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## G = D77order 154 = 2·7·11

### Dihedral group

Aliases: D77, C7⋊D11, C11⋊D7, C771C2, sometimes denoted D154 or Dih77 or Dih154, SmallGroup(154,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C77 — D77
 Chief series C1 — C11 — C77 — D77
 Lower central C77 — D77
 Upper central C1

Generators and relations for D77
G = < a,b | a77=b2=1, bab=a-1 >

77C2
11D7
7D11

Smallest permutation representation of D77
On 77 points
Generators in S77
```(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)
(1 77)(2 76)(3 75)(4 74)(5 73)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 63)(16 62)(17 61)(18 60)(19 59)(20 58)(21 57)(22 56)(23 55)(24 54)(25 53)(26 52)(27 51)(28 50)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)```

`G:=sub<Sym(77)| (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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,60)(19,59)(20,58)(21,57)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)>;`

`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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,60)(19,59)(20,58)(21,57)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40) );`

`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)], [(1,77),(2,76),(3,75),(4,74),(5,73),(6,72),(7,71),(8,70),(9,69),(10,68),(11,67),(12,66),(13,65),(14,64),(15,63),(16,62),(17,61),(18,60),(19,59),(20,58),(21,57),(22,56),(23,55),(24,54),(25,53),(26,52),(27,51),(28,50),(29,49),(30,48),(31,47),(32,46),(33,45),(34,44),(35,43),(36,42),(37,41),(38,40)])`

D77 is a maximal subgroup of   D7×D11  C11⋊F7  D231
D77 is a maximal quotient of   Dic77  D231

40 conjugacy classes

 class 1 2 7A 7B 7C 11A ··· 11E 77A ··· 77AD order 1 2 7 7 7 11 ··· 11 77 ··· 77 size 1 77 2 2 2 2 ··· 2 2 ··· 2

40 irreducible representations

 dim 1 1 2 2 2 type + + + + + image C1 C2 D7 D11 D77 kernel D77 C77 C11 C7 C1 # reps 1 1 3 5 30

Matrix representation of D77 in GL2(𝔽463) generated by

 105 399 64 424
,
 105 399 288 358
`G:=sub<GL(2,GF(463))| [105,64,399,424],[105,288,399,358] >;`

D77 in GAP, Magma, Sage, TeX

`D_{77}`
`% in TeX`

`G:=Group("D77");`
`// GroupNames label`

`G:=SmallGroup(154,3);`
`// by ID`

`G=gap.SmallGroup(154,3);`
`# by ID`

`G:=PCGroup([3,-2,-7,-11,73,1262]);`
`// Polycyclic`

`G:=Group<a,b|a^77=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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