direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C9xD11, C11:C18, C99:2C2, C33.C6, C3.(C3xD11), (C3xD11).C3, SmallGroup(198,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C9xD11 |
Generators and relations for C9xD11
G = < a,b,c | a9=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >
Quotients: C1, C2, C3, C6, C9, C18, D11, C3xD11, C9xD11
Smallest permutation representation of C9xD11
►On 99 points
Generators in S99
(1 98 65 32 87 54 21 76 43)(2 99 66 33 88 55 22 77 44)(3 89 56 23 78 45 12 67 34)(4 90 57 24 79 46 13 68 35)(5 91 58 25 80 47 14 69 36)(6 92 59 26 81 48 15 70 37)(7 93 60 27 82 49 16 71 38)(8 94 61 28 83 50 17 72 39)(9 95 62 29 84 51 18 73 40)(10 96 63 30 85 52 19 74 41)(11 97 64 31 86 53 20 75 42)
(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)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 51)(46 50)(47 49)(52 55)(53 54)(56 62)(57 61)(58 60)(63 66)(64 65)(67 73)(68 72)(69 71)(74 77)(75 76)(78 84)(79 83)(80 82)(85 88)(86 87)(89 95)(90 94)(91 93)(96 99)(97 98)
G:=sub<Sym(99)| (1,98,65,32,87,54,21,76,43)(2,99,66,33,88,55,22,77,44)(3,89,56,23,78,45,12,67,34)(4,90,57,24,79,46,13,68,35)(5,91,58,25,80,47,14,69,36)(6,92,59,26,81,48,15,70,37)(7,93,60,27,82,49,16,71,38)(8,94,61,28,83,50,17,72,39)(9,95,62,29,84,51,18,73,40)(10,96,63,30,85,52,19,74,41)(11,97,64,31,86,53,20,75,42), (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)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)(56,62)(57,61)(58,60)(63,66)(64,65)(67,73)(68,72)(69,71)(74,77)(75,76)(78,84)(79,83)(80,82)(85,88)(86,87)(89,95)(90,94)(91,93)(96,99)(97,98)>;
G:=Group( (1,98,65,32,87,54,21,76,43)(2,99,66,33,88,55,22,77,44)(3,89,56,23,78,45,12,67,34)(4,90,57,24,79,46,13,68,35)(5,91,58,25,80,47,14,69,36)(6,92,59,26,81,48,15,70,37)(7,93,60,27,82,49,16,71,38)(8,94,61,28,83,50,17,72,39)(9,95,62,29,84,51,18,73,40)(10,96,63,30,85,52,19,74,41)(11,97,64,31,86,53,20,75,42), (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)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)(56,62)(57,61)(58,60)(63,66)(64,65)(67,73)(68,72)(69,71)(74,77)(75,76)(78,84)(79,83)(80,82)(85,88)(86,87)(89,95)(90,94)(91,93)(96,99)(97,98) );
G=PermutationGroup([[(1,98,65,32,87,54,21,76,43),(2,99,66,33,88,55,22,77,44),(3,89,56,23,78,45,12,67,34),(4,90,57,24,79,46,13,68,35),(5,91,58,25,80,47,14,69,36),(6,92,59,26,81,48,15,70,37),(7,93,60,27,82,49,16,71,38),(8,94,61,28,83,50,17,72,39),(9,95,62,29,84,51,18,73,40),(10,96,63,30,85,52,19,74,41),(11,97,64,31,86,53,20,75,42)], [(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),(12,18),(13,17),(14,16),(19,22),(20,21),(23,29),(24,28),(25,27),(30,33),(31,32),(34,40),(35,39),(36,38),(41,44),(42,43),(45,51),(46,50),(47,49),(52,55),(53,54),(56,62),(57,61),(58,60),(63,66),(64,65),(67,73),(68,72),(69,71),(74,77),(75,76),(78,84),(79,83),(80,82),(85,88),(86,87),(89,95),(90,94),(91,93),(96,99),(97,98)]])
63 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 9A | ··· | 9F | 11A | ··· | 11E | 18A | ··· | 18F | 33A | ··· | 33J | 99A | ··· | 99AD |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 11 | ··· | 11 | 18 | ··· | 18 | 33 | ··· | 33 | 99 | ··· | 99 |
| size | 1 | 11 | 1 | 1 | 11 | 11 | 1 | ··· | 1 | 2 | ··· | 2 | 11 | ··· | 11 | 2 | ··· | 2 | 2 | ··· | 2 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | D11 | C3xD11 | C9xD11 |
| kernel | C9xD11 | C99 | C3xD11 | C33 | D11 | C11 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 5 | 10 | 30 |
Matrix representation of C9xD11 ►in GL2(F199) generated by
| 178 | 0 |
| 0 | 178 |
| 123 | 1 |
| 198 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(199))| [178,0,0,178],[123,198,1,0],[0,1,1,0] >;
C9xD11 in GAP, Magma, Sage, TeX
C_9\times D_{11} % in TeX
G:=Group("C9xD11"); // GroupNames label
G:=SmallGroup(198,2);
// by ID
G=gap.SmallGroup(198,2);
# by ID
G:=PCGroup([4,-2,-3,-3,-11,29,2883]);
// Polycyclic
G:=Group<a,b,c|a^9=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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