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## G = D91order 182 = 2·7·13

### Dihedral group

Aliases: D91, C13⋊D7, C7⋊D13, C911C2, sometimes denoted D182 or Dih91 or Dih182, SmallGroup(182,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C91 — D91
 Chief series C1 — C13 — C91 — D91
 Lower central C91 — D91
 Upper central C1

Generators and relations for D91
G = < a,b | a91=b2=1, bab=a-1 >

91C2
13D7
7D13

Smallest permutation representation of D91
On 91 points
Generators in S91
(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)
(1 91)(2 90)(3 89)(4 88)(5 87)(6 86)(7 85)(8 84)(9 83)(10 82)(11 81)(12 80)(13 79)(14 78)(15 77)(16 76)(17 75)(18 74)(19 73)(20 72)(21 71)(22 70)(23 69)(24 68)(25 67)(26 66)(27 65)(28 64)(29 63)(30 62)(31 61)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)

G:=sub<Sym(91)| (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), (1,91)(2,90)(3,89)(4,88)(5,87)(6,86)(7,85)(8,84)(9,83)(10,82)(11,81)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,73)(20,72)(21,71)(22,70)(23,69)(24,68)(25,67)(26,66)(27,65)(28,64)(29,63)(30,62)(31,61)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)>;

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), (1,91)(2,90)(3,89)(4,88)(5,87)(6,86)(7,85)(8,84)(9,83)(10,82)(11,81)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,73)(20,72)(21,71)(22,70)(23,69)(24,68)(25,67)(26,66)(27,65)(28,64)(29,63)(30,62)(31,61)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47) );

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)], [(1,91),(2,90),(3,89),(4,88),(5,87),(6,86),(7,85),(8,84),(9,83),(10,82),(11,81),(12,80),(13,79),(14,78),(15,77),(16,76),(17,75),(18,74),(19,73),(20,72),(21,71),(22,70),(23,69),(24,68),(25,67),(26,66),(27,65),(28,64),(29,63),(30,62),(31,61),(32,60),(33,59),(34,58),(35,57),(36,56),(37,55),(38,54),(39,53),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47)]])

D91 is a maximal subgroup of   D7×D13
D91 is a maximal quotient of   Dic91

47 conjugacy classes

 class 1 2 7A 7B 7C 13A ··· 13F 91A ··· 91AJ order 1 2 7 7 7 13 ··· 13 91 ··· 91 size 1 91 2 2 2 2 ··· 2 2 ··· 2

47 irreducible representations

 dim 1 1 2 2 2 type + + + + + image C1 C2 D7 D13 D91 kernel D91 C91 C13 C7 C1 # reps 1 1 3 6 36

Matrix representation of D91 in GL2(𝔽547) generated by

 219 77 470 445
,
 219 77 393 328
G:=sub<GL(2,GF(547))| [219,470,77,445],[219,393,77,328] >;

D91 in GAP, Magma, Sage, TeX

D_{91}
% in TeX

G:=Group("D91");
// GroupNames label

G:=SmallGroup(182,3);
// by ID

G=gap.SmallGroup(182,3);
# by ID

G:=PCGroup([3,-2,-7,-13,73,1514]);
// Polycyclic

G:=Group<a,b|a^91=b^2=1,b*a*b=a^-1>;
// generators/relations

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