direct product, metabelian, supersoluble, monomial
Aliases: D9×D12, C36⋊1D6, D6⋊1D18, C12⋊4D18, Dic9⋊3D6, D18.10D6, C3⋊1(D4×D9), C4⋊1(S3×D9), C12.19S32, (C3×D9)⋊1D4, (C4×D9)⋊3S3, C9⋊1(C2×D12), (C12×D9)⋊3C2, (C9×D12)⋊5C2, (S3×C6).3D6, C3.2(S3×D12), C36⋊S3⋊8C2, C9⋊D12⋊3C2, (C3×C36)⋊2C22, (C3×D12).7S3, (C3×C12).99D6, C32.2(S3×D4), (S3×C18)⋊1C22, C6.13(C22×D9), C18.13(C22×S3), (C3×C18).13C23, (C3×Dic9)⋊4C22, (C6×D9).10C22, (C2×S3×D9)⋊2C2, (C3×C9)⋊2(C2×D4), C6.32(C2×S32), C2.16(C2×S3×D9), (C2×C9⋊S3)⋊2C22, (C3×C6).81(C22×S3), SmallGroup(432,292)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D9×D12
G = < a,b,c,d | a9=b2=c12=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1452 in 178 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, C23, C9, C9, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×D4, D9, D9, C18, C18, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, D12, C3⋊D4, C2×C12, C3×D4, C22×S3, C3×C9, Dic9, C36, C36, D18, D18, C2×C18, C3×Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×D12, S3×D4, C3×D9, S3×C9, C9⋊S3, C3×C18, C4×D9, D36, C9⋊D4, D4×C9, C22×D9, C3⋊D12, S3×C12, C3×D12, C12⋊S3, C2×S32, C3×Dic9, C3×C36, S3×D9, C6×D9, S3×C18, C2×C9⋊S3, D4×D9, S3×D12, C9⋊D12, C12×D9, C9×D12, C36⋊S3, C2×S3×D9, D9×D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, D12, C22×S3, D18, S32, C2×D12, S3×D4, C22×D9, C2×S32, S3×D9, D4×D9, S3×D12, C2×S3×D9, D9×D12
►On 72 points
(1 49 63 9 57 71 5 53 67)(2 50 64 10 58 72 6 54 68)(3 51 65 11 59 61 7 55 69)(4 52 66 12 60 62 8 56 70)(13 29 46 17 33 38 21 25 42)(14 30 47 18 34 39 22 26 43)(15 31 48 19 35 40 23 27 44)(16 32 37 20 36 41 24 28 45)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 25)(10 26)(11 27)(12 28)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(37 70)(38 71)(39 72)(40 61)(41 62)(42 63)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)
(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)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 55)(14 54)(15 53)(16 52)(17 51)(18 50)(19 49)(20 60)(21 59)(22 58)(23 57)(24 56)(25 61)(26 72)(27 71)(28 70)(29 69)(30 68)(31 67)(32 66)(33 65)(34 64)(35 63)(36 62)
G:=sub<Sym(72)| (1,49,63,9,57,71,5,53,67)(2,50,64,10,58,72,6,54,68)(3,51,65,11,59,61,7,55,69)(4,52,66,12,60,62,8,56,70)(13,29,46,17,33,38,21,25,42)(14,30,47,18,34,39,22,26,43)(15,31,48,19,35,40,23,27,44)(16,32,37,20,36,41,24,28,45), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,25)(10,26)(11,27)(12,28)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(37,70)(38,71)(39,72)(40,61)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69), (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)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,55)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,60)(21,59)(22,58)(23,57)(24,56)(25,61)(26,72)(27,71)(28,70)(29,69)(30,68)(31,67)(32,66)(33,65)(34,64)(35,63)(36,62)>;
G:=Group( (1,49,63,9,57,71,5,53,67)(2,50,64,10,58,72,6,54,68)(3,51,65,11,59,61,7,55,69)(4,52,66,12,60,62,8,56,70)(13,29,46,17,33,38,21,25,42)(14,30,47,18,34,39,22,26,43)(15,31,48,19,35,40,23,27,44)(16,32,37,20,36,41,24,28,45), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,25)(10,26)(11,27)(12,28)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(37,70)(38,71)(39,72)(40,61)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69), (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)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,55)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,60)(21,59)(22,58)(23,57)(24,56)(25,61)(26,72)(27,71)(28,70)(29,69)(30,68)(31,67)(32,66)(33,65)(34,64)(35,63)(36,62) );
G=PermutationGroup([[(1,49,63,9,57,71,5,53,67),(2,50,64,10,58,72,6,54,68),(3,51,65,11,59,61,7,55,69),(4,52,66,12,60,62,8,56,70),(13,29,46,17,33,38,21,25,42),(14,30,47,18,34,39,22,26,43),(15,31,48,19,35,40,23,27,44),(16,32,37,20,36,41,24,28,45)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,25),(10,26),(11,27),(12,28),(13,49),(14,50),(15,51),(16,52),(17,53),(18,54),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(37,70),(38,71),(39,72),(40,61),(41,62),(42,63),(43,64),(44,65),(45,66),(46,67),(47,68),(48,69)], [(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),(2,47),(3,46),(4,45),(5,44),(6,43),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(13,55),(14,54),(15,53),(16,52),(17,51),(18,50),(19,49),(20,60),(21,59),(22,58),(23,57),(24,56),(25,61),(26,72),(27,71),(28,70),(29,69),(30,68),(31,67),(32,66),(33,65),(34,64),(35,63),(36,62)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 18A | 18B | 18C | 18D | 18E | 18F | 18G | ··· | 18L | 36A | ··· | 36I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 6 | 6 | 9 | 9 | 54 | 54 | 2 | 2 | 4 | 2 | 18 | 2 | 2 | 4 | 12 | 12 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | D6 | D6 | D6 | D9 | D12 | D18 | D18 | S32 | S3×D4 | C2×S32 | S3×D9 | D4×D9 | S3×D12 | C2×S3×D9 | D9×D12 |
| kernel | D9×D12 | C9⋊D12 | C12×D9 | C9×D12 | C36⋊S3 | C2×S3×D9 | C4×D9 | C3×D12 | C3×D9 | Dic9 | C36 | D18 | C3×C12 | S3×C6 | D12 | D9 | C12 | D6 | C12 | C32 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 3 | 4 | 3 | 6 | 1 | 1 | 1 | 3 | 3 | 2 | 3 | 6 |
Matrix representation of D9×D12 ►in GL4(𝔽37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 26 | 31 |
| 0 | 0 | 6 | 20 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 31 | 17 |
| 0 | 0 | 11 | 6 |
| 32 | 32 | 0 | 0 |
| 5 | 27 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 |
| 1 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,26,6,0,0,31,20],[1,0,0,0,0,1,0,0,0,0,31,11,0,0,17,6],[32,5,0,0,32,27,0,0,0,0,36,0,0,0,0,36],[1,1,0,0,0,36,0,0,0,0,36,0,0,0,0,36] >;
D9×D12 in GAP, Magma, Sage, TeX
D_9\times D_{12} % in TeX
G:=Group("D9xD12"); // GroupNames label
G:=SmallGroup(432,292);
// by ID
G=gap.SmallGroup(432,292);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,58,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^2=c^12=d^2=1,b*a*b=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