metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊2D9, C9⋊3SD16, C4.4D18, C12.4D6, D36.2C2, C18.10D4, C36.4C22, C9⋊C8⋊3C2, (Q8×C9)⋊1C2, (C3×Q8).6S3, C2.7(C9⋊D4), C3.(Q8⋊2S3), C6.17(C3⋊D4), SmallGroup(144,18)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊2D9
G = < a,b,c,d | a4=c9=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >
Character table of Q8⋊2D9
| class | 1 | 2A | 2B | 3 | 4A | 4B | 6 | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 12C | 18A | 18B | 18C | 36A | 36B | 36C | 36D | 36E | 36F | 36G | 36H | 36I | |
| size | 1 | 1 | 36 | 2 | 2 | 4 | 2 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | orthogonal lifted from D4 |
| ρ7 | 2 | 2 | 0 | 2 | 2 | -2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -2 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | 0 | -1 | 2 | -2 | -1 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | 1 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D18 |
| ρ9 | 2 | 2 | 0 | -1 | 2 | 2 | -1 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | -1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ10 | 2 | 2 | 0 | -1 | 2 | -2 | -1 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | 1 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D18 |
| ρ11 | 2 | 2 | 0 | -1 | 2 | -2 | -1 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | 1 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D18 |
| ρ12 | 2 | 2 | 0 | -1 | 2 | 2 | -1 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | -1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ13 | 2 | 2 | 0 | -1 | 2 | 2 | -1 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | -1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ14 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | √-2 | -√-2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ15 | 2 | 2 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | -1 | -1 | -1 | -2 | 0 | 0 | -1 | -1 | -1 | -√-3 | -√-3 | -√-3 | √-3 | √-3 | √-3 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ16 | 2 | 2 | 0 | -1 | -2 | 0 | -1 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | 1 | √-3 | -√-3 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98-ζ9 | -ζ97+ζ92 | ζ95-ζ94 | ζ97-ζ92 | -ζ98+ζ9 | -ζ95+ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | complex lifted from C9⋊D4 |
| ρ17 | 2 | 2 | 0 | -1 | -2 | 0 | -1 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | 1 | √-3 | -√-3 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95-ζ94 | ζ98-ζ9 | -ζ97+ζ92 | -ζ98+ζ9 | -ζ95+ζ94 | ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | complex lifted from C9⋊D4 |
| ρ18 | 2 | 2 | 0 | -1 | -2 | 0 | -1 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | 1 | -√-3 | √-3 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97-ζ92 | -ζ95+ζ94 | -ζ98+ζ9 | ζ95-ζ94 | -ζ97+ζ92 | ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | complex lifted from C9⋊D4 |
| ρ19 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | -√-2 | √-2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ20 | 2 | 2 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | -1 | -1 | -1 | -2 | 0 | 0 | -1 | -1 | -1 | √-3 | √-3 | √-3 | -√-3 | -√-3 | -√-3 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ21 | 2 | 2 | 0 | -1 | -2 | 0 | -1 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | 1 | -√-3 | √-3 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ95+ζ94 | -ζ98+ζ9 | ζ97-ζ92 | ζ98-ζ9 | ζ95-ζ94 | -ζ97+ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | complex lifted from C9⋊D4 |
| ρ22 | 2 | 2 | 0 | -1 | -2 | 0 | -1 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | 1 | √-3 | -√-3 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ97+ζ92 | ζ95-ζ94 | ζ98-ζ9 | -ζ95+ζ94 | ζ97-ζ92 | -ζ98+ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | complex lifted from C9⋊D4 |
| ρ23 | 2 | 2 | 0 | -1 | -2 | 0 | -1 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | 1 | -√-3 | √-3 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ98+ζ9 | ζ97-ζ92 | -ζ95+ζ94 | -ζ97+ζ92 | ζ98-ζ9 | ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | complex lifted from C9⋊D4 |
| ρ24 | 4 | -4 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊2S3 |
| ρ25 | 4 | -4 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 2ζ95+2ζ94 | 2ζ98+2ζ9 | 2ζ97+2ζ92 | 0 | 0 | 0 | -2ζ98-2ζ9 | -2ζ97-2ζ92 | -2ζ95-2ζ94 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ26 | 4 | -4 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 2ζ98+2ζ9 | 2ζ97+2ζ92 | 2ζ95+2ζ94 | 0 | 0 | 0 | -2ζ97-2ζ92 | -2ζ95-2ζ94 | -2ζ98-2ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ27 | 4 | -4 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 2ζ97+2ζ92 | 2ζ95+2ζ94 | 2ζ98+2ζ9 | 0 | 0 | 0 | -2ζ95-2ζ94 | -2ζ98-2ζ9 | -2ζ97-2ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 72 points
Generators in S72
(1 32 14 23)(2 33 15 24)(3 34 16 25)(4 35 17 26)(5 36 18 27)(6 28 10 19)(7 29 11 20)(8 30 12 21)(9 31 13 22)(37 55 46 64)(38 56 47 65)(39 57 48 66)(40 58 49 67)(41 59 50 68)(42 60 51 69)(43 61 52 70)(44 62 53 71)(45 63 54 72)
(1 50 14 41)(2 51 15 42)(3 52 16 43)(4 53 17 44)(5 54 18 45)(6 46 10 37)(7 47 11 38)(8 48 12 39)(9 49 13 40)(19 64 28 55)(20 65 29 56)(21 66 30 57)(22 67 31 58)(23 68 32 59)(24 69 33 60)(25 70 34 61)(26 71 35 62)(27 72 36 63)
(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 17)(11 16)(12 15)(13 14)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(27 36)(37 62)(38 61)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 63)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 72)
G:=sub<Sym(72)| (1,32,14,23)(2,33,15,24)(3,34,16,25)(4,35,17,26)(5,36,18,27)(6,28,10,19)(7,29,11,20)(8,30,12,21)(9,31,13,22)(37,55,46,64)(38,56,47,65)(39,57,48,66)(40,58,49,67)(41,59,50,68)(42,60,51,69)(43,61,52,70)(44,62,53,71)(45,63,54,72), (1,50,14,41)(2,51,15,42)(3,52,16,43)(4,53,17,44)(5,54,18,45)(6,46,10,37)(7,47,11,38)(8,48,12,39)(9,49,13,40)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63), (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)(10,17)(11,16)(12,15)(13,14)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(27,36)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,63)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,72)>;
G:=Group( (1,32,14,23)(2,33,15,24)(3,34,16,25)(4,35,17,26)(5,36,18,27)(6,28,10,19)(7,29,11,20)(8,30,12,21)(9,31,13,22)(37,55,46,64)(38,56,47,65)(39,57,48,66)(40,58,49,67)(41,59,50,68)(42,60,51,69)(43,61,52,70)(44,62,53,71)(45,63,54,72), (1,50,14,41)(2,51,15,42)(3,52,16,43)(4,53,17,44)(5,54,18,45)(6,46,10,37)(7,47,11,38)(8,48,12,39)(9,49,13,40)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63), (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)(10,17)(11,16)(12,15)(13,14)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(27,36)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,63)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,72) );
G=PermutationGroup([[(1,32,14,23),(2,33,15,24),(3,34,16,25),(4,35,17,26),(5,36,18,27),(6,28,10,19),(7,29,11,20),(8,30,12,21),(9,31,13,22),(37,55,46,64),(38,56,47,65),(39,57,48,66),(40,58,49,67),(41,59,50,68),(42,60,51,69),(43,61,52,70),(44,62,53,71),(45,63,54,72)], [(1,50,14,41),(2,51,15,42),(3,52,16,43),(4,53,17,44),(5,54,18,45),(6,46,10,37),(7,47,11,38),(8,48,12,39),(9,49,13,40),(19,64,28,55),(20,65,29,56),(21,66,30,57),(22,67,31,58),(23,68,32,59),(24,69,33,60),(25,70,34,61),(26,71,35,62),(27,72,36,63)], [(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),(10,17),(11,16),(12,15),(13,14),(19,35),(20,34),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(27,36),(37,62),(38,61),(39,60),(40,59),(41,58),(42,57),(43,56),(44,55),(45,63),(46,71),(47,70),(48,69),(49,68),(50,67),(51,66),(52,65),(53,64),(54,72)]])
Q8⋊2D9 is a maximal subgroup of
SD16×D9 D72⋊C2 Q16⋊D9 D72⋊5C2 C36.C23 D4⋊D18 D4.9D18 Q8⋊2D27 Q8⋊D27 Dic6⋊D9 C18.D12 D36.C6 C36.20D6 C18.6S4
Q8⋊2D9 is a maximal quotient of
C4.Dic18 C18.D8 Q8⋊2Dic9 Q8⋊2D27 Dic6⋊D9 C18.D12 C36.20D6
Matrix representation of Q8⋊2D9 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 71 |
| 0 | 0 | 1 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 61 | 12 |
| 0 | 0 | 67 | 12 |
| 70 | 45 | 0 | 0 |
| 28 | 42 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 28 | 42 | 0 | 0 |
| 70 | 45 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 72 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,71,72],[1,0,0,0,0,1,0,0,0,0,61,67,0,0,12,12],[70,28,0,0,45,42,0,0,0,0,1,0,0,0,0,1],[28,70,0,0,42,45,0,0,0,0,1,1,0,0,0,72] >;
Q8⋊2D9 in GAP, Magma, Sage, TeX
Q_8\rtimes_2D_9
% in TeX
G:=Group("Q8:2D9"); // GroupNames label
G:=SmallGroup(144,18);
// by ID
G=gap.SmallGroup(144,18);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,55,218,116,50,2404,208,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^9=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of Q8⋊2D9 in TeX
Character table of Q8⋊2D9 in TeX