metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D68, C4⋊D17, C17⋊1D4, C68⋊1C2, D34⋊1C2, C2.4D34, C34.3C22, sometimes denoted D136 or Dih68 or Dih136, SmallGroup(136,6)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D68
G = < a,b | a68=b2=1, bab=a-1 >
Smallest permutation representation of D68
►On 68 points
Generators in S68
(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)
(1 68)(2 67)(3 66)(4 65)(5 64)(6 63)(7 62)(8 61)(9 60)(10 59)(11 58)(12 57)(13 56)(14 55)(15 54)(16 53)(17 52)(18 51)(19 50)(20 49)(21 48)(22 47)(23 46)(24 45)(25 44)(26 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)
G:=sub<Sym(68)| (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), (1,68)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)>;
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), (1,68)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35) );
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)], [(1,68),(2,67),(3,66),(4,65),(5,64),(6,63),(7,62),(8,61),(9,60),(10,59),(11,58),(12,57),(13,56),(14,55),(15,54),(16,53),(17,52),(18,51),(19,50),(20,49),(21,48),(22,47),(23,46),(24,45),(25,44),(26,43),(27,42),(28,41),(29,40),(30,39),(31,38),(32,37),(33,36),(34,35)]])
D68 is a maximal subgroup of
C136⋊C2 D136 D4⋊D17 Q8⋊D17 D68⋊5C2 D4×D17 D68⋊C2 C3⋊D68 D204
D68 is a maximal quotient of
C136⋊C2 D136 Dic68 C68⋊3C4 D34⋊C4 C3⋊D68 D204
37 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 17A | ··· | 17H | 34A | ··· | 34H | 68A | ··· | 68P |
| order | 1 | 2 | 2 | 2 | 4 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 34 | 34 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
37 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D17 | D34 | D68 |
| kernel | D68 | C68 | D34 | C17 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 8 | 8 | 16 |
Matrix representation of D68 ►in GL2(𝔽137) generated by
| 37 | 7 |
| 130 | 132 |
| 37 | 7 |
| 59 | 100 |
G:=sub<GL(2,GF(137))| [37,130,7,132],[37,59,7,100] >;
D68 in GAP, Magma, Sage, TeX
D_{68} % in TeX
G:=Group("D68"); // GroupNames label
G:=SmallGroup(136,6);
// by ID
G=gap.SmallGroup(136,6);
# by ID
G:=PCGroup([4,-2,-2,-2,-17,49,21,2051]);
// Polycyclic
G:=Group<a,b|a^68=b^2=1,b*a*b=a^-1>;
// generators/relations
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