metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊20D4, C10.412+ 1+4, C4⋊C4⋊24D10, (C2×D4)⋊8D10, C4⋊D4⋊15D5, C5⋊6(D4⋊5D4), C4.110(D4×D5), D10⋊9(C4○D4), C20⋊2D4⋊21C2, C22⋊C4⋊28D10, D10.42(C2×D4), C20.229(C2×D4), (C22×C4)⋊19D10, D20⋊8C4⋊21C2, C23⋊D10⋊11C2, D10⋊2Q8⋊22C2, (D4×C10)⋊14C22, C4⋊Dic5⋊32C22, C10.71(C22×D4), C20.17D4⋊17C2, (C2×C10).156C24, (C2×C20).595C23, (C22×C20)⋊22C22, (C4×Dic5)⋊23C22, D10.12D4⋊20C2, C23.D5⋊24C22, C2.43(D4⋊6D10), D10⋊C4⋊18C22, (C2×Dic10)⋊62C22, (C2×D20).272C22, C10.D4⋊65C22, (C22×C10).23C23, (C2×Dic5).75C23, (C23×D5).48C22, C22.177(C23×D5), C23.113(C22×D5), (C22×D5).200C23, (C2×D4×D5)⋊13C2, C2.44(C2×D4×D5), (C4×C5⋊D4)⋊18C2, (D5×C22⋊C4)⋊6C2, C2.40(D5×C4○D4), (C2×C4×D5)⋊15C22, (C2×C4○D20)⋊22C2, (C5×C4⋊D4)⋊18C2, (C5×C4⋊C4)⋊13C22, C10.153(C2×C4○D4), (C2×C5⋊D4)⋊40C22, (C2×C4).39(C22×D5), (C5×C22⋊C4)⋊15C22, SmallGroup(320,1284)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊20D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, cac-1=dad=a11, bc=cb, dbd=a10b, dcd=c-1 >
Subgroups: 1414 in 334 conjugacy classes, 105 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×D5, C22×C10, C22×C10, D4⋊5D4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C4○D20, D4×D5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, C23×D5, D5×C22⋊C4, D10.12D4, D20⋊8C4, D10⋊2Q8, C4×C5⋊D4, C20.17D4, C23⋊D10, C20⋊2D4, C5×C4⋊D4, C2×C4○D20, C2×D4×D5, D20⋊20D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, 2+ 1+4, C22×D5, D4⋊5D4, D4×D5, C23×D5, C2×D4×D5, D4⋊6D10, D5×C4○D4, D20⋊20D4
►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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 30)(39 40)(41 42)(43 60)(44 59)(45 58)(46 57)(47 56)(48 55)(49 54)(50 53)(51 52)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(75 80)(76 79)(77 78)
(1 35 78 47)(2 26 79 58)(3 37 80 49)(4 28 61 60)(5 39 62 51)(6 30 63 42)(7 21 64 53)(8 32 65 44)(9 23 66 55)(10 34 67 46)(11 25 68 57)(12 36 69 48)(13 27 70 59)(14 38 71 50)(15 29 72 41)(16 40 73 52)(17 31 74 43)(18 22 75 54)(19 33 76 45)(20 24 77 56)
(1 78)(2 69)(3 80)(4 71)(5 62)(6 73)(7 64)(8 75)(9 66)(10 77)(11 68)(12 79)(13 70)(14 61)(15 72)(16 63)(17 74)(18 65)(19 76)(20 67)(22 32)(24 34)(26 36)(28 38)(30 40)(42 52)(44 54)(46 56)(48 58)(50 60)
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(39,40)(41,42)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(75,80)(76,79)(77,78), (1,35,78,47)(2,26,79,58)(3,37,80,49)(4,28,61,60)(5,39,62,51)(6,30,63,42)(7,21,64,53)(8,32,65,44)(9,23,66,55)(10,34,67,46)(11,25,68,57)(12,36,69,48)(13,27,70,59)(14,38,71,50)(15,29,72,41)(16,40,73,52)(17,31,74,43)(18,22,75,54)(19,33,76,45)(20,24,77,56), (1,78)(2,69)(3,80)(4,71)(5,62)(6,73)(7,64)(8,75)(9,66)(10,77)(11,68)(12,79)(13,70)(14,61)(15,72)(16,63)(17,74)(18,65)(19,76)(20,67)(22,32)(24,34)(26,36)(28,38)(30,40)(42,52)(44,54)(46,56)(48,58)(50,60)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(39,40)(41,42)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(75,80)(76,79)(77,78), (1,35,78,47)(2,26,79,58)(3,37,80,49)(4,28,61,60)(5,39,62,51)(6,30,63,42)(7,21,64,53)(8,32,65,44)(9,23,66,55)(10,34,67,46)(11,25,68,57)(12,36,69,48)(13,27,70,59)(14,38,71,50)(15,29,72,41)(16,40,73,52)(17,31,74,43)(18,22,75,54)(19,33,76,45)(20,24,77,56), (1,78)(2,69)(3,80)(4,71)(5,62)(6,73)(7,64)(8,75)(9,66)(10,77)(11,68)(12,79)(13,70)(14,61)(15,72)(16,63)(17,74)(18,65)(19,76)(20,67)(22,32)(24,34)(26,36)(28,38)(30,40)(42,52)(44,54)(46,56)(48,58)(50,60) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,38),(22,37),(23,36),(24,35),(25,34),(26,33),(27,32),(28,31),(29,30),(39,40),(41,42),(43,60),(44,59),(45,58),(46,57),(47,56),(48,55),(49,54),(50,53),(51,52),(61,74),(62,73),(63,72),(64,71),(65,70),(66,69),(67,68),(75,80),(76,79),(77,78)], [(1,35,78,47),(2,26,79,58),(3,37,80,49),(4,28,61,60),(5,39,62,51),(6,30,63,42),(7,21,64,53),(8,32,65,44),(9,23,66,55),(10,34,67,46),(11,25,68,57),(12,36,69,48),(13,27,70,59),(14,38,71,50),(15,29,72,41),(16,40,73,52),(17,31,74,43),(18,22,75,54),(19,33,76,45),(20,24,77,56)], [(1,78),(2,69),(3,80),(4,71),(5,62),(6,73),(7,64),(8,75),(9,66),(10,77),(11,68),(12,79),(13,70),(14,61),(15,72),(16,63),(17,74),(18,65),(19,76),(20,67),(22,32),(24,34),(26,36),(28,38),(30,40),(42,52),(44,54),(46,56),(48,58),(50,60)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | ··· | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 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 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ 1+4 | D4×D5 | D4⋊6D10 | D5×C4○D4 |
| kernel | D20⋊20D4 | D5×C22⋊C4 | D10.12D4 | D20⋊8C4 | D10⋊2Q8 | C4×C5⋊D4 | C20.17D4 | C23⋊D10 | C20⋊2D4 | C5×C4⋊D4 | C2×C4○D20 | C2×D4×D5 | D20 | C4⋊D4 | D10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 2 | 2 | 6 | 1 | 4 | 4 | 4 |
Matrix representation of D20⋊20D4 ►in GL6(𝔽41)
| 0 | 32 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 33 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 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 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [0,32,0,0,0,0,32,0,0,0,0,0,0,0,7,33,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,9,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1] >;
D20⋊20D4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_{20}D_4 % in TeX
G:=Group("D20:20D4"); // GroupNames label
G:=SmallGroup(320,1284);
// by ID
G=gap.SmallGroup(320,1284);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,1571,570,297,12550]);
// 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=d*a*d=a^11,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations