direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D9×D11, D99⋊C2, C9⋊1D22, C99⋊C22, C33.D6, C11⋊1D18, (C9×D11)⋊C2, (C11×D9)⋊C2, C3.(S3×D11), (C3×D11).S3, SmallGroup(396,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C99 — D9×D11 |
Generators and relations for D9×D11
G = < a,b,c,d | a9=b2=c11=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of D9×D11
►On 99 points
Generators in S99
(1 91 58 35 30 86 53 18 68)(2 92 59 36 31 87 54 19 69)(3 93 60 37 32 88 55 20 70)(4 94 61 38 33 78 45 21 71)(5 95 62 39 23 79 46 22 72)(6 96 63 40 24 80 47 12 73)(7 97 64 41 25 81 48 13 74)(8 98 65 42 26 82 49 14 75)(9 99 66 43 27 83 50 15 76)(10 89 56 44 28 84 51 16 77)(11 90 57 34 29 85 52 17 67)
(1 68)(2 69)(3 70)(4 71)(5 72)(6 73)(7 74)(8 75)(9 76)(10 77)(11 67)(12 96)(13 97)(14 98)(15 99)(16 89)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(34 85)(35 86)(36 87)(37 88)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 61)(46 62)(47 63)(48 64)(49 65)(50 66)(51 56)(52 57)(53 58)(54 59)(55 60)
(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 11)(2 10)(3 9)(4 8)(5 7)(13 22)(14 21)(15 20)(16 19)(17 18)(23 25)(26 33)(27 32)(28 31)(29 30)(34 35)(36 44)(37 43)(38 42)(39 41)(45 49)(46 48)(50 55)(51 54)(52 53)(56 59)(57 58)(60 66)(61 65)(62 64)(67 68)(69 77)(70 76)(71 75)(72 74)(78 82)(79 81)(83 88)(84 87)(85 86)(89 92)(90 91)(93 99)(94 98)(95 97)
G:=sub<Sym(99)| (1,91,58,35,30,86,53,18,68)(2,92,59,36,31,87,54,19,69)(3,93,60,37,32,88,55,20,70)(4,94,61,38,33,78,45,21,71)(5,95,62,39,23,79,46,22,72)(6,96,63,40,24,80,47,12,73)(7,97,64,41,25,81,48,13,74)(8,98,65,42,26,82,49,14,75)(9,99,66,43,27,83,50,15,76)(10,89,56,44,28,84,51,16,77)(11,90,57,34,29,85,52,17,67), (1,68)(2,69)(3,70)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,67)(12,96)(13,97)(14,98)(15,99)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(34,85)(35,86)(36,87)(37,88)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66)(51,56)(52,57)(53,58)(54,59)(55,60), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(13,22)(14,21)(15,20)(16,19)(17,18)(23,25)(26,33)(27,32)(28,31)(29,30)(34,35)(36,44)(37,43)(38,42)(39,41)(45,49)(46,48)(50,55)(51,54)(52,53)(56,59)(57,58)(60,66)(61,65)(62,64)(67,68)(69,77)(70,76)(71,75)(72,74)(78,82)(79,81)(83,88)(84,87)(85,86)(89,92)(90,91)(93,99)(94,98)(95,97)>;
G:=Group( (1,91,58,35,30,86,53,18,68)(2,92,59,36,31,87,54,19,69)(3,93,60,37,32,88,55,20,70)(4,94,61,38,33,78,45,21,71)(5,95,62,39,23,79,46,22,72)(6,96,63,40,24,80,47,12,73)(7,97,64,41,25,81,48,13,74)(8,98,65,42,26,82,49,14,75)(9,99,66,43,27,83,50,15,76)(10,89,56,44,28,84,51,16,77)(11,90,57,34,29,85,52,17,67), (1,68)(2,69)(3,70)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,67)(12,96)(13,97)(14,98)(15,99)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(34,85)(35,86)(36,87)(37,88)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66)(51,56)(52,57)(53,58)(54,59)(55,60), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(13,22)(14,21)(15,20)(16,19)(17,18)(23,25)(26,33)(27,32)(28,31)(29,30)(34,35)(36,44)(37,43)(38,42)(39,41)(45,49)(46,48)(50,55)(51,54)(52,53)(56,59)(57,58)(60,66)(61,65)(62,64)(67,68)(69,77)(70,76)(71,75)(72,74)(78,82)(79,81)(83,88)(84,87)(85,86)(89,92)(90,91)(93,99)(94,98)(95,97) );
G=PermutationGroup([[(1,91,58,35,30,86,53,18,68),(2,92,59,36,31,87,54,19,69),(3,93,60,37,32,88,55,20,70),(4,94,61,38,33,78,45,21,71),(5,95,62,39,23,79,46,22,72),(6,96,63,40,24,80,47,12,73),(7,97,64,41,25,81,48,13,74),(8,98,65,42,26,82,49,14,75),(9,99,66,43,27,83,50,15,76),(10,89,56,44,28,84,51,16,77),(11,90,57,34,29,85,52,17,67)], [(1,68),(2,69),(3,70),(4,71),(5,72),(6,73),(7,74),(8,75),(9,76),(10,77),(11,67),(12,96),(13,97),(14,98),(15,99),(16,89),(17,90),(18,91),(19,92),(20,93),(21,94),(22,95),(34,85),(35,86),(36,87),(37,88),(38,78),(39,79),(40,80),(41,81),(42,82),(43,83),(44,84),(45,61),(46,62),(47,63),(48,64),(49,65),(50,66),(51,56),(52,57),(53,58),(54,59),(55,60)], [(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)], [(1,11),(2,10),(3,9),(4,8),(5,7),(13,22),(14,21),(15,20),(16,19),(17,18),(23,25),(26,33),(27,32),(28,31),(29,30),(34,35),(36,44),(37,43),(38,42),(39,41),(45,49),(46,48),(50,55),(51,54),(52,53),(56,59),(57,58),(60,66),(61,65),(62,64),(67,68),(69,77),(70,76),(71,75),(72,74),(78,82),(79,81),(83,88),(84,87),(85,86),(89,92),(90,91),(93,99),(94,98),(95,97)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 9A | 9B | 9C | 11A | ··· | 11E | 18A | 18B | 18C | 22A | ··· | 22E | 33A | ··· | 33E | 99A | ··· | 99O |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 9 | 9 | 9 | 11 | ··· | 11 | 18 | 18 | 18 | 22 | ··· | 22 | 33 | ··· | 33 | 99 | ··· | 99 |
| size | 1 | 9 | 11 | 99 | 2 | 22 | 2 | 2 | 2 | 2 | ··· | 2 | 22 | 22 | 22 | 18 | ··· | 18 | 4 | ··· | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | D6 | D9 | D11 | D18 | D22 | S3×D11 | D9×D11 |
| kernel | D9×D11 | C11×D9 | C9×D11 | D99 | C3×D11 | C33 | D11 | D9 | C11 | C9 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 5 | 3 | 5 | 5 | 15 |
Matrix representation of D9×D11 ►in GL4(𝔽199) generated by
| 57 | 108 | 0 | 0 |
| 91 | 148 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 57 | 108 | 0 | 0 |
| 51 | 142 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 198 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(199))| [57,91,0,0,108,148,0,0,0,0,1,0,0,0,0,1],[57,51,0,0,108,142,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,198,0,0,1,3],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;
D9×D11 in GAP, Magma, Sage, TeX
D_9\times D_{11} % in TeX
G:=Group("D9xD11"); // GroupNames label
G:=SmallGroup(396,5);
// by ID
G=gap.SmallGroup(396,5);
# by ID
G:=PCGroup([5,-2,-2,-3,-11,-3,1057,1002,2403,3309]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^2=c^11=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
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