metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D109, C109:C2, sometimes denoted D218 or Dih109 or Dih218, SmallGroup(218,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C109 — D109 |
Generators and relations for D109
G = < a,b | a109=b2=1, bab=a-1 >
Quotients: C1, C2, D109
Smallest permutation representation of D109
►On 109 points: primitive
Generators in S109
(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)
(1 109)(2 108)(3 107)(4 106)(5 105)(6 104)(7 103)(8 102)(9 101)(10 100)(11 99)(12 98)(13 97)(14 96)(15 95)(16 94)(17 93)(18 92)(19 91)(20 90)(21 89)(22 88)(23 87)(24 86)(25 85)(26 84)(27 83)(28 82)(29 81)(30 80)(31 79)(32 78)(33 77)(34 76)(35 75)(36 74)(37 73)(38 72)(39 71)(40 70)(41 69)(42 68)(43 67)(44 66)(45 65)(46 64)(47 63)(48 62)(49 61)(50 60)(51 59)(52 58)(53 57)(54 56)
G:=sub<Sym(109)| (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), (1,109)(2,108)(3,107)(4,106)(5,105)(6,104)(7,103)(8,102)(9,101)(10,100)(11,99)(12,98)(13,97)(14,96)(15,95)(16,94)(17,93)(18,92)(19,91)(20,90)(21,89)(22,88)(23,87)(24,86)(25,85)(26,84)(27,83)(28,82)(29,81)(30,80)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,70)(41,69)(42,68)(43,67)(44,66)(45,65)(46,64)(47,63)(48,62)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109), (1,109)(2,108)(3,107)(4,106)(5,105)(6,104)(7,103)(8,102)(9,101)(10,100)(11,99)(12,98)(13,97)(14,96)(15,95)(16,94)(17,93)(18,92)(19,91)(20,90)(21,89)(22,88)(23,87)(24,86)(25,85)(26,84)(27,83)(28,82)(29,81)(30,80)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,70)(41,69)(42,68)(43,67)(44,66)(45,65)(46,64)(47,63)(48,62)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109)], [(1,109),(2,108),(3,107),(4,106),(5,105),(6,104),(7,103),(8,102),(9,101),(10,100),(11,99),(12,98),(13,97),(14,96),(15,95),(16,94),(17,93),(18,92),(19,91),(20,90),(21,89),(22,88),(23,87),(24,86),(25,85),(26,84),(27,83),(28,82),(29,81),(30,80),(31,79),(32,78),(33,77),(34,76),(35,75),(36,74),(37,73),(38,72),(39,71),(40,70),(41,69),(42,68),(43,67),(44,66),(45,65),(46,64),(47,63),(48,62),(49,61),(50,60),(51,59),(52,58),(53,57),(54,56)]])
D109 is a maximal subgroup of
C109:C4
D109 is a maximal quotient of Dic109
56 conjugacy classes
| class | 1 | 2 | 109A | ··· | 109BB |
| order | 1 | 2 | 109 | ··· | 109 |
| size | 1 | 109 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 2 |
| type | + | + | + |
| image | C1 | C2 | D109 |
| kernel | D109 | C109 | C1 |
| # reps | 1 | 1 | 54 |
Matrix representation of D109 ►in GL2(F1091) generated by
| 300 | 1090 |
| 1 | 0 |
| 300 | 1090 |
| 537 | 791 |
G:=sub<GL(2,GF(1091))| [300,1,1090,0],[300,537,1090,791] >;
D109 in GAP, Magma, Sage, TeX
D_{109} % in TeX
G:=Group("D109"); // GroupNames label
G:=SmallGroup(218,1);
// by ID
G=gap.SmallGroup(218,1);
# by ID
G:=PCGroup([2,-2,-109,865]);
// Polycyclic
G:=Group<a,b|a^109=b^2=1,b*a*b=a^-1>;
// generators/relations
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