direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D78, C2×D39, C26⋊S3, C6⋊D13, C13⋊2D6, C3⋊2D26, C78⋊1C2, C39⋊2C22, sometimes denoted D156 or Dih78 or Dih156, SmallGroup(156,17)
Series: Derived ►Chief ►Lower central ►Upper central
| C39 — D78 |
Generators and relations for D78
G = < a,b | a78=b2=1, bab=a-1 >
Smallest permutation representation of D78
►On 78 points
Generators in S78
(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)
(1 78)(2 77)(3 76)(4 75)(5 74)(6 73)(7 72)(8 71)(9 70)(10 69)(11 68)(12 67)(13 66)(14 65)(15 64)(16 63)(17 62)(18 61)(19 60)(20 59)(21 58)(22 57)(23 56)(24 55)(25 54)(26 53)(27 52)(28 51)(29 50)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)
G:=sub<Sym(78)| (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), (1,78)(2,77)(3,76)(4,75)(5,74)(6,73)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,60)(20,59)(21,58)(22,57)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)>;
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), (1,78)(2,77)(3,76)(4,75)(5,74)(6,73)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,60)(20,59)(21,58)(22,57)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40) );
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)], [(1,78),(2,77),(3,76),(4,75),(5,74),(6,73),(7,72),(8,71),(9,70),(10,69),(11,68),(12,67),(13,66),(14,65),(15,64),(16,63),(17,62),(18,61),(19,60),(20,59),(21,58),(22,57),(23,56),(24,55),(25,54),(26,53),(27,52),(28,51),(29,50),(30,49),(31,48),(32,47),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,40)]])
D78 is a maximal subgroup of
D78.C2 C3⋊D52 C13⋊D12 D156 C39⋊7D4 C2×S3×D13
D78 is a maximal quotient of Dic78 D156 C39⋊7D4
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 13A | ··· | 13F | 26A | ··· | 26F | 39A | ··· | 39L | 78A | ··· | 78L |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 39 | 39 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D6 | D13 | D26 | D39 | D78 |
| kernel | D78 | D39 | C78 | C26 | C13 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 6 | 6 | 12 | 12 |
Matrix representation of D78 ►in GL2(𝔽79) generated by
| 76 | 12 |
| 44 | 8 |
| 63 | 15 |
| 62 | 16 |
G:=sub<GL(2,GF(79))| [76,44,12,8],[63,62,15,16] >;
D78 in GAP, Magma, Sage, TeX
D_{78} % in TeX
G:=Group("D78"); // GroupNames label
G:=SmallGroup(156,17);
// by ID
G=gap.SmallGroup(156,17);
# by ID
G:=PCGroup([4,-2,-2,-3,-13,98,2307]);
// Polycyclic
G:=Group<a,b|a^78=b^2=1,b*a*b=a^-1>;
// generators/relations
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