direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C11×D9, C9⋊C22, C99⋊3C2, C33.2S3, C3.(S3×C11), SmallGroup(198,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C11×D9 |
Generators and relations for C11×D9
G = < a,b,c | a11=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C11×D9
►On 99 points
Generators in S99
(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)
(1 71 22 32 41 45 60 90 79)(2 72 12 33 42 46 61 91 80)(3 73 13 23 43 47 62 92 81)(4 74 14 24 44 48 63 93 82)(5 75 15 25 34 49 64 94 83)(6 76 16 26 35 50 65 95 84)(7 77 17 27 36 51 66 96 85)(8 67 18 28 37 52 56 97 86)(9 68 19 29 38 53 57 98 87)(10 69 20 30 39 54 58 99 88)(11 70 21 31 40 55 59 89 78)
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 78)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 56)(19 57)(20 58)(21 59)(22 60)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 45)(33 46)(67 97)(68 98)(69 99)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)
G:=sub<Sym(99)| (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), (1,71,22,32,41,45,60,90,79)(2,72,12,33,42,46,61,91,80)(3,73,13,23,43,47,62,92,81)(4,74,14,24,44,48,63,93,82)(5,75,15,25,34,49,64,94,83)(6,76,16,26,35,50,65,95,84)(7,77,17,27,36,51,66,96,85)(8,67,18,28,37,52,56,97,86)(9,68,19,29,38,53,57,98,87)(10,69,20,30,39,54,58,99,88)(11,70,21,31,40,55,59,89,78), (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,78)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,56)(19,57)(20,58)(21,59)(22,60)(23,47)(24,48)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,45)(33,46)(67,97)(68,98)(69,99)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)>;
G:=Group( (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), (1,71,22,32,41,45,60,90,79)(2,72,12,33,42,46,61,91,80)(3,73,13,23,43,47,62,92,81)(4,74,14,24,44,48,63,93,82)(5,75,15,25,34,49,64,94,83)(6,76,16,26,35,50,65,95,84)(7,77,17,27,36,51,66,96,85)(8,67,18,28,37,52,56,97,86)(9,68,19,29,38,53,57,98,87)(10,69,20,30,39,54,58,99,88)(11,70,21,31,40,55,59,89,78), (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,78)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,56)(19,57)(20,58)(21,59)(22,60)(23,47)(24,48)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,45)(33,46)(67,97)(68,98)(69,99)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96) );
G=PermutationGroup([[(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)], [(1,71,22,32,41,45,60,90,79),(2,72,12,33,42,46,61,91,80),(3,73,13,23,43,47,62,92,81),(4,74,14,24,44,48,63,93,82),(5,75,15,25,34,49,64,94,83),(6,76,16,26,35,50,65,95,84),(7,77,17,27,36,51,66,96,85),(8,67,18,28,37,52,56,97,86),(9,68,19,29,38,53,57,98,87),(10,69,20,30,39,54,58,99,88),(11,70,21,31,40,55,59,89,78)], [(1,79),(2,80),(3,81),(4,82),(5,83),(6,84),(7,85),(8,86),(9,87),(10,88),(11,78),(12,61),(13,62),(14,63),(15,64),(16,65),(17,66),(18,56),(19,57),(20,58),(21,59),(22,60),(23,47),(24,48),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,45),(33,46),(67,97),(68,98),(69,99),(70,89),(71,90),(72,91),(73,92),(74,93),(75,94),(76,95),(77,96)]])
66 conjugacy classes
| class | 1 | 2 | 3 | 9A | 9B | 9C | 11A | ··· | 11J | 22A | ··· | 22J | 33A | ··· | 33J | 99A | ··· | 99AD |
| order | 1 | 2 | 3 | 9 | 9 | 9 | 11 | ··· | 11 | 22 | ··· | 22 | 33 | ··· | 33 | 99 | ··· | 99 |
| size | 1 | 9 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C11 | C22 | S3 | D9 | S3×C11 | C11×D9 |
| kernel | C11×D9 | C99 | D9 | C9 | C33 | C11 | C3 | C1 |
| # reps | 1 | 1 | 10 | 10 | 1 | 3 | 10 | 30 |
Matrix representation of C11×D9 ►in GL2(𝔽199) generated by
| 188 | 0 |
| 0 | 188 |
| 108 | 142 |
| 57 | 51 |
| 57 | 91 |
| 148 | 142 |
G:=sub<GL(2,GF(199))| [188,0,0,188],[108,57,142,51],[57,148,91,142] >;
C11×D9 in GAP, Magma, Sage, TeX
C_{11}\times D_9 % in TeX
G:=Group("C11xD9"); // GroupNames label
G:=SmallGroup(198,1);
// by ID
G=gap.SmallGroup(198,1);
# by ID
G:=PCGroup([4,-2,-11,-3,-3,1322,82,2115]);
// Polycyclic
G:=Group<a,b,c|a^11=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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