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G = D74order 148 = 22·37

Dihedral group

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

C1C37 — D74
C1C37D37 — D74
C37 — D74
C1C2

Generators and relations for D74
 G = < a,b | a74=b2=1, bab=a-1 >

37C2
37C2
37C22

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 2A2B2C37A···37R74A···74R
order122237···3774···74
size1137372···22···2

40 irreducible representations

dim11122
type+++++
imageC1C2C2D37D74
kernelD74D37C74C2C1
# reps1211818

Matrix representation of D74 in GL2(𝔽149) generated by

13299
12920
,
83132
266
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

Export

Subgroup lattice of D74 in TeX

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