metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.D9, C9⋊2SD16, C4.1D18, C18.7D4, C12.1D6, Dic18⋊2C2, C36.1C22, C9⋊C8⋊1C2, (C3×D4).1S3, (D4×C9).1C2, C3.(D4.S3), C2.4(C9⋊D4), C6.14(C3⋊D4), SmallGroup(144,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D9
G = < a,b,c,d | a4=b2=c9=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >
Character table of D4.D9
| class | 1 | 2A | 2B | 3 | 4A | 4B | 6A | 6B | 6C | 8A | 8B | 9A | 9B | 9C | 12 | 18A | 18B | 18C | 18D | 18E | 18F | 18G | 18H | 18I | 36A | 36B | 36C | |
| size | 1 | 1 | 4 | 2 | 2 | 36 | 2 | 4 | 4 | 18 | 18 | 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 | 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 | -2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | orthogonal lifted from D4 |
| ρ8 | 2 | 2 | -2 | -1 | 2 | 0 | -1 | 1 | 1 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D18 |
| ρ9 | 2 | 2 | 2 | -1 | 2 | 0 | -1 | -1 | -1 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ10 | 2 | 2 | -2 | -1 | 2 | 0 | -1 | 1 | 1 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D18 |
| ρ11 | 2 | 2 | 2 | -1 | 2 | 0 | -1 | -1 | -1 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ12 | 2 | 2 | 2 | -1 | 2 | 0 | -1 | -1 | -1 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ13 | 2 | 2 | -2 | -1 | 2 | 0 | -1 | 1 | 1 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D18 |
| ρ14 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | √-2 | -√-2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ15 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | -√-2 | √-2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ16 | 2 | 2 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -2 | -1 | -1 | -1 | -√-3 | -√-3 | √-3 | √-3 | √-3 | -√-3 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ17 | 2 | 2 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -2 | -1 | -1 | -1 | √-3 | √-3 | -√-3 | -√-3 | -√-3 | √-3 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | 0 | -1 | -2 | 0 | -1 | √-3 | -√-3 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ98+ζ9 | ζ97-ζ92 | ζ98-ζ9 | ζ95-ζ94 | -ζ97+ζ92 | -ζ95+ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | complex lifted from C9⋊D4 |
| ρ19 | 2 | 2 | 0 | -1 | -2 | 0 | -1 | -√-3 | √-3 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95-ζ94 | ζ98-ζ9 | -ζ95+ζ94 | ζ97-ζ92 | -ζ98+ζ9 | -ζ97+ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | complex lifted from C9⋊D4 |
| ρ20 | 2 | 2 | 0 | -1 | -2 | 0 | -1 | √-3 | -√-3 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97-ζ92 | -ζ95+ζ94 | -ζ97+ζ92 | ζ98-ζ9 | ζ95-ζ94 | -ζ98+ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | complex lifted from C9⋊D4 |
| ρ21 | 2 | 2 | 0 | -1 | -2 | 0 | -1 | √-3 | -√-3 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ95+ζ94 | -ζ98+ζ9 | ζ95-ζ94 | -ζ97+ζ92 | ζ98-ζ9 | ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | complex lifted from C9⋊D4 |
| ρ22 | 2 | 2 | 0 | -1 | -2 | 0 | -1 | -√-3 | √-3 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98-ζ9 | -ζ97+ζ92 | -ζ98+ζ9 | -ζ95+ζ94 | ζ97-ζ92 | ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | complex lifted from C9⋊D4 |
| ρ23 | 2 | 2 | 0 | -1 | -2 | 0 | -1 | -√-3 | √-3 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ97+ζ92 | ζ95-ζ94 | ζ97-ζ92 | -ζ98+ζ9 | -ζ95+ζ94 | ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | complex lifted from C9⋊D4 |
| ρ24 | 4 | -4 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4.S3, Schur index 2 |
| ρ25 | 4 | -4 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2ζ97+2ζ92 | 2ζ95+2ζ94 | 2ζ98+2ζ9 | 0 | -2ζ97-2ζ92 | -2ζ95-2ζ94 | -2ζ98-2ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ26 | 4 | -4 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2ζ95+2ζ94 | 2ζ98+2ζ9 | 2ζ97+2ζ92 | 0 | -2ζ95-2ζ94 | -2ζ98-2ζ9 | -2ζ97-2ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ27 | 4 | -4 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2ζ98+2ζ9 | 2ζ97+2ζ92 | 2ζ95+2ζ94 | 0 | -2ζ98-2ζ9 | -2ζ97-2ζ92 | -2ζ95-2ζ94 | 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 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 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 48 14 39)(2 47 15 38)(3 46 16 37)(4 54 17 45)(5 53 18 44)(6 52 10 43)(7 51 11 42)(8 50 12 41)(9 49 13 40)(19 70 28 61)(20 69 29 60)(21 68 30 59)(22 67 31 58)(23 66 32 57)(24 65 33 56)(25 64 34 55)(26 72 35 63)(27 71 36 62)
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), (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54), (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), (1,48,14,39)(2,47,15,38)(3,46,16,37)(4,54,17,45)(5,53,18,44)(6,52,10,43)(7,51,11,42)(8,50,12,41)(9,49,13,40)(19,70,28,61)(20,69,29,60)(21,68,30,59)(22,67,31,58)(23,66,32,57)(24,65,33,56)(25,64,34,55)(26,72,35,63)(27,71,36,62)>;
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), (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54), (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), (1,48,14,39)(2,47,15,38)(3,46,16,37)(4,54,17,45)(5,53,18,44)(6,52,10,43)(7,51,11,42)(8,50,12,41)(9,49,13,40)(19,70,28,61)(20,69,29,60)(21,68,30,59)(22,67,31,58)(23,66,32,57)(24,65,33,56)(25,64,34,55)(26,72,35,63)(27,71,36,62) );
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)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,19),(7,20),(8,21),(9,22),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(37,46),(38,47),(39,48),(40,49),(41,50),(42,51),(43,52),(44,53),(45,54)], [(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)], [(1,48,14,39),(2,47,15,38),(3,46,16,37),(4,54,17,45),(5,53,18,44),(6,52,10,43),(7,51,11,42),(8,50,12,41),(9,49,13,40),(19,70,28,61),(20,69,29,60),(21,68,30,59),(22,67,31,58),(23,66,32,57),(24,65,33,56),(25,64,34,55),(26,72,35,63),(27,71,36,62)]])
D4.D9 is a maximal subgroup of
D8⋊D9 D8⋊3D9 SD16×D9 SD16⋊D9 D36⋊6C22 D4.D18 D4.9D18 D4.D27 D12.D9 C36.D6 Dic18⋊C6 C36.17D6
D4.D9 is a maximal quotient of
C4.Dic18 C18.Q16 D4⋊Dic9 D4.D27 D12.D9 C36.D6 C36.17D6
Matrix representation of D4.D9 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 70 |
| 0 | 0 | 25 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 48 | 72 |
| 32 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 16 | 0 | 0 |
| 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 55 |
| 0 | 0 | 69 | 0 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,25,0,0,70,1],[72,0,0,0,0,1,0,0,0,0,1,48,0,0,0,72],[32,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[0,32,0,0,16,0,0,0,0,0,0,69,0,0,55,0] >;
D4.D9 in GAP, Magma, Sage, TeX
D_4.D_9
% in TeX
G:=Group("D4.D9"); // GroupNames label
G:=SmallGroup(144,15);
// by ID
G=gap.SmallGroup(144,15);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,73,218,116,50,2404,208,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^9=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of D4.D9 in TeX
Character table of D4.D9 in TeX