metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4:2D9, C12.6D6, C4.5D18, Dic18:3C2, C36.5C22, C18.6C23, C22.1D18, D18.2C22, Dic9.3C22, (D4xC9):3C2, (C4xD9):2C2, C9:2(C4oD4), C9:D4:2C2, (C2xC6).3D6, (C3xD4).4S3, (C2xC18).C22, C3.(D4:2S3), (C2xDic9):3C2, C2.7(C22xD9), C6.24(C22xS3), SmallGroup(144,42)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4:2D9
G = < a,b,c,d | a4=b2=c9=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 199 in 60 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, D4, D4, Q8, C9, Dic3, C12, D6, C2xC6, C4oD4, D9, C18, C18, Dic6, C4xS3, C2xDic3, C3:D4, C3xD4, Dic9, Dic9, C36, D18, C2xC18, D4:2S3, Dic18, C4xD9, C2xDic9, C9:D4, D4xC9, D4:2D9
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, D9, C22xS3, D18, D4:2S3, C22xD9, D4:2D9
Character table of D4:2D9
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 9A | 9B | 9C | 12 | 18A | 18B | 18C | 18D | 18E | 18F | 18G | 18H | 18I | 36A | 36B | 36C | |
| size | 1 | 1 | 2 | 2 | 18 | 2 | 2 | 9 | 9 | 18 | 18 | 2 | 4 | 4 | 2 | 2 | 2 | 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 | -2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -1 | -1 | -1 | -2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -1 | -1 | -1 | -2 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D18 |
| ρ16 | 2 | 2 | -2 | 2 | 0 | -1 | -2 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | 1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | ζ95+ζ94 | ζ98+ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ17 | 2 | 2 | 2 | -2 | 0 | -1 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | 1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ98-ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ18 | 2 | 2 | 2 | -2 | 0 | -1 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | 1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ97-ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D18 |
| ρ20 | 2 | 2 | -2 | 2 | 0 | -1 | -2 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | 1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | ζ97+ζ92 | ζ95+ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ21 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ22 | 2 | 2 | -2 | 2 | 0 | -1 | -2 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | 1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | ζ98+ζ9 | ζ97+ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ23 | 2 | 2 | 2 | -2 | 0 | -1 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | 1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ95-ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ24 | 2 | 2 | -2 | -2 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D18 |
| ρ25 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 2i | -2i | 0 | 0 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ26 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | -2i | 2i | 0 | 0 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | -2 | -2 | -2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4:2S3, Schur index 2 |
| ρ28 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2ζ98+2ζ9 | 2ζ95+2ζ94 | 2ζ97+2ζ92 | 0 | -2ζ97-2ζ92 | -2ζ98-2ζ9 | -2ζ95-2ζ94 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ29 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2ζ97+2ζ92 | 2ζ98+2ζ9 | 2ζ95+2ζ94 | 0 | -2ζ95-2ζ94 | -2ζ97-2ζ92 | -2ζ98-2ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ30 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2ζ95+2ζ94 | 2ζ97+2ζ92 | 2ζ98+2ζ9 | 0 | -2ζ98-2ζ9 | -2ζ95-2ζ94 | -2ζ97-2ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 72 points
Generators in S72
(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 59)(2 60)(3 61)(4 62)(5 63)(6 55)(7 56)(8 57)(9 58)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)
(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)| (1,50,14,41)(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), (1,59)(2,60)(3,61)(4,62)(5,63)(6,55)(7,56)(8,57)(9,58)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54), (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( (1,50,14,41)(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), (1,59)(2,60)(3,61)(4,62)(5,63)(6,55)(7,56)(8,57)(9,58)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54), (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([[(1,50,14,41),(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)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,55),(7,56),(8,57),(9,58),(10,64),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,49),(32,50),(33,51),(34,52),(35,53),(36,54)], [(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)]])
D4:2D9 is a maximal subgroup of
D8:D9 D8:3D9 SD16:D9 SD16:3D9 D4:6D18 C4oD4xD9 D4.10D18 D4:2D27 D12:5D9 D12:D9 Dic3.D18 D18.4D6 Dic18:2C6 C36.27D6
D4:2D9 is a maximal quotient of
C23.16D18 C22:2Dic18 C23.8D18 Dic9:4D4 C23.9D18 Dic9.D4 C22.4D36 Dic9:3Q8 Dic9.Q8 C36.3Q8 C4:C4:7D9 D18:2Q8 C4:C4:D9 D4xDic9 C23.23D18 C36.17D4 C36:2D4 Dic9:D4 D4:2D27 D12:5D9 D12:D9 Dic3.D18 D18.4D6 C36.27D6
Matrix representation of D4:2D9 ►in GL4(F37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 6 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 6 |
| 0 | 0 | 31 | 0 |
| 20 | 6 | 0 | 0 |
| 31 | 26 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 17 | 26 | 0 | 0 |
| 6 | 20 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 36 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,31,0,0,0,0,6],[36,0,0,0,0,36,0,0,0,0,0,31,0,0,6,0],[20,31,0,0,6,26,0,0,0,0,1,0,0,0,0,1],[17,6,0,0,26,20,0,0,0,0,1,0,0,0,0,36] >;
D4:2D9 in GAP, Magma, Sage, TeX
D_4\rtimes_2D_9
% in TeX
G:=Group("D4:2D9"); // GroupNames label
G:=SmallGroup(144,42);
// by ID
G=gap.SmallGroup(144,42);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,218,116,2404,208,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^9=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
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