direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D114, C2×D57, C38⋊S3, C6⋊D19, C19⋊2D6, C3⋊2D38, C114⋊1C2, C57⋊2C22, sometimes denoted D228 or Dih114 or Dih228, SmallGroup(228,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C57 — D114 |
Generators and relations for D114
G = < a,b | a114=b2=1, bab=a-1 >
Smallest permutation representation of D114
►On 114 points
Generators in S114
(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)
(1 114)(2 113)(3 112)(4 111)(5 110)(6 109)(7 108)(8 107)(9 106)(10 105)(11 104)(12 103)(13 102)(14 101)(15 100)(16 99)(17 98)(18 97)(19 96)(20 95)(21 94)(22 93)(23 92)(24 91)(25 90)(26 89)(27 88)(28 87)(29 86)(30 85)(31 84)(32 83)(33 82)(34 81)(35 80)(36 79)(37 78)(38 77)(39 76)(40 75)(41 74)(42 73)(43 72)(44 71)(45 70)(46 69)(47 68)(48 67)(49 66)(50 65)(51 64)(52 63)(53 62)(54 61)(55 60)(56 59)(57 58)
G:=sub<Sym(114)| (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), (1,114)(2,113)(3,112)(4,111)(5,110)(6,109)(7,108)(8,107)(9,106)(10,105)(11,104)(12,103)(13,102)(14,101)(15,100)(16,99)(17,98)(18,97)(19,96)(20,95)(21,94)(22,93)(23,92)(24,91)(25,90)(26,89)(27,88)(28,87)(29,86)(30,85)(31,84)(32,83)(33,82)(34,81)(35,80)(36,79)(37,78)(38,77)(39,76)(40,75)(41,74)(42,73)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)>;
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), (1,114)(2,113)(3,112)(4,111)(5,110)(6,109)(7,108)(8,107)(9,106)(10,105)(11,104)(12,103)(13,102)(14,101)(15,100)(16,99)(17,98)(18,97)(19,96)(20,95)(21,94)(22,93)(23,92)(24,91)(25,90)(26,89)(27,88)(28,87)(29,86)(30,85)(31,84)(32,83)(33,82)(34,81)(35,80)(36,79)(37,78)(38,77)(39,76)(40,75)(41,74)(42,73)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58) );
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)], [(1,114),(2,113),(3,112),(4,111),(5,110),(6,109),(7,108),(8,107),(9,106),(10,105),(11,104),(12,103),(13,102),(14,101),(15,100),(16,99),(17,98),(18,97),(19,96),(20,95),(21,94),(22,93),(23,92),(24,91),(25,90),(26,89),(27,88),(28,87),(29,86),(30,85),(31,84),(32,83),(33,82),(34,81),(35,80),(36,79),(37,78),(38,77),(39,76),(40,75),(41,74),(42,73),(43,72),(44,71),(45,70),(46,69),(47,68),(48,67),(49,66),(50,65),(51,64),(52,63),(53,62),(54,61),(55,60),(56,59),(57,58)]])
D114 is a maximal subgroup of
D57⋊C4 C3⋊D76 C19⋊D12 D228 C57⋊7D4 C2×S3×D19
D114 is a maximal quotient of Dic114 D228 C57⋊7D4
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 19A | ··· | 19I | 38A | ··· | 38I | 57A | ··· | 57R | 114A | ··· | 114R |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 114 | ··· | 114 |
| size | 1 | 1 | 57 | 57 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D6 | D19 | D38 | D57 | D114 |
| kernel | D114 | D57 | C114 | C38 | C19 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 9 | 9 | 18 | 18 |
Matrix representation of D114 ►in GL3(𝔽229) generated by
| 228 | 0 | 0 |
| 0 | 175 | 132 |
| 0 | 197 | 57 |
| 1 | 0 | 0 |
| 0 | 86 | 147 |
| 0 | 160 | 143 |
G:=sub<GL(3,GF(229))| [228,0,0,0,175,197,0,132,57],[1,0,0,0,86,160,0,147,143] >;
D114 in GAP, Magma, Sage, TeX
D_{114} % in TeX
G:=Group("D114"); // GroupNames label
G:=SmallGroup(228,14);
// by ID
G=gap.SmallGroup(228,14);
# by ID
G:=PCGroup([4,-2,-2,-3,-19,98,3459]);
// Polycyclic
G:=Group<a,b|a^114=b^2=1,b*a*b=a^-1>;
// generators/relations
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