metabelian, supersoluble, monomial
Aliases: C32⋊1Dic9, C33.1Dic3, C3.(C9⋊C12), (C3×C9)⋊1C12, C32⋊C9⋊2C4, C6.1(C3×D9), (C3×C6).1D9, C6.1(C9⋊C6), (C3×C18).1C6, C9⋊Dic3⋊1C3, (C32×C6).1S3, C3.1(C3×Dic9), C6.1(C32⋊C6), C2.(C32⋊D9), C3.1(C32⋊C12), C32.13(C3×Dic3), (C3×C6).27(C3×S3), (C2×C32⋊C9).2C2, SmallGroup(324,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — C32⋊Dic9 |
Generators and relations for C32⋊Dic9
G = < a,b,c,d | a3=b3=c18=1, d2=c9, cac-1=ab=ba, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Smallest permutation representation of C32⋊Dic9
►On 108 points
Generators in S108
(1 13 7)(2 84 97)(3 104 79)(4 16 10)(5 87 100)(6 107 82)(8 90 103)(9 92 85)(11 75 106)(12 95 88)(14 78 91)(15 98 73)(17 81 94)(18 101 76)(19 38 67)(20 26 32)(21 63 46)(22 41 70)(23 29 35)(24 66 49)(25 44 55)(27 69 52)(28 47 58)(30 72 37)(31 50 61)(33 57 40)(34 53 64)(36 60 43)(39 45 51)(42 48 54)(56 62 68)(59 65 71)(74 86 80)(77 89 83)(93 105 99)(96 108 102)
(1 89 108)(2 90 91)(3 73 92)(4 74 93)(5 75 94)(6 76 95)(7 77 96)(8 78 97)(9 79 98)(10 80 99)(11 81 100)(12 82 101)(13 83 102)(14 84 103)(15 85 104)(16 86 105)(17 87 106)(18 88 107)(19 55 50)(20 56 51)(21 57 52)(22 58 53)(23 59 54)(24 60 37)(25 61 38)(26 62 39)(27 63 40)(28 64 41)(29 65 42)(30 66 43)(31 67 44)(32 68 45)(33 69 46)(34 70 47)(35 71 48)(36 72 49)
(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 26 10 35)(2 25 11 34)(3 24 12 33)(4 23 13 32)(5 22 14 31)(6 21 15 30)(7 20 16 29)(8 19 17 28)(9 36 18 27)(37 82 46 73)(38 81 47 90)(39 80 48 89)(40 79 49 88)(41 78 50 87)(42 77 51 86)(43 76 52 85)(44 75 53 84)(45 74 54 83)(55 106 64 97)(56 105 65 96)(57 104 66 95)(58 103 67 94)(59 102 68 93)(60 101 69 92)(61 100 70 91)(62 99 71 108)(63 98 72 107)
G:=sub<Sym(108)| (1,13,7)(2,84,97)(3,104,79)(4,16,10)(5,87,100)(6,107,82)(8,90,103)(9,92,85)(11,75,106)(12,95,88)(14,78,91)(15,98,73)(17,81,94)(18,101,76)(19,38,67)(20,26,32)(21,63,46)(22,41,70)(23,29,35)(24,66,49)(25,44,55)(27,69,52)(28,47,58)(30,72,37)(31,50,61)(33,57,40)(34,53,64)(36,60,43)(39,45,51)(42,48,54)(56,62,68)(59,65,71)(74,86,80)(77,89,83)(93,105,99)(96,108,102), (1,89,108)(2,90,91)(3,73,92)(4,74,93)(5,75,94)(6,76,95)(7,77,96)(8,78,97)(9,79,98)(10,80,99)(11,81,100)(12,82,101)(13,83,102)(14,84,103)(15,85,104)(16,86,105)(17,87,106)(18,88,107)(19,55,50)(20,56,51)(21,57,52)(22,58,53)(23,59,54)(24,60,37)(25,61,38)(26,62,39)(27,63,40)(28,64,41)(29,65,42)(30,66,43)(31,67,44)(32,68,45)(33,69,46)(34,70,47)(35,71,48)(36,72,49), (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), (1,26,10,35)(2,25,11,34)(3,24,12,33)(4,23,13,32)(5,22,14,31)(6,21,15,30)(7,20,16,29)(8,19,17,28)(9,36,18,27)(37,82,46,73)(38,81,47,90)(39,80,48,89)(40,79,49,88)(41,78,50,87)(42,77,51,86)(43,76,52,85)(44,75,53,84)(45,74,54,83)(55,106,64,97)(56,105,65,96)(57,104,66,95)(58,103,67,94)(59,102,68,93)(60,101,69,92)(61,100,70,91)(62,99,71,108)(63,98,72,107)>;
G:=Group( (1,13,7)(2,84,97)(3,104,79)(4,16,10)(5,87,100)(6,107,82)(8,90,103)(9,92,85)(11,75,106)(12,95,88)(14,78,91)(15,98,73)(17,81,94)(18,101,76)(19,38,67)(20,26,32)(21,63,46)(22,41,70)(23,29,35)(24,66,49)(25,44,55)(27,69,52)(28,47,58)(30,72,37)(31,50,61)(33,57,40)(34,53,64)(36,60,43)(39,45,51)(42,48,54)(56,62,68)(59,65,71)(74,86,80)(77,89,83)(93,105,99)(96,108,102), (1,89,108)(2,90,91)(3,73,92)(4,74,93)(5,75,94)(6,76,95)(7,77,96)(8,78,97)(9,79,98)(10,80,99)(11,81,100)(12,82,101)(13,83,102)(14,84,103)(15,85,104)(16,86,105)(17,87,106)(18,88,107)(19,55,50)(20,56,51)(21,57,52)(22,58,53)(23,59,54)(24,60,37)(25,61,38)(26,62,39)(27,63,40)(28,64,41)(29,65,42)(30,66,43)(31,67,44)(32,68,45)(33,69,46)(34,70,47)(35,71,48)(36,72,49), (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), (1,26,10,35)(2,25,11,34)(3,24,12,33)(4,23,13,32)(5,22,14,31)(6,21,15,30)(7,20,16,29)(8,19,17,28)(9,36,18,27)(37,82,46,73)(38,81,47,90)(39,80,48,89)(40,79,49,88)(41,78,50,87)(42,77,51,86)(43,76,52,85)(44,75,53,84)(45,74,54,83)(55,106,64,97)(56,105,65,96)(57,104,66,95)(58,103,67,94)(59,102,68,93)(60,101,69,92)(61,100,70,91)(62,99,71,108)(63,98,72,107) );
G=PermutationGroup([[(1,13,7),(2,84,97),(3,104,79),(4,16,10),(5,87,100),(6,107,82),(8,90,103),(9,92,85),(11,75,106),(12,95,88),(14,78,91),(15,98,73),(17,81,94),(18,101,76),(19,38,67),(20,26,32),(21,63,46),(22,41,70),(23,29,35),(24,66,49),(25,44,55),(27,69,52),(28,47,58),(30,72,37),(31,50,61),(33,57,40),(34,53,64),(36,60,43),(39,45,51),(42,48,54),(56,62,68),(59,65,71),(74,86,80),(77,89,83),(93,105,99),(96,108,102)], [(1,89,108),(2,90,91),(3,73,92),(4,74,93),(5,75,94),(6,76,95),(7,77,96),(8,78,97),(9,79,98),(10,80,99),(11,81,100),(12,82,101),(13,83,102),(14,84,103),(15,85,104),(16,86,105),(17,87,106),(18,88,107),(19,55,50),(20,56,51),(21,57,52),(22,58,53),(23,59,54),(24,60,37),(25,61,38),(26,62,39),(27,63,40),(28,64,41),(29,65,42),(30,66,43),(31,67,44),(32,68,45),(33,69,46),(34,70,47),(35,71,48),(36,72,49)], [(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)], [(1,26,10,35),(2,25,11,34),(3,24,12,33),(4,23,13,32),(5,22,14,31),(6,21,15,30),(7,20,16,29),(8,19,17,28),(9,36,18,27),(37,82,46,73),(38,81,47,90),(39,80,48,89),(40,79,49,88),(41,78,50,87),(42,77,51,86),(43,76,52,85),(44,75,53,84),(45,74,54,83),(55,106,64,97),(56,105,65,96),(57,104,66,95),(58,103,67,94),(59,102,68,93),(60,101,69,92),(61,100,70,91),(62,99,71,108),(63,98,72,107)]])
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 9A | ··· | 9I | 12A | 12B | 12C | 12D | 18A | ··· | 18I |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 27 | 27 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | ··· | 6 | 27 | 27 | 27 | 27 | 6 | ··· | 6 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 |
| type | + | + | + | - | + | - | + | + | - | - | ||||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | S3 | Dic3 | D9 | C3×S3 | Dic9 | C3×Dic3 | C3×D9 | C3×Dic9 | C32⋊C6 | C9⋊C6 | C32⋊C12 | C9⋊C12 |
| kernel | C32⋊Dic9 | C2×C32⋊C9 | C9⋊Dic3 | C32⋊C9 | C3×C18 | C3×C9 | C32×C6 | C33 | C3×C6 | C3×C6 | C32 | C32 | C6 | C3 | C6 | C6 | C3 | C3 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 3 | 2 | 3 | 2 | 6 | 6 | 1 | 2 | 1 | 2 |
Matrix representation of C32⋊Dic9 ►in GL8(𝔽37)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 36 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 1 | 0 | 0 | 36 | 36 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 36 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 | 36 | 36 |
| 34 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 26 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 36 | 0 | 0 |
| 0 | 0 | 36 | 36 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 36 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 1 | 0 | 0 | 0 |
| 36 | 35 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 0 | 0 | 6 | 6 |
| 0 | 0 | 0 | 6 | 0 | 0 | 0 | 31 |
| 0 | 0 | 31 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 31 | 0 | 0 |
G:=sub<GL(8,GF(37))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,1,0,0,0,1,1,0,0,1,0,0,0,0,36,1,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,36,36,0,36,0,0,0,1,0,0,1,0,1,0,0,0,0,0,36,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36],[34,26,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,36,0,0,0,36,0,0,0,36,0,0,0,0,0,0,1,2,1,1,1,1,0,0,36,1,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0],[36,0,0,0,0,0,0,0,35,1,0,0,0,0,0,0,0,0,31,0,31,0,31,0,0,0,6,6,0,6,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,31,0,0,0,0,6,0,0,0,0,0,0,0,6,31,0,0] >;
C32⋊Dic9 in GAP, Magma, Sage, TeX
C_3^2\rtimes {\rm Dic}_9 % in TeX
G:=Group("C3^2:Dic9"); // GroupNames label
G:=SmallGroup(324,8);
// by ID
G=gap.SmallGroup(324,8);
# by ID
G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,3171,585,453,2164,7781]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^18=1,d^2=c^9,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations
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