metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D105, C3⋊D35, C5⋊D21, C7⋊D15, C35⋊1S3, C21⋊1D5, C15⋊1D7, C105⋊1C2, sometimes denoted D210 or Dih105 or Dih210, SmallGroup(210,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C105 — D105 |
Generators and relations for D105
G = < a,b | a105=b2=1, bab=a-1 >
Smallest permutation representation of D105
►On 105 points
Generators in S105
(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)
(2 105)(3 104)(4 103)(5 102)(6 101)(7 100)(8 99)(9 98)(10 97)(11 96)(12 95)(13 94)(14 93)(15 92)(16 91)(17 90)(18 89)(19 88)(20 87)(21 86)(22 85)(23 84)(24 83)(25 82)(26 81)(27 80)(28 79)(29 78)(30 77)(31 76)(32 75)(33 74)(34 73)(35 72)(36 71)(37 70)(38 69)(39 68)(40 67)(41 66)(42 65)(43 64)(44 63)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)
G:=sub<Sym(105)| (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), (2,105)(3,104)(4,103)(5,102)(6,101)(7,100)(8,99)(9,98)(10,97)(11,96)(12,95)(13,94)(14,93)(15,92)(16,91)(17,90)(18,89)(19,88)(20,87)(21,86)(22,85)(23,84)(24,83)(25,82)(26,81)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,74)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)>;
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), (2,105)(3,104)(4,103)(5,102)(6,101)(7,100)(8,99)(9,98)(10,97)(11,96)(12,95)(13,94)(14,93)(15,92)(16,91)(17,90)(18,89)(19,88)(20,87)(21,86)(22,85)(23,84)(24,83)(25,82)(26,81)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,74)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54) );
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)], [(2,105),(3,104),(4,103),(5,102),(6,101),(7,100),(8,99),(9,98),(10,97),(11,96),(12,95),(13,94),(14,93),(15,92),(16,91),(17,90),(18,89),(19,88),(20,87),(21,86),(22,85),(23,84),(24,83),(25,82),(26,81),(27,80),(28,79),(29,78),(30,77),(31,76),(32,75),(33,74),(34,73),(35,72),(36,71),(37,70),(38,69),(39,68),(40,67),(41,66),(42,65),(43,64),(44,63),(45,62),(46,61),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,54)]])
D105 is a maximal subgroup of
D7×D15 D5×D21 S3×D35
D105 is a maximal quotient of Dic105
54 conjugacy classes
| class | 1 | 2 | 3 | 5A | 5B | 7A | 7B | 7C | 15A | 15B | 15C | 15D | 21A | ··· | 21F | 35A | ··· | 35L | 105A | ··· | 105X |
| order | 1 | 2 | 3 | 5 | 5 | 7 | 7 | 7 | 15 | 15 | 15 | 15 | 21 | ··· | 21 | 35 | ··· | 35 | 105 | ··· | 105 |
| size | 1 | 105 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
54 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | S3 | D5 | D7 | D15 | D21 | D35 | D105 |
| kernel | D105 | C105 | C35 | C21 | C15 | C7 | C5 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 3 | 4 | 6 | 12 | 24 |
Matrix representation of D105 ►in GL2(𝔽211) generated by
| 181 | 42 |
| 121 | 133 |
| 0 | 193 |
| 82 | 0 |
G:=sub<GL(2,GF(211))| [181,121,42,133],[0,82,193,0] >;
D105 in GAP, Magma, Sage, TeX
D_{105} % in TeX
G:=Group("D105"); // GroupNames label
G:=SmallGroup(210,11);
// by ID
G=gap.SmallGroup(210,11);
# by ID
G:=PCGroup([4,-2,-3,-5,-7,33,290,2883]);
// Polycyclic
G:=Group<a,b|a^105=b^2=1,b*a*b=a^-1>;
// generators/relations
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