metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D80:2C2, C16:1D10, C8.3D20, C40.2D4, C80:1C22, C20.14D8, C4.14D40, D40:8C22, M5(2):1D5, C22.5D40, C40.59C23, Dic20:7C22, C16:D5:1C2, (C2xC10).6D8, (C2xD40):11C2, C5:1(C16:C22), C10.13(C2xD8), (C2xC4).41D20, C4.40(C2xD20), (C2xC8).73D10, C2.15(C2xD40), D40:7C2:9C2, (C2xC20).128D4, C20.283(C2xD4), (C5xM5(2)):1C2, C8.49(C22xD5), (C2xC40).59C22, SmallGroup(320,535)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D80:C2
G = < a,b,c | a80=b2=c2=1, bab=a-1, cac=a41, cbc=a40b >
Subgroups: 622 in 90 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2xC4, C2xC4, D4, Q8, C23, D5, C10, C10, C16, C2xC8, D8, SD16, Q16, C2xD4, C4oD4, Dic5, C20, D10, C2xC10, M5(2), D16, SD32, C2xD8, C4oD8, C40, Dic10, C4xD5, D20, C5:D4, C2xC20, C22xD5, C16:C22, C80, C40:C2, D40, D40, D40, Dic20, C2xC40, C2xD20, C4oD20, D80, C16:D5, C5xM5(2), C2xD40, D40:7C2, D80:C2
Quotients: C1, C2, C22, D4, C23, D5, D8, C2xD4, D10, C2xD8, D20, C22xD5, C16:C22, D40, C2xD20, C2xD40, D80:C2
►On 80 points
(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 79 80)
(1 80)(2 79)(3 78)(4 77)(5 76)(6 75)(7 74)(8 73)(9 72)(10 71)(11 70)(12 69)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 61)(21 60)(22 59)(23 58)(24 57)(25 56)(26 55)(27 54)(28 53)(29 52)(30 51)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)
(2 42)(4 44)(6 46)(8 48)(10 50)(12 52)(14 54)(16 56)(18 58)(20 60)(22 62)(24 64)(26 66)(28 68)(30 70)(32 72)(34 74)(36 76)(38 78)(40 80)
G:=sub<Sym(80)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,80)(2,79)(3,78)(4,77)(5,76)(6,75)(7,74)(8,73)(9,72)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41), (2,42)(4,44)(6,46)(8,48)(10,50)(12,52)(14,54)(16,56)(18,58)(20,60)(22,62)(24,64)(26,66)(28,68)(30,70)(32,72)(34,74)(36,76)(38,78)(40,80)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,80)(2,79)(3,78)(4,77)(5,76)(6,75)(7,74)(8,73)(9,72)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41), (2,42)(4,44)(6,46)(8,48)(10,50)(12,52)(14,54)(16,56)(18,58)(20,60)(22,62)(24,64)(26,66)(28,68)(30,70)(32,72)(34,74)(36,76)(38,78)(40,80) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,80),(2,79),(3,78),(4,77),(5,76),(6,75),(7,74),(8,73),(9,72),(10,71),(11,70),(12,69),(13,68),(14,67),(15,66),(16,65),(17,64),(18,63),(19,62),(20,61),(21,60),(22,59),(23,58),(24,57),(25,56),(26,55),(27,54),(28,53),(29,52),(30,51),(31,50),(32,49),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41)], [(2,42),(4,44),(6,46),(8,48),(10,50),(12,52),(14,54),(16,56),(18,58),(20,60),(22,62),(24,64),(26,66),(28,68),(30,70),(32,72),(34,74),(36,76),(38,78),(40,80)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 8C | 10A | 10B | 10C | 10D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H | 40I | 40J | 40K | 40L | 80A | ··· | 80P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 40 | 40 | 40 | 2 | 2 | 40 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D8 | D8 | D10 | D10 | D20 | D20 | D40 | D40 | C16:C22 | D80:C2 |
| kernel | D80:C2 | D80 | C16:D5 | C5xM5(2) | C2xD40 | D40:7C2 | C40 | C2xC20 | M5(2) | C20 | C2xC10 | C16 | C2xC8 | C8 | C2xC4 | C4 | C22 | C5 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 2 | 8 |
Matrix representation of D80:C2 ►in GL4(F241) generated by
| 9 | 51 | 0 | 2 |
| 17 | 227 | 239 | 104 |
| 147 | 42 | 15 | 215 |
| 18 | 104 | 51 | 231 |
| 202 | 65 | 3 | 82 |
| 33 | 142 | 238 | 238 |
| 173 | 204 | 69 | 209 |
| 237 | 2 | 63 | 69 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 31 | 226 | 240 | 0 |
| 232 | 190 | 0 | 240 |
G:=sub<GL(4,GF(241))| [9,17,147,18,51,227,42,104,0,239,15,51,2,104,215,231],[202,33,173,237,65,142,204,2,3,238,69,63,82,238,209,69],[1,0,31,232,0,1,226,190,0,0,240,0,0,0,0,240] >;
D80:C2 in GAP, Magma, Sage, TeX
D_{80}\rtimes C_2 % in TeX
G:=Group("D80:C2"); // GroupNames label
G:=SmallGroup(320,535);
// by ID
G=gap.SmallGroup(320,535);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,387,142,675,192,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^80=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^41,c*b*c=a^40*b>;
// generators/relations