metabelian, supersoluble, monomial
Aliases: D20.2D5, C52⋊2SD16, Dic10⋊2D5, C20.11D10, C4.9D52, C5⋊2(Q8⋊D5), (C5×C10).8D4, C5⋊2(D4.D5), C52⋊7C8⋊2C2, (C5×D20).1C2, (C5×Dic10)⋊1C2, C10.8(C5⋊D4), (C5×C20).3C22, C2.4(C52⋊2D4), SmallGroup(400,66)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.D5
G = < a,b,c,d | a20=b2=c5=1, d2=a10, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a15b, dcd-1=c-1 >
20C2
2C5
2C5
10C4
10C22
2C10
2C10
4D5
20C10
5D4
5Q8
25C8
2Dic5
2C20
2C20
2D10
10C20
10C2×C10
25SD16
10C5⋊2C8
10C5⋊2C8
Smallest permutation representation of D20.D5
►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 50)(2 49)(3 48)(4 47)(5 46)(6 45)(7 44)(8 43)(9 42)(10 41)(11 60)(12 59)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 66)(22 65)(23 64)(24 63)(25 62)(26 61)(27 80)(28 79)(29 78)(30 77)(31 76)(32 75)(33 74)(34 73)(35 72)(36 71)(37 70)(38 69)(39 68)(40 67)
(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 35 11 25)(2 26 12 36)(3 37 13 27)(4 28 14 38)(5 39 15 29)(6 30 16 40)(7 21 17 31)(8 32 18 22)(9 23 19 33)(10 34 20 24)(41 78 51 68)(42 69 52 79)(43 80 53 70)(44 71 54 61)(45 62 55 72)(46 73 56 63)(47 64 57 74)(48 75 58 65)(49 66 59 76)(50 77 60 67)
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), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,66)(22,65)(23,64)(24,63)(25,62)(26,61)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,74)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67), (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), (1,35,11,25)(2,26,12,36)(3,37,13,27)(4,28,14,38)(5,39,15,29)(6,30,16,40)(7,21,17,31)(8,32,18,22)(9,23,19,33)(10,34,20,24)(41,78,51,68)(42,69,52,79)(43,80,53,70)(44,71,54,61)(45,62,55,72)(46,73,56,63)(47,64,57,74)(48,75,58,65)(49,66,59,76)(50,77,60,67)>;
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), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,66)(22,65)(23,64)(24,63)(25,62)(26,61)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,74)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67), (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), (1,35,11,25)(2,26,12,36)(3,37,13,27)(4,28,14,38)(5,39,15,29)(6,30,16,40)(7,21,17,31)(8,32,18,22)(9,23,19,33)(10,34,20,24)(41,78,51,68)(42,69,52,79)(43,80,53,70)(44,71,54,61)(45,62,55,72)(46,73,56,63)(47,64,57,74)(48,75,58,65)(49,66,59,76)(50,77,60,67) );
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)], [(1,50),(2,49),(3,48),(4,47),(5,46),(6,45),(7,44),(8,43),(9,42),(10,41),(11,60),(12,59),(13,58),(14,57),(15,56),(16,55),(17,54),(18,53),(19,52),(20,51),(21,66),(22,65),(23,64),(24,63),(25,62),(26,61),(27,80),(28,79),(29,78),(30,77),(31,76),(32,75),(33,74),(34,73),(35,72),(36,71),(37,70),(38,69),(39,68),(40,67)], [(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)], [(1,35,11,25),(2,26,12,36),(3,37,13,27),(4,28,14,38),(5,39,15,29),(6,30,16,40),(7,21,17,31),(8,32,18,22),(9,23,19,33),(10,34,20,24),(41,78,51,68),(42,69,52,79),(43,80,53,70),(44,71,54,61),(45,62,55,72),(46,73,56,63),(47,64,57,74),(48,75,58,65),(49,66,59,76),(50,77,60,67)]])
43 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 20 | 2 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 50 | 50 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | ··· | 4 | 20 | 20 | 20 | 20 |
43 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | D4 | D5 | D5 | SD16 | D10 | C5⋊D4 | D4.D5 | Q8⋊D5 | D52 | C52⋊2D4 | D20.D5 |
| kernel | D20.D5 | C52⋊7C8 | C5×Dic10 | C5×D20 | C5×C10 | Dic10 | D20 | C52 | C20 | C10 | C5 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of D20.D5 ►in GL6(𝔽41)
| 1 | 39 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 36 | 15 | 0 | 0 | 0 | 0 |
| 23 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 26 | 0 | 0 |
| 0 | 0 | 2 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 40 |
| 0 | 0 | 0 | 0 | 1 | 24 |
| 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 | 34 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 21 | 13 | 0 | 0 | 0 | 0 |
| 7 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 1 | 0 | 0 |
| 0 | 0 | 40 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 39 |
| 0 | 0 | 0 | 0 | 38 | 35 |
G:=sub<GL(6,GF(41))| [1,1,0,0,0,0,39,40,0,0,0,0,0,0,0,40,0,0,0,0,1,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[36,23,0,0,0,0,15,5,0,0,0,0,0,0,20,2,0,0,0,0,26,21,0,0,0,0,0,0,17,1,0,0,0,0,40,24],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,1,0,0,0,0,40,0],[21,7,0,0,0,0,13,20,0,0,0,0,0,0,17,40,0,0,0,0,1,24,0,0,0,0,0,0,6,38,0,0,0,0,39,35] >;
D20.D5 in GAP, Magma, Sage, TeX
D_{20}.D_5 % in TeX
G:=Group("D20.D5"); // GroupNames label
G:=SmallGroup(400,66);
// by ID
G=gap.SmallGroup(400,66);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,73,55,218,116,50,970,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^5=1,d^2=a^10,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=c^-1>;
// generators/relations
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