metabelian, supersoluble, monomial
Aliases: D20⋊5D5, D10.1D10, C20.21D10, Dic5.16D10, C4.6D52, (C4×D5)⋊1D5, (C5×D20)⋊4C2, (D5×C20)⋊2C2, C5⋊3(C4○D20), (D5×Dic5)⋊4C2, C52⋊2(C4○D4), C52⋊2D4⋊1C2, C5⋊2(D4⋊2D5), C52⋊4Q8⋊5C2, (C5×C10).2C23, C10.2(C22×D5), (C5×C20).17C22, (D5×C10).1C22, C52⋊6C4.2C22, (C5×Dic5).20C22, C2.5(C2×D52), SmallGroup(400,164)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊5D5
G = < a,b,c,d | a20=b2=c5=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a10b, dcd=c-1 >
Subgroups: 516 in 92 conjugacy classes, 32 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, D4, Q8, D5, C10, C10, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C52, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×D5, C5×C10, C4○D20, D4⋊2D5, C5×Dic5, C52⋊6C4, C5×C20, D5×C10, D5×C10, D5×Dic5, C52⋊2D4, D5×C20, C5×D20, C52⋊4Q8, D20⋊5D5
Quotients: C1, C2, C22, C23, D5, C4○D4, D10, C22×D5, C4○D20, D4⋊2D5, D52, C2×D52, D20⋊5D5
►On 80 points
Generators in S80
(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 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 48)(9 47)(10 46)(11 45)(12 44)(13 43)(14 42)(15 41)(16 60)(17 59)(18 58)(19 57)(20 56)(21 68)(22 67)(23 66)(24 65)(25 64)(26 63)(27 62)(28 61)(29 80)(30 79)(31 78)(32 77)(33 76)(34 75)(35 74)(36 73)(37 72)(38 71)(39 70)(40 69)
(1 9 17 5 13)(2 10 18 6 14)(3 11 19 7 15)(4 12 20 8 16)(21 33 25 37 29)(22 34 26 38 30)(23 35 27 39 31)(24 36 28 40 32)(41 53 45 57 49)(42 54 46 58 50)(43 55 47 59 51)(44 56 48 60 52)(61 69 77 65 73)(62 70 78 66 74)(63 71 79 67 75)(64 72 80 68 76)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 21)(41 63)(42 64)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(49 71)(50 72)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 61)(60 62)
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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,60)(17,59)(18,58)(19,57)(20,56)(21,68)(22,67)(23,66)(24,65)(25,64)(26,63)(27,62)(28,61)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69), (1,9,17,5,13)(2,10,18,6,14)(3,11,19,7,15)(4,12,20,8,16)(21,33,25,37,29)(22,34,26,38,30)(23,35,27,39,31)(24,36,28,40,32)(41,53,45,57,49)(42,54,46,58,50)(43,55,47,59,51)(44,56,48,60,52)(61,69,77,65,73)(62,70,78,66,74)(63,71,79,67,75)(64,72,80,68,76), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,21)(41,63)(42,64)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,61)(60,62)>;
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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,60)(17,59)(18,58)(19,57)(20,56)(21,68)(22,67)(23,66)(24,65)(25,64)(26,63)(27,62)(28,61)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69), (1,9,17,5,13)(2,10,18,6,14)(3,11,19,7,15)(4,12,20,8,16)(21,33,25,37,29)(22,34,26,38,30)(23,35,27,39,31)(24,36,28,40,32)(41,53,45,57,49)(42,54,46,58,50)(43,55,47,59,51)(44,56,48,60,52)(61,69,77,65,73)(62,70,78,66,74)(63,71,79,67,75)(64,72,80,68,76), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,21)(41,63)(42,64)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,61)(60,62) );
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)], [(1,55),(2,54),(3,53),(4,52),(5,51),(6,50),(7,49),(8,48),(9,47),(10,46),(11,45),(12,44),(13,43),(14,42),(15,41),(16,60),(17,59),(18,58),(19,57),(20,56),(21,68),(22,67),(23,66),(24,65),(25,64),(26,63),(27,62),(28,61),(29,80),(30,79),(31,78),(32,77),(33,76),(34,75),(35,74),(36,73),(37,72),(38,71),(39,70),(40,69)], [(1,9,17,5,13),(2,10,18,6,14),(3,11,19,7,15),(4,12,20,8,16),(21,33,25,37,29),(22,34,26,38,30),(23,35,27,39,31),(24,36,28,40,32),(41,53,45,57,49),(42,54,46,58,50),(43,55,47,59,51),(44,56,48,60,52),(61,69,77,65,73),(62,70,78,66,74),(63,71,79,67,75),(64,72,80,68,76)], [(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,21),(41,63),(42,64),(43,65),(44,66),(45,67),(46,68),(47,69),(48,70),(49,71),(50,72),(51,73),(52,74),(53,75),(54,76),(55,77),(56,78),(57,79),(58,80),(59,61),(60,62)]])
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 10O | 10P | 20A | 20B | 20C | 20D | 20E | ··· | 20N | 20O | 20P | 20Q | 20R |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 10 | 10 | 10 | 2 | 5 | 5 | 50 | 50 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 |
52 irreducible representations
| dim | 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 | D5 | D5 | C4○D4 | D10 | D10 | D10 | C4○D20 | D4⋊2D5 | D52 | C2×D52 | D20⋊5D5 |
| kernel | D20⋊5D5 | D5×Dic5 | C52⋊2D4 | D5×C20 | C5×D20 | C52⋊4Q8 | C4×D5 | D20 | C52 | Dic5 | C20 | D10 | C5 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 6 | 8 | 2 | 4 | 4 | 8 |
Matrix representation of D20⋊5D5 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 28 | 2 |
| 0 | 0 | 39 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 35 |
| 0 | 0 | 0 | 1 |
| 0 | 40 | 0 | 0 |
| 1 | 6 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 35 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 18 | 6 |
| 0 | 0 | 35 | 23 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,28,39,0,0,2,16],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,35,1],[0,1,0,0,40,6,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,35,1,0,0,0,0,18,35,0,0,6,23] >;
D20⋊5D5 in GAP, Magma, Sage, TeX
D_{20}\rtimes_5D_5 % in TeX
G:=Group("D20:5D5"); // GroupNames label
G:=SmallGroup(400,164);
// by ID
G=gap.SmallGroup(400,164);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,55,218,50,970,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^5=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations