metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D101, C101:C2, sometimes denoted D202 or Dih101 or Dih202, SmallGroup(202,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C101 — D101 |
Generators and relations for D101
G = < a,b | a101=b2=1, bab=a-1 >
Quotients: C1, C2, D101
Smallest permutation representation of D101
►On 101 points: primitive
Generators in S101
(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)
(1 101)(2 100)(3 99)(4 98)(5 97)(6 96)(7 95)(8 94)(9 93)(10 92)(11 91)(12 90)(13 89)(14 88)(15 87)(16 86)(17 85)(18 84)(19 83)(20 82)(21 81)(22 80)(23 79)(24 78)(25 77)(26 76)(27 75)(28 74)(29 73)(30 72)(31 71)(32 70)(33 69)(34 68)(35 67)(36 66)(37 65)(38 64)(39 63)(40 62)(41 61)(42 60)(43 59)(44 58)(45 57)(46 56)(47 55)(48 54)(49 53)(50 52)
G:=sub<Sym(101)| (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), (1,101)(2,100)(3,99)(4,98)(5,97)(6,96)(7,95)(8,94)(9,93)(10,92)(11,91)(12,90)(13,89)(14,88)(15,87)(16,86)(17,85)(18,84)(19,83)(20,82)(21,81)(22,80)(23,79)(24,78)(25,77)(26,76)(27,75)(28,74)(29,73)(30,72)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(37,65)(38,64)(39,63)(40,62)(41,61)(42,60)(43,59)(44,58)(45,57)(46,56)(47,55)(48,54)(49,53)(50,52)>;
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), (1,101)(2,100)(3,99)(4,98)(5,97)(6,96)(7,95)(8,94)(9,93)(10,92)(11,91)(12,90)(13,89)(14,88)(15,87)(16,86)(17,85)(18,84)(19,83)(20,82)(21,81)(22,80)(23,79)(24,78)(25,77)(26,76)(27,75)(28,74)(29,73)(30,72)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(37,65)(38,64)(39,63)(40,62)(41,61)(42,60)(43,59)(44,58)(45,57)(46,56)(47,55)(48,54)(49,53)(50,52) );
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)], [(1,101),(2,100),(3,99),(4,98),(5,97),(6,96),(7,95),(8,94),(9,93),(10,92),(11,91),(12,90),(13,89),(14,88),(15,87),(16,86),(17,85),(18,84),(19,83),(20,82),(21,81),(22,80),(23,79),(24,78),(25,77),(26,76),(27,75),(28,74),(29,73),(30,72),(31,71),(32,70),(33,69),(34,68),(35,67),(36,66),(37,65),(38,64),(39,63),(40,62),(41,61),(42,60),(43,59),(44,58),(45,57),(46,56),(47,55),(48,54),(49,53),(50,52)]])
D101 is a maximal subgroup of
C101:C4
D101 is a maximal quotient of Dic101
52 conjugacy classes
| class | 1 | 2 | 101A | ··· | 101AX |
| order | 1 | 2 | 101 | ··· | 101 |
| size | 1 | 101 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 2 |
| type | + | + | + |
| image | C1 | C2 | D101 |
| kernel | D101 | C101 | C1 |
| # reps | 1 | 1 | 50 |
Matrix representation of D101 ►in GL2(F607) generated by
| 425 | 606 |
| 1 | 0 |
| 425 | 606 |
| 345 | 182 |
G:=sub<GL(2,GF(607))| [425,1,606,0],[425,345,606,182] >;
D101 in GAP, Magma, Sage, TeX
D_{101} % in TeX
G:=Group("D101"); // GroupNames label
G:=SmallGroup(202,1);
// by ID
G=gap.SmallGroup(202,1);
# by ID
G:=PCGroup([2,-2,-101,801]);
// Polycyclic
G:=Group<a,b|a^101=b^2=1,b*a*b=a^-1>;
// generators/relations
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