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G = D76order 152 = 23·19

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D76, C4⋊D19, C191D4, C761C2, D381C2, C2.4D38, C38.3C22, sometimes denoted D152 or Dih76 or Dih152, SmallGroup(152,5)

Series: Derived Chief Lower central Upper central

C1C38 — D76
C1C19C38D38 — D76
C19C38 — D76
C1C2C4

Generators and relations for D76
 G = < a,b | a76=b2=1, bab=a-1 >

38C2
38C2
19C22
19C22
2D19
2D19
19D4

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  D765C2  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 2A2B2C 4 19A···19I38A···38I76A···76R
order1222419···1938···3876···76
size11383822···22···22···2

41 irreducible representations

dim1112222
type+++++++
imageC1C2C2D4D19D38D76
kernelD76C76D38C19C4C2C1
# reps11219918

Matrix representation of D76 in GL2(𝔽229) generated by

184110
15697
,
21135
10218
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

Export

Subgroup lattice of D76 in TeX

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