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G = D75order 150 = 2·3·52

Dihedral group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D75, C25⋊S3, C3⋊D25, C751C2, C5.D15, C15.1D5, sometimes denoted D150 or Dih75 or Dih150, SmallGroup(150,3)

Series: Derived Chief Lower central Upper central

C1C75 — D75
C1C5C25C75 — D75
C75 — D75
C1

Generators and relations for D75
 G = < a,b | a75=b2=1, bab=a-1 >

75C2
25S3
15D5
5D15
3D25

Smallest permutation representation of D75
On 75 points
Generators in S75
(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)
(2 75)(3 74)(4 73)(5 72)(6 71)(7 70)(8 69)(9 68)(10 67)(11 66)(12 65)(13 64)(14 63)(15 62)(16 61)(17 60)(18 59)(19 58)(20 57)(21 56)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)

G:=sub<Sym(75)| (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), (2,75)(3,74)(4,73)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)>;

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), (2,75)(3,74)(4,73)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39) );

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)], [(2,75),(3,74),(4,73),(5,72),(6,71),(7,70),(8,69),(9,68),(10,67),(11,66),(12,65),(13,64),(14,63),(15,62),(16,61),(17,60),(18,59),(19,58),(20,57),(21,56),(22,55),(23,54),(24,53),(25,52),(26,51),(27,50),(28,49),(29,48),(30,47),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41),(37,40),(38,39)])

D75 is a maximal subgroup of   S3×D25  D225  C3⋊D75
D75 is a maximal quotient of   Dic75  D225  C3⋊D75

39 conjugacy classes

class 1  2  3 5A5B15A15B15C15D25A···25J75A···75T
order123551515151525···2575···75
size17522222222···22···2

39 irreducible representations

dim1122222
type+++++++
imageC1C2S3D5D15D25D75
kernelD75C75C25C15C5C3C1
# reps111241020

Matrix representation of D75 in GL2(𝔽151) generated by

11291
60131
,
10
123150
G:=sub<GL(2,GF(151))| [112,60,91,131],[1,123,0,150] >;

D75 in GAP, Magma, Sage, TeX

D_{75}
% in TeX

G:=Group("D75");
// GroupNames label

G:=SmallGroup(150,3);
// by ID

G=gap.SmallGroup(150,3);
# by ID

G:=PCGroup([4,-2,-3,-5,-5,33,650,250,1923]);
// Polycyclic

G:=Group<a,b|a^75=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D75 in TeX

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