metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D107, C107⋊C2, sometimes denoted D214 or Dih107 or Dih214, SmallGroup(214,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C107 — D107 |
Generators and relations for D107
G = < a,b | a107=b2=1, bab=a-1 >
Smallest permutation representation of D107
►On 107 points: primitive
Generators in S107
(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)
(1 107)(2 106)(3 105)(4 104)(5 103)(6 102)(7 101)(8 100)(9 99)(10 98)(11 97)(12 96)(13 95)(14 94)(15 93)(16 92)(17 91)(18 90)(19 89)(20 88)(21 87)(22 86)(23 85)(24 84)(25 83)(26 82)(27 81)(28 80)(29 79)(30 78)(31 77)(32 76)(33 75)(34 74)(35 73)(36 72)(37 71)(38 70)(39 69)(40 68)(41 67)(42 66)(43 65)(44 64)(45 63)(46 62)(47 61)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)
G:=sub<Sym(107)| (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), (1,107)(2,106)(3,105)(4,104)(5,103)(6,102)(7,101)(8,100)(9,99)(10,98)(11,97)(12,96)(13,95)(14,94)(15,93)(16,92)(17,91)(18,90)(19,89)(20,88)(21,87)(22,86)(23,85)(24,84)(25,83)(26,82)(27,81)(28,80)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,70)(39,69)(40,68)(41,67)(42,66)(43,65)(44,64)(45,63)(46,62)(47,61)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)>;
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), (1,107)(2,106)(3,105)(4,104)(5,103)(6,102)(7,101)(8,100)(9,99)(10,98)(11,97)(12,96)(13,95)(14,94)(15,93)(16,92)(17,91)(18,90)(19,89)(20,88)(21,87)(22,86)(23,85)(24,84)(25,83)(26,82)(27,81)(28,80)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,70)(39,69)(40,68)(41,67)(42,66)(43,65)(44,64)(45,63)(46,62)(47,61)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55) );
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)], [(1,107),(2,106),(3,105),(4,104),(5,103),(6,102),(7,101),(8,100),(9,99),(10,98),(11,97),(12,96),(13,95),(14,94),(15,93),(16,92),(17,91),(18,90),(19,89),(20,88),(21,87),(22,86),(23,85),(24,84),(25,83),(26,82),(27,81),(28,80),(29,79),(30,78),(31,77),(32,76),(33,75),(34,74),(35,73),(36,72),(37,71),(38,70),(39,69),(40,68),(41,67),(42,66),(43,65),(44,64),(45,63),(46,62),(47,61),(48,60),(49,59),(50,58),(51,57),(52,56),(53,55)]])
D107 is a maximal quotient of Dic107
55 conjugacy classes
| class | 1 | 2 | 107A | ··· | 107BA |
| order | 1 | 2 | 107 | ··· | 107 |
| size | 1 | 107 | 2 | ··· | 2 |
55 irreducible representations
| dim | 1 | 1 | 2 |
| type | + | + | + |
| image | C1 | C2 | D107 |
| kernel | D107 | C107 | C1 |
| # reps | 1 | 1 | 53 |
Matrix representation of D107 ►in GL2(𝔽643) generated by
| 92 | 642 |
| 1 | 0 |
| 92 | 642 |
| 104 | 551 |
G:=sub<GL(2,GF(643))| [92,1,642,0],[92,104,642,551] >;
D107 in GAP, Magma, Sage, TeX
D_{107} % in TeX
G:=Group("D107"); // GroupNames label
G:=SmallGroup(214,1);
// by ID
G=gap.SmallGroup(214,1);
# by ID
G:=PCGroup([2,-2,-107,849]);
// Polycyclic
G:=Group<a,b|a^107=b^2=1,b*a*b=a^-1>;
// generators/relations
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