metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D79, C79⋊C2, sometimes denoted D158 or Dih79 or Dih158, SmallGroup(158,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C79 — D79 |
Generators and relations for D79
G = < a,b | a79=b2=1, bab=a-1 >
Smallest permutation representation of D79
►On 79 points: primitive
Generators in S79
(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)
(1 79)(2 78)(3 77)(4 76)(5 75)(6 74)(7 73)(8 72)(9 71)(10 70)(11 69)(12 68)(13 67)(14 66)(15 65)(16 64)(17 63)(18 62)(19 61)(20 60)(21 59)(22 58)(23 57)(24 56)(25 55)(26 54)(27 53)(28 52)(29 51)(30 50)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)
G:=sub<Sym(79)| (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), (1,79)(2,78)(3,77)(4,76)(5,75)(6,74)(7,73)(8,72)(9,71)(10,70)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,60)(21,59)(22,58)(23,57)(24,56)(25,55)(26,54)(27,53)(28,52)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)>;
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), (1,79)(2,78)(3,77)(4,76)(5,75)(6,74)(7,73)(8,72)(9,71)(10,70)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,60)(21,59)(22,58)(23,57)(24,56)(25,55)(26,54)(27,53)(28,52)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41) );
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)], [(1,79),(2,78),(3,77),(4,76),(5,75),(6,74),(7,73),(8,72),(9,71),(10,70),(11,69),(12,68),(13,67),(14,66),(15,65),(16,64),(17,63),(18,62),(19,61),(20,60),(21,59),(22,58),(23,57),(24,56),(25,55),(26,54),(27,53),(28,52),(29,51),(30,50),(31,49),(32,48),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41)]])
D79 is a maximal subgroup of
C79⋊C6 D237
D79 is a maximal quotient of Dic79 D237
41 conjugacy classes
| class | 1 | 2 | 79A | ··· | 79AM |
| order | 1 | 2 | 79 | ··· | 79 |
| size | 1 | 79 | 2 | ··· | 2 |
41 irreducible representations
| dim | 1 | 1 | 2 |
| type | + | + | + |
| image | C1 | C2 | D79 |
| kernel | D79 | C79 | C1 |
| # reps | 1 | 1 | 39 |
Matrix representation of D79 ►in GL2(𝔽317) generated by
| 117 | 316 |
| 159 | 42 |
| 308 | 80 |
| 316 | 9 |
G:=sub<GL(2,GF(317))| [117,159,316,42],[308,316,80,9] >;
D79 in GAP, Magma, Sage, TeX
D_{79} % in TeX
G:=Group("D79"); // GroupNames label
G:=SmallGroup(158,1);
// by ID
G=gap.SmallGroup(158,1);
# by ID
G:=PCGroup([2,-2,-79,625]);
// Polycyclic
G:=Group<a,b|a^79=b^2=1,b*a*b=a^-1>;
// generators/relations
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