metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D84, C4⋊D21, C21⋊4D4, C3⋊1D28, C7⋊1D12, C84⋊1C2, C28⋊1S3, C12⋊1D7, D42⋊1C2, 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
Generators and relations for D84
G = < a,b | a84=b2=1, bab=a-1 >
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 Q8⋊2D21 D84⋊C2 D14.D6 D7×D12 S3×D28 D84⋊11C2 D4×D21 Q8⋊3D21
D84 is a maximal quotient of
C8⋊D21 D168 Dic84 C84⋊C4 C2.D84
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6 | 7A | 7B | 7C | 12A | 12B | 14A | 14B | 14C | 21A | ··· | 21F | 28A | ··· | 28F | 42A | ··· | 42F | 84A | ··· | 84L |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 7 | 7 | 7 | 12 | 12 | 14 | 14 | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 42 | 42 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
45 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D4 | D6 | D7 | D12 | D14 | D21 | D28 | D42 | D84 |
| kernel | D84 | C84 | D42 | C28 | C21 | C14 | C12 | C7 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 3 | 2 | 3 | 6 | 6 | 6 | 12 |
Matrix representation of D84 ►in GL4(𝔽337) generated by
| 194 | 228 | 0 | 0 |
| 109 | 109 | 0 | 0 |
| 0 | 0 | 292 | 93 |
| 0 | 0 | 308 | 0 |
| 194 | 228 | 0 | 0 |
| 336 | 143 | 0 | 0 |
| 0 | 0 | 292 | 93 |
| 0 | 0 | 279 | 45 |
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