direct product, metabelian, supersoluble, monomial, A-group
Aliases: Dic3×C3×C9, C32⋊4C36, C33.6C12, C3⋊(C3×C36), C6.(C3×C18), (C32×C9)⋊3C4, (C3×C9)⋊13C12, C6.10(S3×C9), (C3×C6).7C18, C18.12(C3×S3), (C3×C18).26S3, (C3×C18).23C6, C6.8(S3×C32), C32.6(C3×C12), (C32×C18).1C2, (C32×C6).15C6, C3.4(C32×Dic3), (C32×Dic3).2C3, (C3×Dic3).1C32, C32.19(C3×Dic3), C2.(S3×C3×C9), (C3×C6).15(C3×C6), (C3×C6).43(C3×S3), SmallGroup(324,91)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C3×C9 |
Generators and relations for Dic3×C3×C9
G = < a,b,c,d | a3=b9=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 140 in 90 conjugacy classes, 50 normal (20 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C9, C9, C32, C32, C32, Dic3, C12, C18, C18, C3×C6, C3×C6, C3×C6, C3×C9, C3×C9, C3×C9, C33, C36, C3×Dic3, C3×Dic3, C3×C12, C3×C18, C3×C18, C3×C18, C32×C6, C32×C9, C9×Dic3, C3×C36, C32×Dic3, C32×C18, Dic3×C3×C9
Quotients: C1, C2, C3, C4, S3, C6, C9, C32, Dic3, C12, C18, C3×S3, C3×C6, C3×C9, C36, C3×Dic3, C3×C12, S3×C9, C3×C18, S3×C32, C9×Dic3, C3×C36, C32×Dic3, S3×C3×C9, Dic3×C3×C9
►On 108 points
Generators in S108
(1 53 62)(2 54 63)(3 46 55)(4 47 56)(5 48 57)(6 49 58)(7 50 59)(8 51 60)(9 52 61)(10 31 77)(11 32 78)(12 33 79)(13 34 80)(14 35 81)(15 36 73)(16 28 74)(17 29 75)(18 30 76)(19 95 102)(20 96 103)(21 97 104)(22 98 105)(23 99 106)(24 91 107)(25 92 108)(26 93 100)(27 94 101)(37 71 83)(38 72 84)(39 64 85)(40 65 86)(41 66 87)(42 67 88)(43 68 89)(44 69 90)(45 70 82)
(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)(82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99)(100 101 102 103 104 105 106 107 108)
(1 80 53 13 62 34)(2 81 54 14 63 35)(3 73 46 15 55 36)(4 74 47 16 56 28)(5 75 48 17 57 29)(6 76 49 18 58 30)(7 77 50 10 59 31)(8 78 51 11 60 32)(9 79 52 12 61 33)(19 71 102 37 95 83)(20 72 103 38 96 84)(21 64 104 39 97 85)(22 65 105 40 98 86)(23 66 106 41 99 87)(24 67 107 42 91 88)(25 68 108 43 92 89)(26 69 100 44 93 90)(27 70 101 45 94 82)
(1 67 13 91)(2 68 14 92)(3 69 15 93)(4 70 16 94)(5 71 17 95)(6 72 18 96)(7 64 10 97)(8 65 11 98)(9 66 12 99)(19 57 37 75)(20 58 38 76)(21 59 39 77)(22 60 40 78)(23 61 41 79)(24 62 42 80)(25 63 43 81)(26 55 44 73)(27 56 45 74)(28 101 47 82)(29 102 48 83)(30 103 49 84)(31 104 50 85)(32 105 51 86)(33 106 52 87)(34 107 53 88)(35 108 54 89)(36 100 46 90)
G:=sub<Sym(108)| (1,53,62)(2,54,63)(3,46,55)(4,47,56)(5,48,57)(6,49,58)(7,50,59)(8,51,60)(9,52,61)(10,31,77)(11,32,78)(12,33,79)(13,34,80)(14,35,81)(15,36,73)(16,28,74)(17,29,75)(18,30,76)(19,95,102)(20,96,103)(21,97,104)(22,98,105)(23,99,106)(24,91,107)(25,92,108)(26,93,100)(27,94,101)(37,71,83)(38,72,84)(39,64,85)(40,65,86)(41,66,87)(42,67,88)(43,68,89)(44,69,90)(45,70,82), (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)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108), (1,80,53,13,62,34)(2,81,54,14,63,35)(3,73,46,15,55,36)(4,74,47,16,56,28)(5,75,48,17,57,29)(6,76,49,18,58,30)(7,77,50,10,59,31)(8,78,51,11,60,32)(9,79,52,12,61,33)(19,71,102,37,95,83)(20,72,103,38,96,84)(21,64,104,39,97,85)(22,65,105,40,98,86)(23,66,106,41,99,87)(24,67,107,42,91,88)(25,68,108,43,92,89)(26,69,100,44,93,90)(27,70,101,45,94,82), (1,67,13,91)(2,68,14,92)(3,69,15,93)(4,70,16,94)(5,71,17,95)(6,72,18,96)(7,64,10,97)(8,65,11,98)(9,66,12,99)(19,57,37,75)(20,58,38,76)(21,59,39,77)(22,60,40,78)(23,61,41,79)(24,62,42,80)(25,63,43,81)(26,55,44,73)(27,56,45,74)(28,101,47,82)(29,102,48,83)(30,103,49,84)(31,104,50,85)(32,105,51,86)(33,106,52,87)(34,107,53,88)(35,108,54,89)(36,100,46,90)>;
G:=Group( (1,53,62)(2,54,63)(3,46,55)(4,47,56)(5,48,57)(6,49,58)(7,50,59)(8,51,60)(9,52,61)(10,31,77)(11,32,78)(12,33,79)(13,34,80)(14,35,81)(15,36,73)(16,28,74)(17,29,75)(18,30,76)(19,95,102)(20,96,103)(21,97,104)(22,98,105)(23,99,106)(24,91,107)(25,92,108)(26,93,100)(27,94,101)(37,71,83)(38,72,84)(39,64,85)(40,65,86)(41,66,87)(42,67,88)(43,68,89)(44,69,90)(45,70,82), (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)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108), (1,80,53,13,62,34)(2,81,54,14,63,35)(3,73,46,15,55,36)(4,74,47,16,56,28)(5,75,48,17,57,29)(6,76,49,18,58,30)(7,77,50,10,59,31)(8,78,51,11,60,32)(9,79,52,12,61,33)(19,71,102,37,95,83)(20,72,103,38,96,84)(21,64,104,39,97,85)(22,65,105,40,98,86)(23,66,106,41,99,87)(24,67,107,42,91,88)(25,68,108,43,92,89)(26,69,100,44,93,90)(27,70,101,45,94,82), (1,67,13,91)(2,68,14,92)(3,69,15,93)(4,70,16,94)(5,71,17,95)(6,72,18,96)(7,64,10,97)(8,65,11,98)(9,66,12,99)(19,57,37,75)(20,58,38,76)(21,59,39,77)(22,60,40,78)(23,61,41,79)(24,62,42,80)(25,63,43,81)(26,55,44,73)(27,56,45,74)(28,101,47,82)(29,102,48,83)(30,103,49,84)(31,104,50,85)(32,105,51,86)(33,106,52,87)(34,107,53,88)(35,108,54,89)(36,100,46,90) );
G=PermutationGroup([[(1,53,62),(2,54,63),(3,46,55),(4,47,56),(5,48,57),(6,49,58),(7,50,59),(8,51,60),(9,52,61),(10,31,77),(11,32,78),(12,33,79),(13,34,80),(14,35,81),(15,36,73),(16,28,74),(17,29,75),(18,30,76),(19,95,102),(20,96,103),(21,97,104),(22,98,105),(23,99,106),(24,91,107),(25,92,108),(26,93,100),(27,94,101),(37,71,83),(38,72,84),(39,64,85),(40,65,86),(41,66,87),(42,67,88),(43,68,89),(44,69,90),(45,70,82)], [(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),(82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108)], [(1,80,53,13,62,34),(2,81,54,14,63,35),(3,73,46,15,55,36),(4,74,47,16,56,28),(5,75,48,17,57,29),(6,76,49,18,58,30),(7,77,50,10,59,31),(8,78,51,11,60,32),(9,79,52,12,61,33),(19,71,102,37,95,83),(20,72,103,38,96,84),(21,64,104,39,97,85),(22,65,105,40,98,86),(23,66,106,41,99,87),(24,67,107,42,91,88),(25,68,108,43,92,89),(26,69,100,44,93,90),(27,70,101,45,94,82)], [(1,67,13,91),(2,68,14,92),(3,69,15,93),(4,70,16,94),(5,71,17,95),(6,72,18,96),(7,64,10,97),(8,65,11,98),(9,66,12,99),(19,57,37,75),(20,58,38,76),(21,59,39,77),(22,60,40,78),(23,61,41,79),(24,62,42,80),(25,63,43,81),(26,55,44,73),(27,56,45,74),(28,101,47,82),(29,102,48,83),(30,103,49,84),(31,104,50,85),(32,105,51,86),(33,106,52,87),(34,107,53,88),(35,108,54,89),(36,100,46,90)]])
162 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 6A | ··· | 6H | 6I | ··· | 6Q | 9A | ··· | 9R | 9S | ··· | 9AJ | 12A | ··· | 12P | 18A | ··· | 18R | 18S | ··· | 18AJ | 36A | ··· | 36AJ |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
162 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||||||||
| image | C1 | C2 | C3 | C3 | C4 | C6 | C6 | C9 | C12 | C12 | C18 | C36 | S3 | Dic3 | C3×S3 | C3×S3 | C3×Dic3 | C3×Dic3 | S3×C9 | C9×Dic3 |
| kernel | Dic3×C3×C9 | C32×C18 | C9×Dic3 | C32×Dic3 | C32×C9 | C3×C18 | C32×C6 | C3×Dic3 | C3×C9 | C33 | C3×C6 | C32 | C3×C18 | C3×C9 | C18 | C3×C6 | C9 | C32 | C6 | C3 |
| # reps | 1 | 1 | 6 | 2 | 2 | 6 | 2 | 18 | 12 | 4 | 18 | 36 | 1 | 1 | 6 | 2 | 6 | 2 | 18 | 18 |
Matrix representation of Dic3×C3×C9 ►in GL3(𝔽37) generated by
| 10 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 16 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 36 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 28 | 27 |
| 6 | 0 | 0 |
| 0 | 16 | 25 |
| 0 | 6 | 21 |
G:=sub<GL(3,GF(37))| [10,0,0,0,10,0,0,0,10],[16,0,0,0,1,0,0,0,1],[36,0,0,0,11,28,0,0,27],[6,0,0,0,16,6,0,25,21] >;
Dic3×C3×C9 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_3\times C_9 % in TeX
G:=Group("Dic3xC3xC9"); // GroupNames label
G:=SmallGroup(324,91);
// by ID
G=gap.SmallGroup(324,91);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,122,7781]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^9=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations