metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D76, C4⋊D19, C19⋊1D4, C76⋊1C2, D38⋊1C2, C2.4D38, C38.3C22, sometimes denoted D152 or Dih76 or Dih152, SmallGroup(152,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D76
G = < a,b | a76=b2=1, bab=a-1 >
Smallest permutation representation of D76
►On 76 points
Generators in S76
(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)
(1 76)(2 75)(3 74)(4 73)(5 72)(6 71)(7 70)(8 69)(9 68)(10 67)(11 66)(12 65)(13 64)(14 63)(15 62)(16 61)(17 60)(18 59)(19 58)(20 57)(21 56)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)
G:=sub<Sym(76)| (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), (1,76)(2,75)(3,74)(4,73)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)>;
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), (1,76)(2,75)(3,74)(4,73)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39) );
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)], [(1,76),(2,75),(3,74),(4,73),(5,72),(6,71),(7,70),(8,69),(9,68),(10,67),(11,66),(12,65),(13,64),(14,63),(15,62),(16,61),(17,60),(18,59),(19,58),(20,57),(21,56),(22,55),(23,54),(24,53),(25,52),(26,51),(27,50),(28,49),(29,48),(30,47),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41),(37,40),(38,39)]])
D76 is a maximal subgroup of
C152⋊C2 D152 D4⋊D19 Q8⋊D19 D76⋊5C2 D4×D19 D76⋊C2 D76⋊C3 C3⋊D76 D228
D76 is a maximal quotient of
C152⋊C2 D152 Dic76 C76⋊C4 D38⋊C4 C3⋊D76 D228
41 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 19A | ··· | 19I | 38A | ··· | 38I | 76A | ··· | 76R |
| order | 1 | 2 | 2 | 2 | 4 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 38 | 38 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
41 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D19 | D38 | D76 |
| kernel | D76 | C76 | D38 | C19 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 9 | 9 | 18 |
Matrix representation of D76 ►in GL2(𝔽229) generated by
| 184 | 110 |
| 156 | 97 |
| 211 | 35 |
| 102 | 18 |
G:=sub<GL(2,GF(229))| [184,156,110,97],[211,102,35,18] >;
D76 in GAP, Magma, Sage, TeX
D_{76} % in TeX
G:=Group("D76"); // GroupNames label
G:=SmallGroup(152,5);
// by ID
G=gap.SmallGroup(152,5);
# by ID
G:=PCGroup([4,-2,-2,-2,-19,49,21,2307]);
// Polycyclic
G:=Group<a,b|a^76=b^2=1,b*a*b=a^-1>;
// generators/relations
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