metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.6D20, D20.8D4, D10:5SD16, C4:C4:2D10, (C2xC8):16D10, (C5xD4).1D4, C4.85(D4xD5), C4.2(C2xD20), C20.1(C2xD4), D20:6C4:7C2, D10:1C8:9C2, D4:C4:10D5, (C2xC40):15C22, D10:2Q8:1C2, C5:2(C22:SD16), C2.11(D5xSD16), C10.20C22wrC2, (C2xD4).135D10, (C2xDic5).28D4, C10.23(C2xSD16), C22.174(D4xD5), C2.13(D8:D5), C10.31(C8:C22), (C2xC20).216C23, (D4xC10).37C22, (C22xD5).109D4, (C2xD20).54C22, C2.23(C22:D20), (C2xDic10):13C22, (C2xD4xD5).5C2, (C5xC4:C4):4C22, (C2xD4.D5):3C2, (C2xC40:C2):14C2, (C2xC5:2C8):3C22, (C5xD4:C4):10C2, (C2xC4xD5).13C22, (C2xC10).229(C2xD4), (C2xC4).323(C22xD5), SmallGroup(320,403)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.6D20
G = < a,b,c,d | a4=b2=c20=1, d2=a2, bab=cac-1=dad-1=a-1, cbc-1=dbd-1=a-1b, dcd-1=a2c-1 >
Subgroups: 1022 in 188 conjugacy classes, 45 normal (37 characteristic)
C1, C2 [x3], C2 [x6], C4 [x2], C4 [x3], C22, C22 [x20], C5, C8 [x2], C2xC4, C2xC4 [x5], D4 [x2], D4 [x8], Q8 [x2], C23 [x11], D5 [x4], C10 [x3], C10 [x2], C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, SD16 [x4], C22xC4, C2xD4, C2xD4 [x6], C2xQ8, C24, Dic5 [x2], C20 [x2], C20, D10 [x2], D10 [x14], C2xC10, C2xC10 [x4], C22:C8, D4:C4, D4:C4, C22:Q8, C2xSD16 [x2], C22xD4, C5:2C8, C40, Dic10 [x2], C4xD5 [x2], D20 [x2], D20, C2xDic5, C2xDic5, C5:D4 [x4], C2xC20, C2xC20, C5xD4 [x2], C5xD4, C22xD5, C22xD5 [x9], C22xC10, C22:SD16, C40:C2 [x2], C2xC5:2C8, C4:Dic5, D10:C4, D4.D5 [x2], C5xC4:C4, C2xC40, C2xDic10, C2xC4xD5, C2xD20, D4xD5 [x4], C2xC5:D4, D4xC10, C23xD5, D20:6C4, D10:1C8, C5xD4:C4, D10:2Q8, C2xC40:C2, C2xD4.D5, C2xD4xD5, D4.6D20
Quotients: C1, C2 [x7], C22 [x7], D4 [x6], C23, D5, SD16 [x2], C2xD4 [x3], D10 [x3], C22wrC2, C2xSD16, C8:C22, D20 [x2], C22xD5, C22:SD16, C2xD20, D4xD5 [x2], C22:D20, D8:D5, D5xSD16, D4.6D20
►On 80 points
(1 25 76 48)(2 49 77 26)(3 27 78 50)(4 51 79 28)(5 29 80 52)(6 53 61 30)(7 31 62 54)(8 55 63 32)(9 33 64 56)(10 57 65 34)(11 35 66 58)(12 59 67 36)(13 37 68 60)(14 41 69 38)(15 39 70 42)(16 43 71 40)(17 21 72 44)(18 45 73 22)(19 23 74 46)(20 47 75 24)
(1 35)(2 67)(3 37)(4 69)(5 39)(6 71)(7 21)(8 73)(9 23)(10 75)(11 25)(12 77)(13 27)(14 79)(15 29)(16 61)(17 31)(18 63)(19 33)(20 65)(22 32)(24 34)(26 36)(28 38)(30 40)(41 51)(42 80)(43 53)(44 62)(45 55)(46 64)(47 57)(48 66)(49 59)(50 68)(52 70)(54 72)(56 74)(58 76)(60 78)
(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 75 76 20)(2 19 77 74)(3 73 78 18)(4 17 79 72)(5 71 80 16)(6 15 61 70)(7 69 62 14)(8 13 63 68)(9 67 64 12)(10 11 65 66)(21 51 44 28)(22 27 45 50)(23 49 46 26)(24 25 47 48)(29 43 52 40)(30 39 53 42)(31 41 54 38)(32 37 55 60)(33 59 56 36)(34 35 57 58)
G:=sub<Sym(80)| (1,25,76,48)(2,49,77,26)(3,27,78,50)(4,51,79,28)(5,29,80,52)(6,53,61,30)(7,31,62,54)(8,55,63,32)(9,33,64,56)(10,57,65,34)(11,35,66,58)(12,59,67,36)(13,37,68,60)(14,41,69,38)(15,39,70,42)(16,43,71,40)(17,21,72,44)(18,45,73,22)(19,23,74,46)(20,47,75,24), (1,35)(2,67)(3,37)(4,69)(5,39)(6,71)(7,21)(8,73)(9,23)(10,75)(11,25)(12,77)(13,27)(14,79)(15,29)(16,61)(17,31)(18,63)(19,33)(20,65)(22,32)(24,34)(26,36)(28,38)(30,40)(41,51)(42,80)(43,53)(44,62)(45,55)(46,64)(47,57)(48,66)(49,59)(50,68)(52,70)(54,72)(56,74)(58,76)(60,78), (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), (1,75,76,20)(2,19,77,74)(3,73,78,18)(4,17,79,72)(5,71,80,16)(6,15,61,70)(7,69,62,14)(8,13,63,68)(9,67,64,12)(10,11,65,66)(21,51,44,28)(22,27,45,50)(23,49,46,26)(24,25,47,48)(29,43,52,40)(30,39,53,42)(31,41,54,38)(32,37,55,60)(33,59,56,36)(34,35,57,58)>;
G:=Group( (1,25,76,48)(2,49,77,26)(3,27,78,50)(4,51,79,28)(5,29,80,52)(6,53,61,30)(7,31,62,54)(8,55,63,32)(9,33,64,56)(10,57,65,34)(11,35,66,58)(12,59,67,36)(13,37,68,60)(14,41,69,38)(15,39,70,42)(16,43,71,40)(17,21,72,44)(18,45,73,22)(19,23,74,46)(20,47,75,24), (1,35)(2,67)(3,37)(4,69)(5,39)(6,71)(7,21)(8,73)(9,23)(10,75)(11,25)(12,77)(13,27)(14,79)(15,29)(16,61)(17,31)(18,63)(19,33)(20,65)(22,32)(24,34)(26,36)(28,38)(30,40)(41,51)(42,80)(43,53)(44,62)(45,55)(46,64)(47,57)(48,66)(49,59)(50,68)(52,70)(54,72)(56,74)(58,76)(60,78), (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), (1,75,76,20)(2,19,77,74)(3,73,78,18)(4,17,79,72)(5,71,80,16)(6,15,61,70)(7,69,62,14)(8,13,63,68)(9,67,64,12)(10,11,65,66)(21,51,44,28)(22,27,45,50)(23,49,46,26)(24,25,47,48)(29,43,52,40)(30,39,53,42)(31,41,54,38)(32,37,55,60)(33,59,56,36)(34,35,57,58) );
G=PermutationGroup([(1,25,76,48),(2,49,77,26),(3,27,78,50),(4,51,79,28),(5,29,80,52),(6,53,61,30),(7,31,62,54),(8,55,63,32),(9,33,64,56),(10,57,65,34),(11,35,66,58),(12,59,67,36),(13,37,68,60),(14,41,69,38),(15,39,70,42),(16,43,71,40),(17,21,72,44),(18,45,73,22),(19,23,74,46),(20,47,75,24)], [(1,35),(2,67),(3,37),(4,69),(5,39),(6,71),(7,21),(8,73),(9,23),(10,75),(11,25),(12,77),(13,27),(14,79),(15,29),(16,61),(17,31),(18,63),(19,33),(20,65),(22,32),(24,34),(26,36),(28,38),(30,40),(41,51),(42,80),(43,53),(44,62),(45,55),(46,64),(47,57),(48,66),(49,59),(50,68),(52,70),(54,72),(56,74),(58,76),(60,78)], [(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)], [(1,75,76,20),(2,19,77,74),(3,73,78,18),(4,17,79,72),(5,71,80,16),(6,15,61,70),(7,69,62,14),(8,13,63,68),(9,67,64,12),(10,11,65,66),(21,51,44,28),(22,27,45,50),(23,49,46,26),(24,25,47,48),(29,43,52,40),(30,39,53,42),(31,41,54,38),(32,37,55,60),(33,59,56,36),(34,35,57,58)])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 8 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | SD16 | D10 | D10 | D10 | D20 | C8:C22 | D4xD5 | D4xD5 | D8:D5 | D5xSD16 |
| kernel | D4.6D20 | D20:6C4 | D10:1C8 | C5xD4:C4 | D10:2Q8 | C2xC40:C2 | C2xD4.D5 | C2xD4xD5 | D20 | C2xDic5 | C5xD4 | C22xD5 | D4:C4 | D10 | C4:C4 | C2xC8 | C2xD4 | D4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D4.6D20 ►in GL6(F41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 9 |
| 0 | 0 | 0 | 0 | 18 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 9 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 15 | 37 | 0 | 0 | 0 | 0 |
| 36 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 29 |
| 0 | 0 | 0 | 0 | 17 | 30 |
| 15 | 37 | 0 | 0 | 0 | 0 |
| 15 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 35 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 29 |
| 0 | 0 | 0 | 0 | 17 | 30 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,18,0,0,0,0,9,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,9,40],[15,36,0,0,0,0,37,26,0,0,0,0,0,0,35,1,0,0,0,0,40,0,0,0,0,0,0,0,11,17,0,0,0,0,29,30],[15,15,0,0,0,0,37,26,0,0,0,0,0,0,40,0,0,0,0,0,35,1,0,0,0,0,0,0,11,17,0,0,0,0,29,30] >;
D4.6D20 in GAP, Magma, Sage, TeX
D_4._6D_{20} % in TeX
G:=Group("D4.6D20"); // GroupNames label
G:=SmallGroup(320,403);
// by ID
G=gap.SmallGroup(320,403);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,135,268,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^20=1,d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^-1*b,d*c*d^-1=a^2*c^-1>;
// generators/relations