metabelian, supersoluble, monomial
Aliases: C33⋊5D9, C34.6S3, C32⋊C9⋊13S3, C32⋊3(C9⋊S3), (C32×C9)⋊10C6, C3⋊(C32⋊D9), C32⋊4D9⋊2C3, C33.79(C3×S3), C32.17(C3×D9), C33.25(C3⋊S3), C32.10(C9⋊C6), C3.1(He3⋊4S3), C32.46(C32⋊C6), C3.1(C33.S3), C3.2(C3×C9⋊S3), (C3×C9)⋊17(C3×S3), (C3×C32⋊C9)⋊6C2, C32.28(C3×C3⋊S3), SmallGroup(486,137)
Series: Derived ►Chief ►Lower central ►Upper central
| C32×C9 — C33⋊D9 |
Generators and relations for C33⋊D9
G = < a,b,c,d,e | a3=b3=c3=d9=e2=1, ab=ba, dad-1=eae=ac=ca, bc=cb, bd=db, ebe=b-1, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 1520 in 177 conjugacy classes, 44 normal (12 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, C33, C33, C33, C9⋊S3, C3×C3⋊S3, C33⋊C2, C32⋊C9, C32⋊C9, C32×C9, C32×C9, C34, C32⋊D9, C32⋊4D9, C3×C33⋊C2, C3×C32⋊C9, C33⋊D9
Quotients: C1, C2, C3, S3, C6, D9, C3×S3, C3⋊S3, C3×D9, C32⋊C6, C9⋊C6, C9⋊S3, C3×C3⋊S3, C32⋊D9, C3×C9⋊S3, He3⋊4S3, C33.S3, C33⋊D9
►On 81 points
Generators in S81
(2 78 17)(3 18 79)(5 81 11)(6 12 73)(8 75 14)(9 15 76)(19 56 42)(20 43 57)(22 59 45)(23 37 60)(25 62 39)(26 40 63)(28 71 48)(29 49 72)(31 65 51)(32 52 66)(34 68 54)(35 46 69)
(1 24 53)(2 25 54)(3 26 46)(4 27 47)(5 19 48)(6 20 49)(7 21 50)(8 22 51)(9 23 52)(10 41 70)(11 42 71)(12 43 72)(13 44 64)(14 45 65)(15 37 66)(16 38 67)(17 39 68)(18 40 69)(28 81 56)(29 73 57)(30 74 58)(31 75 59)(32 76 60)(33 77 61)(34 78 62)(35 79 63)(36 80 55)
(1 77 16)(2 78 17)(3 79 18)(4 80 10)(5 81 11)(6 73 12)(7 74 13)(8 75 14)(9 76 15)(19 56 42)(20 57 43)(21 58 44)(22 59 45)(23 60 37)(24 61 38)(25 62 39)(26 63 40)(27 55 41)(28 71 48)(29 72 49)(30 64 50)(31 65 51)(32 66 52)(33 67 53)(34 68 54)(35 69 46)(36 70 47)
(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)(73 74 75 76 77 78 79 80 81)
(1 9)(2 8)(3 7)(4 6)(10 73)(11 81)(12 80)(13 79)(14 78)(15 77)(16 76)(17 75)(18 74)(19 48)(20 47)(21 46)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 45)(35 44)(36 43)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)
G:=sub<Sym(81)| (2,78,17)(3,18,79)(5,81,11)(6,12,73)(8,75,14)(9,15,76)(19,56,42)(20,43,57)(22,59,45)(23,37,60)(25,62,39)(26,40,63)(28,71,48)(29,49,72)(31,65,51)(32,52,66)(34,68,54)(35,46,69), (1,24,53)(2,25,54)(3,26,46)(4,27,47)(5,19,48)(6,20,49)(7,21,50)(8,22,51)(9,23,52)(10,41,70)(11,42,71)(12,43,72)(13,44,64)(14,45,65)(15,37,66)(16,38,67)(17,39,68)(18,40,69)(28,81,56)(29,73,57)(30,74,58)(31,75,59)(32,76,60)(33,77,61)(34,78,62)(35,79,63)(36,80,55), (1,77,16)(2,78,17)(3,79,18)(4,80,10)(5,81,11)(6,73,12)(7,74,13)(8,75,14)(9,76,15)(19,56,42)(20,57,43)(21,58,44)(22,59,45)(23,60,37)(24,61,38)(25,62,39)(26,63,40)(27,55,41)(28,71,48)(29,72,49)(30,64,50)(31,65,51)(32,66,52)(33,67,53)(34,68,54)(35,69,46)(36,70,47), (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)(73,74,75,76,77,78,79,80,81), (1,9)(2,8)(3,7)(4,6)(10,73)(11,81)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,48)(20,47)(21,46)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,45)(35,44)(36,43)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)>;
G:=Group( (2,78,17)(3,18,79)(5,81,11)(6,12,73)(8,75,14)(9,15,76)(19,56,42)(20,43,57)(22,59,45)(23,37,60)(25,62,39)(26,40,63)(28,71,48)(29,49,72)(31,65,51)(32,52,66)(34,68,54)(35,46,69), (1,24,53)(2,25,54)(3,26,46)(4,27,47)(5,19,48)(6,20,49)(7,21,50)(8,22,51)(9,23,52)(10,41,70)(11,42,71)(12,43,72)(13,44,64)(14,45,65)(15,37,66)(16,38,67)(17,39,68)(18,40,69)(28,81,56)(29,73,57)(30,74,58)(31,75,59)(32,76,60)(33,77,61)(34,78,62)(35,79,63)(36,80,55), (1,77,16)(2,78,17)(3,79,18)(4,80,10)(5,81,11)(6,73,12)(7,74,13)(8,75,14)(9,76,15)(19,56,42)(20,57,43)(21,58,44)(22,59,45)(23,60,37)(24,61,38)(25,62,39)(26,63,40)(27,55,41)(28,71,48)(29,72,49)(30,64,50)(31,65,51)(32,66,52)(33,67,53)(34,68,54)(35,69,46)(36,70,47), (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)(73,74,75,76,77,78,79,80,81), (1,9)(2,8)(3,7)(4,6)(10,73)(11,81)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,48)(20,47)(21,46)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,45)(35,44)(36,43)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64) );
G=PermutationGroup([[(2,78,17),(3,18,79),(5,81,11),(6,12,73),(8,75,14),(9,15,76),(19,56,42),(20,43,57),(22,59,45),(23,37,60),(25,62,39),(26,40,63),(28,71,48),(29,49,72),(31,65,51),(32,52,66),(34,68,54),(35,46,69)], [(1,24,53),(2,25,54),(3,26,46),(4,27,47),(5,19,48),(6,20,49),(7,21,50),(8,22,51),(9,23,52),(10,41,70),(11,42,71),(12,43,72),(13,44,64),(14,45,65),(15,37,66),(16,38,67),(17,39,68),(18,40,69),(28,81,56),(29,73,57),(30,74,58),(31,75,59),(32,76,60),(33,77,61),(34,78,62),(35,79,63),(36,80,55)], [(1,77,16),(2,78,17),(3,79,18),(4,80,10),(5,81,11),(6,73,12),(7,74,13),(8,75,14),(9,76,15),(19,56,42),(20,57,43),(21,58,44),(22,59,45),(23,60,37),(24,61,38),(25,62,39),(26,63,40),(27,55,41),(28,71,48),(29,72,49),(30,64,50),(31,65,51),(32,66,52),(33,67,53),(34,68,54),(35,69,46),(36,70,47)], [(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),(73,74,75,76,77,78,79,80,81)], [(1,9),(2,8),(3,7),(4,6),(10,73),(11,81),(12,80),(13,79),(14,78),(15,77),(16,76),(17,75),(18,74),(19,48),(20,47),(21,46),(22,54),(23,53),(24,52),(25,51),(26,50),(27,49),(28,42),(29,41),(30,40),(31,39),(32,38),(33,37),(34,45),(35,44),(36,43),(55,72),(56,71),(57,70),(58,69),(59,68),(60,67),(61,66),(62,65),(63,64)]])
54 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3M | 3N | 3O | 3P | ··· | 3W | 6A | 6B | 9A | ··· | 9AA |
| order | 1 | 2 | 3 | ··· | 3 | 3 | 3 | 3 | ··· | 3 | 6 | 6 | 9 | ··· | 9 |
| size | 1 | 81 | 2 | ··· | 2 | 3 | 3 | 6 | ··· | 6 | 81 | 81 | 6 | ··· | 6 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 |
| type | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C3 | C6 | S3 | S3 | C3×S3 | D9 | C3×S3 | C3×D9 | C32⋊C6 | C9⋊C6 |
| kernel | C33⋊D9 | C3×C32⋊C9 | C32⋊4D9 | C32×C9 | C32⋊C9 | C34 | C3×C9 | C33 | C33 | C32 | C32 | C32 |
| # reps | 1 | 1 | 2 | 2 | 3 | 1 | 6 | 9 | 2 | 18 | 3 | 6 |
Matrix representation of C33⋊D9 ►in GL10(𝔽19)
| 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 | 1 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 | 0 | 3 | 17 | 3 |
| 0 | 0 | 0 | 0 | 12 | 18 | 18 | 2 | 18 | 1 |
| 0 | 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 3 | 18 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 | 3 | 0 | 17 | 3 |
| 0 | 0 | 0 | 0 | 13 | 8 | 2 | 18 | 18 | 1 |
| 14 | 17 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 | 3 | 0 | 18 | 1 |
| 0 | 0 | 0 | 0 | 15 | 18 | 11 | 18 | 0 | 18 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 9 | 0 | 1 | 0 |
| 7 | 14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 1 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 10 | 18 | 18 | 8 | 1 | 18 |
G:=sub<GL(10,GF(19))| [11,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,1,0,5,7,17,12,0,0,0,0,0,1,0,0,0,18,0,0,0,0,0,0,0,1,0,18,0,0,0,0,0,0,18,18,3,2,0,0,0,0,0,0,0,0,17,18,0,0,0,0,0,0,0,0,3,1],[0,1,0,0,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,3,17,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,17,18,14,12,8,13,0,0,0,0,3,1,3,12,0,8,0,0,0,0,0,0,18,18,3,2,0,0,0,0,0,0,1,0,0,18,0,0,0,0,0,0,0,0,17,18,0,0,0,0,0,0,0,0,3,1],[14,2,0,0,0,0,0,0,0,0,17,12,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,3,17,0,0,0,0,0,0,0,0,0,0,16,14,13,15,0,4,0,0,0,0,0,0,0,18,0,0,0,0,0,0,3,1,3,11,0,9,0,0,0,0,0,1,0,18,0,0,0,0,0,0,0,0,18,0,1,1,0,0,0,0,0,0,1,18,0,0],[7,2,0,0,0,0,0,0,0,0,14,12,0,0,0,0,0,0,0,0,0,0,2,18,0,0,0,0,0,0,0,0,3,17,0,0,0,0,0,0,0,0,0,0,7,14,13,3,0,10,0,0,0,0,0,0,1,0,0,18,0,0,0,0,0,1,0,0,0,18,0,0,0,0,3,1,3,12,0,8,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,18] >;
C33⋊D9 in GAP, Magma, Sage, TeX
C_3^3\rtimes D_9
% in TeX
G:=Group("C3^3:D9"); // GroupNames label
G:=SmallGroup(486,137);
// by ID
G=gap.SmallGroup(486,137);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,3134,548,986,867,3244,11669]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^9=e^2=1,a*b=b*a,d*a*d^-1=e*a*e=a*c=c*a,b*c=c*b,b*d=d*b,e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations