metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊16D4, C4⋊C4⋊3D10, (C2×C10)⋊2D8, (C2×D4)⋊1D10, C4⋊D4⋊1D5, C4.98(D4×D5), C5⋊3(C22⋊D8), C10.54(C2×D8), (C2×C20).71D4, C22⋊2(D4⋊D5), C20.145(C2×D4), (D4×C10)⋊1C22, D20⋊6C4⋊33C2, C10.44C22≀C2, (C22×D20)⋊13C2, (C22×C10).82D4, C20.55D4⋊10C2, (C2×C20).355C23, (C22×C4).119D10, C23.58(C5⋊D4), C2.12(D4⋊D10), C2.12(C23⋊D10), C10.114(C8⋊C22), (C2×D20).248C22, (C22×C20).159C22, (C2×D4⋊D5)⋊8C2, C2.9(C2×D4⋊D5), (C5×C4⋊D4)⋊1C2, (C5×C4⋊C4)⋊5C22, (C2×C5⋊2C8)⋊5C22, (C2×C10).486(C2×D4), (C2×C4).49(C5⋊D4), (C2×C4).455(C22×D5), C22.161(C2×C5⋊D4), SmallGroup(320,663)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊16D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, bd=db, dcd=c-1 >
Subgroups: 1054 in 198 conjugacy classes, 47 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, C20, C20, D10, C2×C10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, C4⋊D4, C2×D8, C22×D4, C5⋊2C8, D20, D20, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, C22⋊D8, C2×C5⋊2C8, D4⋊D5, C5×C22⋊C4, C5×C4⋊C4, C2×D20, C2×D20, C22×C20, D4×C10, D4×C10, C23×D5, D20⋊6C4, C20.55D4, C2×D4⋊D5, C5×C4⋊D4, C22×D20, D20⋊16D4
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C22≀C2, C2×D8, C8⋊C22, C5⋊D4, C22×D5, C22⋊D8, D4⋊D5, D4×D5, C2×C5⋊D4, C2×D4⋊D5, C23⋊D10, D4⋊D10, D20⋊16D4
►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 73)(2 72)(3 71)(4 70)(5 69)(6 68)(7 67)(8 66)(9 65)(10 64)(11 63)(12 62)(13 61)(14 80)(15 79)(16 78)(17 77)(18 76)(19 75)(20 74)(21 51)(22 50)(23 49)(24 48)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)
(1 33 74 55)(2 24 75 46)(3 35 76 57)(4 26 77 48)(5 37 78 59)(6 28 79 50)(7 39 80 41)(8 30 61 52)(9 21 62 43)(10 32 63 54)(11 23 64 45)(12 34 65 56)(13 25 66 47)(14 36 67 58)(15 27 68 49)(16 38 69 60)(17 29 70 51)(18 40 71 42)(19 31 72 53)(20 22 73 44)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 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), (1,73)(2,72)(3,71)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,64)(11,63)(12,62)(13,61)(14,80)(15,79)(16,78)(17,77)(18,76)(19,75)(20,74)(21,51)(22,50)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52), (1,33,74,55)(2,24,75,46)(3,35,76,57)(4,26,77,48)(5,37,78,59)(6,28,79,50)(7,39,80,41)(8,30,61,52)(9,21,62,43)(10,32,63,54)(11,23,64,45)(12,34,65,56)(13,25,66,47)(14,36,67,58)(15,27,68,49)(16,38,69,60)(17,29,70,51)(18,40,71,42)(19,31,72,53)(20,22,73,44), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)>;
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), (1,73)(2,72)(3,71)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,64)(11,63)(12,62)(13,61)(14,80)(15,79)(16,78)(17,77)(18,76)(19,75)(20,74)(21,51)(22,50)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52), (1,33,74,55)(2,24,75,46)(3,35,76,57)(4,26,77,48)(5,37,78,59)(6,28,79,50)(7,39,80,41)(8,30,61,52)(9,21,62,43)(10,32,63,54)(11,23,64,45)(12,34,65,56)(13,25,66,47)(14,36,67,58)(15,27,68,49)(16,38,69,60)(17,29,70,51)(18,40,71,42)(19,31,72,53)(20,22,73,44), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80) );
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)], [(1,73),(2,72),(3,71),(4,70),(5,69),(6,68),(7,67),(8,66),(9,65),(10,64),(11,63),(12,62),(13,61),(14,80),(15,79),(16,78),(17,77),(18,76),(19,75),(20,74),(21,51),(22,50),(23,49),(24,48),(25,47),(26,46),(27,45),(28,44),(29,43),(30,42),(31,41),(32,60),(33,59),(34,58),(35,57),(36,56),(37,55),(38,54),(39,53),(40,52)], [(1,33,74,55),(2,24,75,46),(3,35,76,57),(4,26,77,48),(5,37,78,59),(6,28,79,50),(7,39,80,41),(8,30,61,52),(9,21,62,43),(10,32,63,54),(11,23,64,45),(12,34,65,56),(13,25,66,47),(14,36,67,58),(15,27,68,49),(16,38,69,60),(17,29,70,51),(18,40,71,42),(19,31,72,53),(20,22,73,44)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,53),(22,54),(23,55),(24,56),(25,57),(26,58),(27,59),(28,60),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(37,49),(38,50),(39,51),(40,52),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 20A | ··· | 20H | 20I | 20J | 20K | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 8 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D8 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | C8⋊C22 | D4×D5 | D4⋊D5 | D4⋊D10 |
| kernel | D20⋊16D4 | D20⋊6C4 | C20.55D4 | C2×D4⋊D5 | C5×C4⋊D4 | C22×D20 | D20 | C2×C20 | C22×C10 | C4⋊D4 | C2×C10 | C4⋊C4 | C22×C4 | C2×D4 | C2×C4 | C23 | C10 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 1 | 4 | 4 | 4 |
Matrix representation of D20⋊16D4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 33 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 33 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 24 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,33,0,0,0,0,1,7,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,33,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,39,1],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,12,0,0,0,0,24,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;
D20⋊16D4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_{16}D_4 % in TeX
G:=Group("D20:16D4"); // GroupNames label
G:=SmallGroup(320,663);
// by ID
G=gap.SmallGroup(320,663);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations