direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D74, C2×D37, C74⋊C2, C37⋊C22, sometimes denoted D148 or Dih74 or Dih148, SmallGroup(148,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C37 — D74 |
Generators and relations for D74
G = < a,b | a74=b2=1, bab=a-1 >
Smallest permutation representation of D74
►On 74 points
Generators in S74
(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)
(1 74)(2 73)(3 72)(4 71)(5 70)(6 69)(7 68)(8 67)(9 66)(10 65)(11 64)(12 63)(13 62)(14 61)(15 60)(16 59)(17 58)(18 57)(19 56)(20 55)(21 54)(22 53)(23 52)(24 51)(25 50)(26 49)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 40)(36 39)(37 38)
G:=sub<Sym(74)| (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), (1,74)(2,73)(3,72)(4,71)(5,70)(6,69)(7,68)(8,67)(9,66)(10,65)(11,64)(12,63)(13,62)(14,61)(15,60)(16,59)(17,58)(18,57)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)>;
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), (1,74)(2,73)(3,72)(4,71)(5,70)(6,69)(7,68)(8,67)(9,66)(10,65)(11,64)(12,63)(13,62)(14,61)(15,60)(16,59)(17,58)(18,57)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38) );
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)], [(1,74),(2,73),(3,72),(4,71),(5,70),(6,69),(7,68),(8,67),(9,66),(10,65),(11,64),(12,63),(13,62),(14,61),(15,60),(16,59),(17,58),(18,57),(19,56),(20,55),(21,54),(22,53),(23,52),(24,51),(25,50),(26,49),(27,48),(28,47),(29,46),(30,45),(31,44),(32,43),(33,42),(34,41),(35,40),(36,39),(37,38)]])
D74 is a maximal subgroup of
D148 C37⋊D4
D74 is a maximal quotient of Dic74 D148 C37⋊D4
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 37A | ··· | 37R | 74A | ··· | 74R |
| order | 1 | 2 | 2 | 2 | 37 | ··· | 37 | 74 | ··· | 74 |
| size | 1 | 1 | 37 | 37 | 2 | ··· | 2 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | D37 | D74 |
| kernel | D74 | D37 | C74 | C2 | C1 |
| # reps | 1 | 2 | 1 | 18 | 18 |
Matrix representation of D74 ►in GL2(𝔽149) generated by
| 132 | 99 |
| 129 | 20 |
| 83 | 132 |
| 2 | 66 |
G:=sub<GL(2,GF(149))| [132,129,99,20],[83,2,132,66] >;
D74 in GAP, Magma, Sage, TeX
D_{74} % in TeX
G:=Group("D74"); // GroupNames label
G:=SmallGroup(148,4);
// by ID
G=gap.SmallGroup(148,4);
# by ID
G:=PCGroup([3,-2,-2,-37,1298]);
// Polycyclic
G:=Group<a,b|a^74=b^2=1,b*a*b=a^-1>;
// generators/relations
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