direct product, metacyclic, supersoluble, monomial
Aliases: C5×D20, C20⋊3D5, C20⋊1C10, C52⋊4D4, D10⋊1C10, C10.19D10, C4⋊(C5×D5), C5⋊1(C5×D4), (C5×C20)⋊2C2, (D5×C10)⋊3C2, C2.4(D5×C10), C10.3(C2×C10), (C5×C10).8C22, SmallGroup(200,29)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D20
G = < a,b,c | a5=b20=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D20
►On 40 points
Generators in S40
(1 5 9 13 17)(2 6 10 14 18)(3 7 11 15 19)(4 8 12 16 20)(21 37 33 29 25)(22 38 34 30 26)(23 39 35 31 27)(24 40 36 32 28)
(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)
(1 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)
G:=sub<Sym(40)| (1,5,9,13,17)(2,6,10,14,18)(3,7,11,15,19)(4,8,12,16,20)(21,37,33,29,25)(22,38,34,30,26)(23,39,35,31,27)(24,40,36,32,28), (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)>;
G:=Group( (1,5,9,13,17)(2,6,10,14,18)(3,7,11,15,19)(4,8,12,16,20)(21,37,33,29,25)(22,38,34,30,26)(23,39,35,31,27)(24,40,36,32,28), (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31) );
G=PermutationGroup([[(1,5,9,13,17),(2,6,10,14,18),(3,7,11,15,19),(4,8,12,16,20),(21,37,33,29,25),(22,38,34,30,26),(23,39,35,31,27),(24,40,36,32,28)], [(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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,40),(12,39),(13,38),(14,37),(15,36),(16,35),(17,34),(18,33),(19,32),(20,31)]])
C5×D20 is a maximal subgroup of
C52⋊2D8 C5⋊D40 D20.D5 C52⋊3SD16 D20⋊5D5 D20⋊D5 C20⋊D10 C5×D4×D5
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 10A | 10B | 10C | 10D | 10E | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 10 | 10 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | D4 | D5 | D10 | D20 | C5×D4 | C5×D5 | D5×C10 | C5×D20 |
| kernel | C5×D20 | C5×C20 | D5×C10 | D20 | C20 | D10 | C52 | C20 | C10 | C5 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C5×D20 ►in GL2(𝔽41) generated by
| 10 | 0 |
| 0 | 10 |
| 36 | 0 |
| 0 | 8 |
| 0 | 8 |
| 36 | 0 |
G:=sub<GL(2,GF(41))| [10,0,0,10],[36,0,0,8],[0,36,8,0] >;
C5×D20 in GAP, Magma, Sage, TeX
C_5\times D_{20} % in TeX
G:=Group("C5xD20"); // GroupNames label
G:=SmallGroup(200,29);
// by ID
G=gap.SmallGroup(200,29);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-5,221,106,4004]);
// Polycyclic
G:=Group<a,b,c|a^5=b^20=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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