direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8×D9, C12.7D6, C4.6D18, Dic18⋊4C2, C36.6C22, C18.7C23, D18.5C22, Dic9.4C22, C9⋊2(C2×Q8), C3.(S3×Q8), (Q8×C9)⋊2C2, (C4×D9).1C2, (C3×Q8).7S3, C2.8(C22×D9), C6.25(C22×S3), SmallGroup(144,43)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×D9
G = < a,b,c,d | a4=c9=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 187 in 57 conjugacy classes, 31 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C2×C4, Q8, Q8, C9, Dic3, C12, D6, C2×Q8, D9, C18, Dic6, C4×S3, C3×Q8, Dic9, C36, D18, S3×Q8, Dic18, C4×D9, Q8×C9, Q8×D9
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, D9, C22×S3, D18, S3×Q8, C22×D9, Q8×D9
Character table of Q8×D9
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6 | 9A | 9B | 9C | 12A | 12B | 12C | 18A | 18B | 18C | 36A | 36B | 36C | 36D | 36E | 36F | 36G | 36H | 36I | |
| size | 1 | 1 | 9 | 9 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 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 | 2 | -2 | 2 | -2 | 0 | 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 |
| ρ10 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 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 |
| ρ11 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 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 | 2 | 2 | -2 | -2 | 0 | 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 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -1 | -1 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ14 | 2 | 2 | 0 | 0 | -1 | -2 | 2 | -2 | 0 | 0 | 0 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -1 | 1 | 1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | ζ98+ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | ζ97+ζ92 | orthogonal lifted from D18 |
| ρ15 | 2 | 2 | 0 | 0 | -1 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | 1 | 1 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ97-ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ16 | 2 | 2 | 0 | 0 | -1 | -2 | 2 | -2 | 0 | 0 | 0 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -1 | 1 | 1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | ζ97+ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | ζ95+ζ94 | orthogonal lifted from D18 |
| ρ17 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -1 | -1 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ18 | 2 | 2 | 0 | 0 | -1 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | 1 | 1 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ98-ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ19 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | 1 | -1 | 1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ98-ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ20 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -1 | -1 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ21 | 2 | 2 | 0 | 0 | -1 | -2 | 2 | -2 | 0 | 0 | 0 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -1 | 1 | 1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | ζ95+ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | ζ98+ζ9 | orthogonal lifted from D18 |
| ρ22 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | 1 | -1 | 1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ97-ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ23 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | 1 | -1 | 1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ95-ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ24 | 2 | 2 | 0 | 0 | -1 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | 1 | 1 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ95-ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ25 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ26 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ27 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | -2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from S3×Q8, Schur index 2 |
| ρ28 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2ζ97+2ζ92 | 2ζ98+2ζ9 | 2ζ95+2ζ94 | 0 | 0 | 0 | -2ζ95-2ζ94 | -2ζ97-2ζ92 | -2ζ98-2ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ29 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2ζ98+2ζ9 | 2ζ95+2ζ94 | 2ζ97+2ζ92 | 0 | 0 | 0 | -2ζ97-2ζ92 | -2ζ98-2ζ9 | -2ζ95-2ζ94 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ30 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2ζ95+2ζ94 | 2ζ97+2ζ92 | 2ζ98+2ζ9 | 0 | 0 | 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 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 13)(2 12)(3 11)(4 10)(5 18)(6 17)(7 16)(8 15)(9 14)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(27 36)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(45 54)(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), (1,13)(2,12)(3,11)(4,10)(5,18)(6,17)(7,16)(8,15)(9,14)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(27,36)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(45,54)(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), (1,13)(2,12)(3,11)(4,10)(5,18)(6,17)(7,16)(8,15)(9,14)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(27,36)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(45,54)(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)], [(1,13),(2,12),(3,11),(4,10),(5,18),(6,17),(7,16),(8,15),(9,14),(19,35),(20,34),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(27,36),(37,53),(38,52),(39,51),(40,50),(41,49),(42,48),(43,47),(44,46),(45,54),(55,71),(56,70),(57,69),(58,68),(59,67),(60,66),(61,65),(62,64),(63,72)]])
Q8×D9 is a maximal subgroup of
SD16⋊D9 Q16⋊D9 Q8.15D18 D4.10D18 D18.A4 Dic18⋊S3
Q8×D9 is a maximal quotient of
Dic9⋊3Q8 C36⋊Q8 Dic9.Q8 D18⋊Q8 D18⋊2Q8 Dic9⋊Q8 D18⋊3Q8 Dic18⋊S3
Matrix representation of Q8×D9 ►in GL4(𝔽37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 36 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 23 | 32 |
| 0 | 0 | 32 | 14 |
| 17 | 11 | 0 | 0 |
| 26 | 6 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 17 | 11 | 0 | 0 |
| 31 | 20 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,0,36,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,23,32,0,0,32,14],[17,26,0,0,11,6,0,0,0,0,1,0,0,0,0,1],[17,31,0,0,11,20,0,0,0,0,36,0,0,0,0,36] >;
Q8×D9 in GAP, Magma, Sage, TeX
Q_8\times D_9
% in TeX
G:=Group("Q8xD9"); // GroupNames label
G:=SmallGroup(144,43);
// by ID
G=gap.SmallGroup(144,43);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,116,50,2404,208,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^9=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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