Aliases: Q8⋊D9, C6.2S4, C3.GL2(𝔽3), Q8⋊C9⋊C2, (C3×Q8).2S3, C2.3(C3.S4), SmallGroup(144,32)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8⋊C9 — Q8⋊D9 |
Generators and relations for Q8⋊D9
G = < a,b,c,d | a4=c9=d2=1, b2=a2, bab-1=a-1, cac-1=b, dad=a-1b, cbc-1=ab, dbd=a2b, dcd=c-1 >
Character table of Q8⋊D9
| class | 1 | 2A | 2B | 3 | 4 | 6 | 8A | 8B | 9A | 9B | 9C | 12 | 18A | 18B | 18C | |
| size | 1 | 1 | 36 | 2 | 6 | 2 | 18 | 18 | 8 | 8 | 8 | 12 | 8 | 8 | 8 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ4 | 2 | 2 | 0 | -1 | 2 | -1 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ5 | 2 | 2 | 0 | -1 | 2 | -1 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ6 | 2 | 2 | 0 | -1 | 2 | -1 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ7 | 2 | -2 | 0 | 2 | 0 | -2 | √-2 | -√-2 | -1 | -1 | -1 | 0 | 1 | 1 | 1 | complex lifted from GL2(𝔽3) |
| ρ8 | 2 | -2 | 0 | 2 | 0 | -2 | -√-2 | √-2 | -1 | -1 | -1 | 0 | 1 | 1 | 1 | complex lifted from GL2(𝔽3) |
| ρ9 | 3 | 3 | 1 | 3 | -1 | 3 | -1 | -1 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ10 | 3 | 3 | -1 | 3 | -1 | 3 | 1 | 1 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ11 | 4 | -4 | 0 | 4 | 0 | -4 | 0 | 0 | 1 | 1 | 1 | 0 | -1 | -1 | -1 | orthogonal lifted from GL2(𝔽3) |
| ρ12 | 4 | -4 | 0 | -2 | 0 | 2 | 0 | 0 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | 0 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | orthogonal faithful |
| ρ13 | 4 | -4 | 0 | -2 | 0 | 2 | 0 | 0 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | 0 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | orthogonal faithful |
| ρ14 | 4 | -4 | 0 | -2 | 0 | 2 | 0 | 0 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | 0 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | orthogonal faithful |
| ρ15 | 6 | 6 | 0 | -3 | -2 | -3 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | orthogonal lifted from C3.S4 |
►On 72 points
Generators in S72
(1 15 52 26)(2 29 53 68)(3 55 54 42)(4 18 46 20)(5 32 47 71)(6 58 48 45)(7 12 49 23)(8 35 50 65)(9 61 51 39)(10 57 21 44)(11 72 22 33)(13 60 24 38)(14 66 25 36)(16 63 27 41)(17 69 19 30)(28 40 67 62)(31 43 70 56)(34 37 64 59)
(1 28 52 67)(2 63 53 41)(3 17 54 19)(4 31 46 70)(5 57 47 44)(6 11 48 22)(7 34 49 64)(8 60 50 38)(9 14 51 25)(10 71 21 32)(12 59 23 37)(13 65 24 35)(15 62 26 40)(16 68 27 29)(18 56 20 43)(30 42 69 55)(33 45 72 58)(36 39 66 61)
(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)
(1 9)(2 8)(3 7)(4 6)(10 71)(11 70)(12 69)(13 68)(14 67)(15 66)(16 65)(17 64)(18 72)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 36)(27 35)(37 55)(38 63)(39 62)(40 61)(41 60)(42 59)(43 58)(44 57)(45 56)(46 48)(49 54)(50 53)(51 52)
G:=sub<Sym(72)| (1,15,52,26)(2,29,53,68)(3,55,54,42)(4,18,46,20)(5,32,47,71)(6,58,48,45)(7,12,49,23)(8,35,50,65)(9,61,51,39)(10,57,21,44)(11,72,22,33)(13,60,24,38)(14,66,25,36)(16,63,27,41)(17,69,19,30)(28,40,67,62)(31,43,70,56)(34,37,64,59), (1,28,52,67)(2,63,53,41)(3,17,54,19)(4,31,46,70)(5,57,47,44)(6,11,48,22)(7,34,49,64)(8,60,50,38)(9,14,51,25)(10,71,21,32)(12,59,23,37)(13,65,24,35)(15,62,26,40)(16,68,27,29)(18,56,20,43)(30,42,69,55)(33,45,72,58)(36,39,66,61), (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), (1,9)(2,8)(3,7)(4,6)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,72)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,36)(27,35)(37,55)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,48)(49,54)(50,53)(51,52)>;
G:=Group( (1,15,52,26)(2,29,53,68)(3,55,54,42)(4,18,46,20)(5,32,47,71)(6,58,48,45)(7,12,49,23)(8,35,50,65)(9,61,51,39)(10,57,21,44)(11,72,22,33)(13,60,24,38)(14,66,25,36)(16,63,27,41)(17,69,19,30)(28,40,67,62)(31,43,70,56)(34,37,64,59), (1,28,52,67)(2,63,53,41)(3,17,54,19)(4,31,46,70)(5,57,47,44)(6,11,48,22)(7,34,49,64)(8,60,50,38)(9,14,51,25)(10,71,21,32)(12,59,23,37)(13,65,24,35)(15,62,26,40)(16,68,27,29)(18,56,20,43)(30,42,69,55)(33,45,72,58)(36,39,66,61), (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), (1,9)(2,8)(3,7)(4,6)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,72)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,36)(27,35)(37,55)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,48)(49,54)(50,53)(51,52) );
G=PermutationGroup([[(1,15,52,26),(2,29,53,68),(3,55,54,42),(4,18,46,20),(5,32,47,71),(6,58,48,45),(7,12,49,23),(8,35,50,65),(9,61,51,39),(10,57,21,44),(11,72,22,33),(13,60,24,38),(14,66,25,36),(16,63,27,41),(17,69,19,30),(28,40,67,62),(31,43,70,56),(34,37,64,59)], [(1,28,52,67),(2,63,53,41),(3,17,54,19),(4,31,46,70),(5,57,47,44),(6,11,48,22),(7,34,49,64),(8,60,50,38),(9,14,51,25),(10,71,21,32),(12,59,23,37),(13,65,24,35),(15,62,26,40),(16,68,27,29),(18,56,20,43),(30,42,69,55),(33,45,72,58),(36,39,66,61)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,71),(11,70),(12,69),(13,68),(14,67),(15,66),(16,65),(17,64),(18,72),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29),(25,28),(26,36),(27,35),(37,55),(38,63),(39,62),(40,61),(41,60),(42,59),(43,58),(44,57),(45,56),(46,48),(49,54),(50,53),(51,52)]])
Q8⋊D9 is a maximal subgroup of
Q8.D18 C12.11S4 C12.4S4 C32.GL2(𝔽3) C18.6S4 C32.3GL2(𝔽3)
Q8⋊D9 is a maximal quotient of Q8⋊Dic9 Q8⋊D27 C32.3GL2(𝔽3)
Matrix representation of Q8⋊D9 ►in GL4(𝔽73) generated by
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 61 | 1 | 0 | 0 |
| 1 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 5 | 0 | 0 |
| 6 | 66 | 0 | 0 |
| 0 | 0 | 31 | 70 |
| 0 | 0 | 3 | 28 |
| 6 | 5 | 0 | 0 |
| 66 | 67 | 0 | 0 |
| 0 | 0 | 3 | 28 |
| 0 | 0 | 31 | 70 |
G:=sub<GL(4,GF(73))| [0,72,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[61,1,0,0,1,12,0,0,0,0,1,0,0,0,0,1],[6,6,0,0,5,66,0,0,0,0,31,3,0,0,70,28],[6,66,0,0,5,67,0,0,0,0,3,31,0,0,28,70] >;
Q8⋊D9 in GAP, Magma, Sage, TeX
Q_8\rtimes D_9
% in TeX
G:=Group("Q8:D9"); // GroupNames label
G:=SmallGroup(144,32);
// by ID
G=gap.SmallGroup(144,32);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,2,-2,121,79,218,867,1305,117,544,820,202,88]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^9=d^2=1,b^2=a^2,b*a*b^-1=a^-1,c*a*c^-1=b,d*a*d=a^-1*b,c*b*c^-1=a*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of Q8⋊D9 in TeX
Character table of Q8⋊D9 in TeX