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G = D84order 168 = 23·3·7

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D84, C4⋊D21, C214D4, C31D28, C71D12, C841C2, C281S3, C121D7, D421C2, C2.4D42, C14.10D6, C6.10D14, C42.10C22, sometimes denoted D168 or Dih84 or Dih168, SmallGroup(168,36)

Series: Derived Chief Lower central Upper central

C1C42 — D84
C1C7C21C42D42 — D84
C21C42 — D84
C1C2C4

Generators and relations for D84
 G = < a,b | a84=b2=1, bab=a-1 >

42C2
42C2
21C22
21C22
14S3
14S3
6D7
6D7
21D4
7D6
7D6
3D14
3D14
2D21
2D21
7D12
3D28

Smallest permutation representation of D84
On 84 points
Generators in S84
(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)
(1 84)(2 83)(3 82)(4 81)(5 80)(6 79)(7 78)(8 77)(9 76)(10 75)(11 74)(12 73)(13 72)(14 71)(15 70)(16 69)(17 68)(18 67)(19 66)(20 65)(21 64)(22 63)(23 62)(24 61)(25 60)(26 59)(27 58)(28 57)(29 56)(30 55)(31 54)(32 53)(33 52)(34 51)(35 50)(36 49)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)

G:=sub<Sym(84)| (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), (1,84)(2,83)(3,82)(4,81)(5,80)(6,79)(7,78)(8,77)(9,76)(10,75)(11,74)(12,73)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,66)(20,65)(21,64)(22,63)(23,62)(24,61)(25,60)(26,59)(27,58)(28,57)(29,56)(30,55)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)>;

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), (1,84)(2,83)(3,82)(4,81)(5,80)(6,79)(7,78)(8,77)(9,76)(10,75)(11,74)(12,73)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,66)(20,65)(21,64)(22,63)(23,62)(24,61)(25,60)(26,59)(27,58)(28,57)(29,56)(30,55)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43) );

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)], [(1,84),(2,83),(3,82),(4,81),(5,80),(6,79),(7,78),(8,77),(9,76),(10,75),(11,74),(12,73),(13,72),(14,71),(15,70),(16,69),(17,68),(18,67),(19,66),(20,65),(21,64),(22,63),(23,62),(24,61),(25,60),(26,59),(27,58),(28,57),(29,56),(30,55),(31,54),(32,53),(33,52),(34,51),(35,50),(36,49),(37,48),(38,47),(39,46),(40,45),(41,44),(42,43)])

D84 is a maximal subgroup of
C3⋊D56  C7⋊D24  C21⋊SD16  Dic6⋊D7  C8⋊D21  D168  D4⋊D21  Q82D21  D84⋊C2  D14.D6  D7×D12  S3×D28  D8411C2  D4×D21  Q83D21
D84 is a maximal quotient of
C8⋊D21  D168  Dic84  C84⋊C4  C2.D84

45 conjugacy classes

class 1 2A2B2C 3  4  6 7A7B7C12A12B14A14B14C21A···21F28A···28F42A···42F84A···84L
order1222346777121214141421···2128···2842···4284···84
size114242222222222222···22···22···22···2

45 irreducible representations

dim1112222222222
type+++++++++++++
imageC1C2C2S3D4D6D7D12D14D21D28D42D84
kernelD84C84D42C28C21C14C12C7C6C4C3C2C1
# reps11211132366612

Matrix representation of D84 in GL4(𝔽337) generated by

19422800
10910900
0029293
003080
,
19422800
33614300
0029293
0027945
G:=sub<GL(4,GF(337))| [194,109,0,0,228,109,0,0,0,0,292,308,0,0,93,0],[194,336,0,0,228,143,0,0,0,0,292,279,0,0,93,45] >;

D84 in GAP, Magma, Sage, TeX

D_{84}
% in TeX

G:=Group("D84");
// GroupNames label

G:=SmallGroup(168,36);
// by ID

G=gap.SmallGroup(168,36);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-7,61,26,323,3604]);
// Polycyclic

G:=Group<a,b|a^84=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D84 in TeX

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