metabelian, supersoluble, monomial
Aliases: C20⋊1D5, C5⋊1D20, C52⋊5D4, C10.14D10, C4⋊(C5⋊D5), (C5×C20)⋊1C2, (C5×C10).13C22, (C2×C5⋊D5)⋊2C2, C2.4(C2×C5⋊D5), SmallGroup(200,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊D5
G = < a,b,c | a20=b5=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Subgroups: 416 in 64 conjugacy classes, 27 normal (7 characteristic)
C1, C2, C2, C4, C22, C5, D4, D5, C10, C20, D10, C52, D20, C5⋊D5, C5×C10, C5×C20, C2×C5⋊D5, C20⋊D5
Quotients: C1, C2, C22, D4, D5, D10, D20, C5⋊D5, C2×C5⋊D5, C20⋊D5
►On 100 points
Generators in S100
(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)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)
(1 57 75 24 98)(2 58 76 25 99)(3 59 77 26 100)(4 60 78 27 81)(5 41 79 28 82)(6 42 80 29 83)(7 43 61 30 84)(8 44 62 31 85)(9 45 63 32 86)(10 46 64 33 87)(11 47 65 34 88)(12 48 66 35 89)(13 49 67 36 90)(14 50 68 37 91)(15 51 69 38 92)(16 52 70 39 93)(17 53 71 40 94)(18 54 72 21 95)(19 55 73 22 96)(20 56 74 23 97)
(1 98)(2 97)(3 96)(4 95)(5 94)(6 93)(7 92)(8 91)(9 90)(10 89)(11 88)(12 87)(13 86)(14 85)(15 84)(16 83)(17 82)(18 81)(19 100)(20 99)(21 60)(22 59)(23 58)(24 57)(25 56)(26 55)(27 54)(28 53)(29 52)(30 51)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(61 69)(62 68)(63 67)(64 66)(70 80)(71 79)(72 78)(73 77)(74 76)
G:=sub<Sym(100)| (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)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,57,75,24,98)(2,58,76,25,99)(3,59,77,26,100)(4,60,78,27,81)(5,41,79,28,82)(6,42,80,29,83)(7,43,61,30,84)(8,44,62,31,85)(9,45,63,32,86)(10,46,64,33,87)(11,47,65,34,88)(12,48,66,35,89)(13,49,67,36,90)(14,50,68,37,91)(15,51,69,38,92)(16,52,70,39,93)(17,53,71,40,94)(18,54,72,21,95)(19,55,73,22,96)(20,56,74,23,97), (1,98)(2,97)(3,96)(4,95)(5,94)(6,93)(7,92)(8,91)(9,90)(10,89)(11,88)(12,87)(13,86)(14,85)(15,84)(16,83)(17,82)(18,81)(19,100)(20,99)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(61,69)(62,68)(63,67)(64,66)(70,80)(71,79)(72,78)(73,77)(74,76)>;
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)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,57,75,24,98)(2,58,76,25,99)(3,59,77,26,100)(4,60,78,27,81)(5,41,79,28,82)(6,42,80,29,83)(7,43,61,30,84)(8,44,62,31,85)(9,45,63,32,86)(10,46,64,33,87)(11,47,65,34,88)(12,48,66,35,89)(13,49,67,36,90)(14,50,68,37,91)(15,51,69,38,92)(16,52,70,39,93)(17,53,71,40,94)(18,54,72,21,95)(19,55,73,22,96)(20,56,74,23,97), (1,98)(2,97)(3,96)(4,95)(5,94)(6,93)(7,92)(8,91)(9,90)(10,89)(11,88)(12,87)(13,86)(14,85)(15,84)(16,83)(17,82)(18,81)(19,100)(20,99)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(61,69)(62,68)(63,67)(64,66)(70,80)(71,79)(72,78)(73,77)(74,76) );
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),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)], [(1,57,75,24,98),(2,58,76,25,99),(3,59,77,26,100),(4,60,78,27,81),(5,41,79,28,82),(6,42,80,29,83),(7,43,61,30,84),(8,44,62,31,85),(9,45,63,32,86),(10,46,64,33,87),(11,47,65,34,88),(12,48,66,35,89),(13,49,67,36,90),(14,50,68,37,91),(15,51,69,38,92),(16,52,70,39,93),(17,53,71,40,94),(18,54,72,21,95),(19,55,73,22,96),(20,56,74,23,97)], [(1,98),(2,97),(3,96),(4,95),(5,94),(6,93),(7,92),(8,91),(9,90),(10,89),(11,88),(12,87),(13,86),(14,85),(15,84),(16,83),(17,82),(18,81),(19,100),(20,99),(21,60),(22,59),(23,58),(24,57),(25,56),(26,55),(27,54),(28,53),(29,52),(30,51),(31,50),(32,49),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(61,69),(62,68),(63,67),(64,66),(70,80),(71,79),(72,78),(73,77),(74,76)]])
C20⋊D5 is a maximal subgroup of
C5⋊D40 C52⋊4SD16 C40⋊2D5 C52⋊5D8 C52⋊7D8 C52⋊10SD16 Dic10⋊5D5 D5×D20 C20.50D10 D4×C5⋊D5 C20.26D10
C20⋊D5 is a maximal quotient of
C40⋊2D5 C52⋊5D8 C40.D5 C20⋊3Dic5 C10.11D20
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | ··· | 5L | 10A | ··· | 10L | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 50 | 50 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
53 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D5 | D10 | D20 |
| kernel | C20⋊D5 | C5×C20 | C2×C5⋊D5 | C52 | C20 | C10 | C5 |
| # reps | 1 | 1 | 2 | 1 | 12 | 12 | 24 |
Matrix representation of C20⋊D5 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 25 |
| 0 | 0 | 16 | 16 |
| 40 | 1 | 0 | 0 |
| 5 | 35 | 0 | 0 |
| 0 | 0 | 6 | 1 |
| 0 | 0 | 40 | 0 |
| 40 | 0 | 0 | 0 |
| 5 | 1 | 0 | 0 |
| 0 | 0 | 1 | 6 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,2,16,0,0,25,16],[40,5,0,0,1,35,0,0,0,0,6,40,0,0,1,0],[40,5,0,0,0,1,0,0,0,0,1,0,0,0,6,40] >;
C20⋊D5 in GAP, Magma, Sage, TeX
C_{20}\rtimes D_5 % in TeX
G:=Group("C20:D5"); // GroupNames label
G:=SmallGroup(200,34);
// by ID
G=gap.SmallGroup(200,34);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-5,61,26,643,4004]);
// Polycyclic
G:=Group<a,b,c|a^20=b^5=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations