46,27,33)(7,61,19,40)(8,11,20,71)(9,49,21,36)(10,34,70,47)(12,63,72,42)(13,28,64,50)(15,57,66,45)(16,31,67,53)(18,60,69,39)(29,56,51,44)(32,59,54,38)(35,62,48,41), (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([[(1,50,22,28),(2,56,23,44),(3,15,24,66),(4,53,25,31),(5,59,26,38),(6,18,27,69),(7,47,19,34),(8,62,20,41),(9,12,21,72),(10,61,70,40),(11,35,71,48),(13,55,64,43),(14,29,65,51),(16,58,67,37),(17,32,68,54),(30,57,52,45),(33,60,46,39),(36,63,49,42)], [(1,55,22,43),(2,14,23,65),(3,52,24,30),(4,58,25,37),(5,17,26,68),(6,46,27,33),(7,61,19,40),(8,11,20,71),(9,49,21,36),(10,34,70,47),(12,63,72,42),(13,28,64,50),(15,57,66,45),(16,31,67,53),(18,60,69,39),(29,56,51,44),(32,59,54,38),(35,62,48,41)], [(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)]])
Q8⋊C9 is a maximal subgroup of
Q8.D9 Q8⋊D9 Q8.C18 C9×SL2(𝔽3) C18.A4 Q8⋊3- 1+2 C22⋊(Q8⋊C9) 2+ 1+4⋊2C9
Q8⋊C9 is a maximal quotient of
Q8⋊C27 C2.(C42⋊C9) C22⋊(Q8⋊C9)
Matrix representation of Q8⋊C9 ►in GL2(𝔽19) generated by
| 8 | 5 |
| 6 | 11 |
| 7 | 16 |
| 4 | 12 |
| 17 | 9 |
| 0 | 16 |
G:=sub<GL(2,GF(19))| [8,6,5,11],[7,4,16,12],[17,0,9,16] >;
Q8⋊C9 in GAP, Magma, Sage, TeX
Q_8\rtimes C_9
% in TeX
G:=Group("Q8:C9"); // GroupNames label
G:=SmallGroup(72,3);
// by ID
G=gap.SmallGroup(72,3);
# by ID
G:=PCGroup([5,-3,-3,-2,2,-2,15,272,72,543,133,58]);
// Polycyclic
G:=Group<a,b,c|a^4=c^9=1,b^2=a^2,b*a*b^-1=a^-1,c*a*c^-1=b,c*b*c^-1=a*b>;
// generators/relations
Export
Subgroup lattice of Q8⋊C9 in TeX
Character table of Q8⋊C9 in TeX