direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C10×D11, C110⋊2C2, C22⋊3C10, C55⋊3C22, C11⋊3(C2×C10), SmallGroup(220,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C10×D11 |
Generators and relations for C10×D11
G = < a,b,c | a10=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C10×D11
►On 110 points
Generators in S110
(1 87 54 76 43 65 32 109 21 98)(2 88 55 77 44 66 33 110 22 99)(3 78 45 67 34 56 23 100 12 89)(4 79 46 68 35 57 24 101 13 90)(5 80 47 69 36 58 25 102 14 91)(6 81 48 70 37 59 26 103 15 92)(7 82 49 71 38 60 27 104 16 93)(8 83 50 72 39 61 28 105 17 94)(9 84 51 73 40 62 29 106 18 95)(10 85 52 74 41 63 30 107 19 96)(11 86 53 75 42 64 31 108 20 97)
(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)(100 101 102 103 104 105 106 107 108 109 110)
(1 64)(2 63)(3 62)(4 61)(5 60)(6 59)(7 58)(8 57)(9 56)(10 66)(11 65)(12 73)(13 72)(14 71)(15 70)(16 69)(17 68)(18 67)(19 77)(20 76)(21 75)(22 74)(23 84)(24 83)(25 82)(26 81)(27 80)(28 79)(29 78)(30 88)(31 87)(32 86)(33 85)(34 95)(35 94)(36 93)(37 92)(38 91)(39 90)(40 89)(41 99)(42 98)(43 97)(44 96)(45 106)(46 105)(47 104)(48 103)(49 102)(50 101)(51 100)(52 110)(53 109)(54 108)(55 107)
G:=sub<Sym(110)| (1,87,54,76,43,65,32,109,21,98)(2,88,55,77,44,66,33,110,22,99)(3,78,45,67,34,56,23,100,12,89)(4,79,46,68,35,57,24,101,13,90)(5,80,47,69,36,58,25,102,14,91)(6,81,48,70,37,59,26,103,15,92)(7,82,49,71,38,60,27,104,16,93)(8,83,50,72,39,61,28,105,17,94)(9,84,51,73,40,62,29,106,18,95)(10,85,52,74,41,63,30,107,19,96)(11,86,53,75,42,64,31,108,20,97), (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)(100,101,102,103,104,105,106,107,108,109,110), (1,64)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,66)(11,65)(12,73)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,77)(20,76)(21,75)(22,74)(23,84)(24,83)(25,82)(26,81)(27,80)(28,79)(29,78)(30,88)(31,87)(32,86)(33,85)(34,95)(35,94)(36,93)(37,92)(38,91)(39,90)(40,89)(41,99)(42,98)(43,97)(44,96)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,110)(53,109)(54,108)(55,107)>;
G:=Group( (1,87,54,76,43,65,32,109,21,98)(2,88,55,77,44,66,33,110,22,99)(3,78,45,67,34,56,23,100,12,89)(4,79,46,68,35,57,24,101,13,90)(5,80,47,69,36,58,25,102,14,91)(6,81,48,70,37,59,26,103,15,92)(7,82,49,71,38,60,27,104,16,93)(8,83,50,72,39,61,28,105,17,94)(9,84,51,73,40,62,29,106,18,95)(10,85,52,74,41,63,30,107,19,96)(11,86,53,75,42,64,31,108,20,97), (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)(100,101,102,103,104,105,106,107,108,109,110), (1,64)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,66)(11,65)(12,73)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,77)(20,76)(21,75)(22,74)(23,84)(24,83)(25,82)(26,81)(27,80)(28,79)(29,78)(30,88)(31,87)(32,86)(33,85)(34,95)(35,94)(36,93)(37,92)(38,91)(39,90)(40,89)(41,99)(42,98)(43,97)(44,96)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,110)(53,109)(54,108)(55,107) );
G=PermutationGroup([[(1,87,54,76,43,65,32,109,21,98),(2,88,55,77,44,66,33,110,22,99),(3,78,45,67,34,56,23,100,12,89),(4,79,46,68,35,57,24,101,13,90),(5,80,47,69,36,58,25,102,14,91),(6,81,48,70,37,59,26,103,15,92),(7,82,49,71,38,60,27,104,16,93),(8,83,50,72,39,61,28,105,17,94),(9,84,51,73,40,62,29,106,18,95),(10,85,52,74,41,63,30,107,19,96),(11,86,53,75,42,64,31,108,20,97)], [(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),(100,101,102,103,104,105,106,107,108,109,110)], [(1,64),(2,63),(3,62),(4,61),(5,60),(6,59),(7,58),(8,57),(9,56),(10,66),(11,65),(12,73),(13,72),(14,71),(15,70),(16,69),(17,68),(18,67),(19,77),(20,76),(21,75),(22,74),(23,84),(24,83),(25,82),(26,81),(27,80),(28,79),(29,78),(30,88),(31,87),(32,86),(33,85),(34,95),(35,94),(36,93),(37,92),(38,91),(39,90),(40,89),(41,99),(42,98),(43,97),(44,96),(45,106),(46,105),(47,104),(48,103),(49,102),(50,101),(51,100),(52,110),(53,109),(54,108),(55,107)]])
C10×D11 is a maximal subgroup of
C55⋊D4 C5⋊D44
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 11A | ··· | 11E | 22A | ··· | 22E | 55A | ··· | 55T | 110A | ··· | 110T |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 11 | ··· | 11 | 22 | ··· | 22 | 55 | ··· | 55 | 110 | ··· | 110 |
| size | 1 | 1 | 11 | 11 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 11 | ··· | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | D11 | D22 | C5×D11 | C10×D11 |
| kernel | C10×D11 | C5×D11 | C110 | D22 | D11 | C22 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 4 | 8 | 4 | 5 | 5 | 20 | 20 |
Matrix representation of C10×D11 ►in GL3(𝔽331) generated by
| 207 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 330 | 200 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(331))| [207,0,0,0,64,0,0,0,64],[1,0,0,0,0,330,0,1,200],[1,0,0,0,0,1,0,1,0] >;
C10×D11 in GAP, Magma, Sage, TeX
C_{10}\times D_{11} % in TeX
G:=Group("C10xD11"); // GroupNames label
G:=SmallGroup(220,12);
// by ID
G=gap.SmallGroup(220,12);
# by ID
G:=PCGroup([4,-2,-2,-5,-11,3203]);
// Polycyclic
G:=Group<a,b,c|a^10=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export