metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D119, C7⋊D17, C17⋊D7, C119⋊1C2, sometimes denoted D238 or Dih119 or Dih238, SmallGroup(238,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C119 — D119 |
Generators and relations for D119
G = < a,b | a119=b2=1, bab=a-1 >
Smallest permutation representation of D119
►On 119 points
Generators in S119
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119)
(1 119)(2 118)(3 117)(4 116)(5 115)(6 114)(7 113)(8 112)(9 111)(10 110)(11 109)(12 108)(13 107)(14 106)(15 105)(16 104)(17 103)(18 102)(19 101)(20 100)(21 99)(22 98)(23 97)(24 96)(25 95)(26 94)(27 93)(28 92)(29 91)(30 90)(31 89)(32 88)(33 87)(34 86)(35 85)(36 84)(37 83)(38 82)(39 81)(40 80)(41 79)(42 78)(43 77)(44 76)(45 75)(46 74)(47 73)(48 72)(49 71)(50 70)(51 69)(52 68)(53 67)(54 66)(55 65)(56 64)(57 63)(58 62)(59 61)
G:=sub<Sym(119)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119), (1,119)(2,118)(3,117)(4,116)(5,115)(6,114)(7,113)(8,112)(9,111)(10,110)(11,109)(12,108)(13,107)(14,106)(15,105)(16,104)(17,103)(18,102)(19,101)(20,100)(21,99)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,84)(37,83)(38,82)(39,81)(40,80)(41,79)(42,78)(43,77)(44,76)(45,75)(46,74)(47,73)(48,72)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119), (1,119)(2,118)(3,117)(4,116)(5,115)(6,114)(7,113)(8,112)(9,111)(10,110)(11,109)(12,108)(13,107)(14,106)(15,105)(16,104)(17,103)(18,102)(19,101)(20,100)(21,99)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,84)(37,83)(38,82)(39,81)(40,80)(41,79)(42,78)(43,77)(44,76)(45,75)(46,74)(47,73)(48,72)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119)], [(1,119),(2,118),(3,117),(4,116),(5,115),(6,114),(7,113),(8,112),(9,111),(10,110),(11,109),(12,108),(13,107),(14,106),(15,105),(16,104),(17,103),(18,102),(19,101),(20,100),(21,99),(22,98),(23,97),(24,96),(25,95),(26,94),(27,93),(28,92),(29,91),(30,90),(31,89),(32,88),(33,87),(34,86),(35,85),(36,84),(37,83),(38,82),(39,81),(40,80),(41,79),(42,78),(43,77),(44,76),(45,75),(46,74),(47,73),(48,72),(49,71),(50,70),(51,69),(52,68),(53,67),(54,66),(55,65),(56,64),(57,63),(58,62),(59,61)]])
D119 is a maximal subgroup of
D7×D17
D119 is a maximal quotient of Dic119
61 conjugacy classes
| class | 1 | 2 | 7A | 7B | 7C | 17A | ··· | 17H | 119A | ··· | 119AV |
| order | 1 | 2 | 7 | 7 | 7 | 17 | ··· | 17 | 119 | ··· | 119 |
| size | 1 | 119 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
61 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | D7 | D17 | D119 |
| kernel | D119 | C119 | C17 | C7 | C1 |
| # reps | 1 | 1 | 3 | 8 | 48 |
Matrix representation of D119 ►in GL2(𝔽239) generated by
| 141 | 19 |
| 220 | 150 |
| 141 | 19 |
| 199 | 98 |
G:=sub<GL(2,GF(239))| [141,220,19,150],[141,199,19,98] >;
D119 in GAP, Magma, Sage, TeX
D_{119} % in TeX
G:=Group("D119"); // GroupNames label
G:=SmallGroup(238,3);
// by ID
G=gap.SmallGroup(238,3);
# by ID
G:=PCGroup([3,-2,-7,-17,73,2018]);
// Polycyclic
G:=Group<a,b|a^119=b^2=1,b*a*b=a^-1>;
// generators/relations
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