metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊3D9, D36⋊4C2, C12.8D6, C4.7D18, C36.7C22, C18.8C23, D18.3C22, Dic9.5C22, (C4×D9)⋊3C2, C9⋊3(C4○D4), (Q8×C9)⋊3C2, (C3×Q8).8S3, C3.(Q8⋊3S3), C2.9(C22×D9), C6.26(C22×S3), SmallGroup(144,44)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊3D9
G = < a,b,c,d | a4=c9=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >
Subgroups: 239 in 60 conjugacy classes, 29 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C2×C4, D4, Q8, C9, Dic3, C12, D6, C4○D4, D9, C18, C4×S3, D12, C3×Q8, Dic9, C36, D18, Q8⋊3S3, C4×D9, D36, Q8×C9, Q8⋊3D9
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C22×S3, D18, Q8⋊3S3, C22×D9, Q8⋊3D9
Character table of Q8⋊3D9
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6 | 9A | 9B | 9C | 12A | 12B | 12C | 18A | 18B | 18C | 36A | 36B | 36C | 36D | 36E | 36F | 36G | 36H | 36I | |
| size | 1 | 1 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 9 | 9 | 2 | 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 | 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 | -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 | 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 | -1 | -1 | 1 | linear of order 2 |
| ρ5 | 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 | 1 | 1 | 1 | linear of order 2 |
| ρ6 | 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 | -1 | -1 | -1 | linear of order 2 |
| ρ7 | 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 | 1 | 1 | -1 | linear of order 2 |
| ρ8 | 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 | -1 | -1 | 1 | linear of order 2 |
| ρ9 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | -2 | 2 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | -2 | -2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | 2 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | 2 | -2 | 0 | 0 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -1 | 1 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ14 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | 1 | 1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ95-ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ15 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | 2 | -2 | 0 | 0 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -1 | 1 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ16 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | 2 | -2 | 0 | 0 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -1 | 1 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ17 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -1 | -1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ18 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | -2 | -2 | 0 | 0 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | 1 | -1 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | ζ98+ζ9 | orthogonal lifted from D18 |
| ρ19 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | 1 | 1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ97-ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ20 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | 1 | 1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ98-ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ21 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -1 | -1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ22 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | -2 | -2 | 0 | 0 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | 1 | -1 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | ζ95+ζ94 | orthogonal lifted from D18 |
| ρ23 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | -2 | -2 | 0 | 0 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | 1 | -1 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | ζ97+ζ92 | orthogonal lifted from D18 |
| ρ24 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -1 | -1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ25 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2i | 2i | -2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2i | -2i | -2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | -2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊3S3, Schur index 2 |
| ρ28 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2ζ95+2ζ94 | 2ζ97+2ζ92 | 2ζ98+2ζ9 | 0 | 0 | 0 | -2ζ95-2ζ94 | -2ζ98-2ζ9 | -2ζ97-2ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
| ρ29 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2ζ97+2ζ92 | 2ζ98+2ζ9 | 2ζ95+2ζ94 | 0 | 0 | 0 | -2ζ97-2ζ92 | -2ζ95-2ζ94 | -2ζ98-2ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
| ρ30 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2ζ98+2ζ9 | 2ζ95+2ζ94 | 2ζ97+2ζ92 | 0 | 0 | 0 | -2ζ98-2ζ9 | -2ζ97-2ζ92 | -2ζ95-2ζ94 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
►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 44)(38 43)(39 42)(40 41)(46 53)(47 52)(48 51)(49 50)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(63 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)(37,44)(38,43)(39,42)(40,41)(46,53)(47,52)(48,51)(49,50)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(63,72)>;
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)(37,44)(38,43)(39,42)(40,41)(46,53)(47,52)(48,51)(49,50)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(63,72) );
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),(37,44),(38,43),(39,42),(40,41),(46,53),(47,52),(48,51),(49,50),(55,71),(56,70),(57,69),(58,68),(59,67),(60,66),(61,65),(62,64),(63,72)]])
Q8⋊3D9 is a maximal subgroup of
D72⋊C2 SD16⋊3D9 Q16⋊D9 D72⋊5C2 Q8.15D18 C4○D4×D9 D4⋊8D18 Q8⋊3D27 Dic9.A4 Dic9.2A4 D18.D6 Dic6⋊5D9 D36⋊3C6 C36.29D6
Q8⋊3D9 is a maximal quotient of
C36.3Q8 C4⋊C4⋊7D9 D36⋊C4 D18.D4 C4⋊D36 C4⋊C4⋊D9 Q8×Dic9 D18⋊3Q8 C36.23D4 Q8⋊3D27 D18.D6 Dic6⋊5D9 C36.29D6
Matrix representation of Q8⋊3D9 ►in GL4(𝔽37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 35 |
| 0 | 0 | 1 | 1 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 31 | 0 |
| 0 | 0 | 6 | 6 |
| 20 | 6 | 0 | 0 |
| 31 | 26 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 |
| 36 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 36 | 36 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,36,1,0,0,35,1],[36,0,0,0,0,36,0,0,0,0,31,6,0,0,0,6],[20,31,0,0,6,26,0,0,0,0,1,0,0,0,0,1],[36,36,0,0,0,1,0,0,0,0,1,36,0,0,0,36] >;
Q8⋊3D9 in GAP, Magma, Sage, TeX
Q_8\rtimes_3D_9
% in TeX
G:=Group("Q8:3D9"); // GroupNames label
G:=SmallGroup(144,44);
// by ID
G=gap.SmallGroup(144,44);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,218,116,50,2404,208,3461]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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