direct product, metacyclic, supersoluble, monomial
Aliases: C5×Dic20, C40.9D5, C40.1C10, C52⋊4Q16, C10.21D20, C20.62D10, Dic10.1C10, C8.(C5×D5), C5⋊1(C5×Q16), (C5×C40).2C2, C10.3(C5×D4), C2.5(C5×D20), C4.10(D5×C10), (C5×C10).19D4, C20.10(C2×C10), (C5×C20).39C22, (C5×Dic10).4C2, SmallGroup(400,80)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Dic20
G = < a,b,c | a5=b40=1, c2=b20, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×Dic20
►On 80 points
Generators in S80
(1 25 9 33 17)(2 26 10 34 18)(3 27 11 35 19)(4 28 12 36 20)(5 29 13 37 21)(6 30 14 38 22)(7 31 15 39 23)(8 32 16 40 24)(41 57 73 49 65)(42 58 74 50 66)(43 59 75 51 67)(44 60 76 52 68)(45 61 77 53 69)(46 62 78 54 70)(47 63 79 55 71)(48 64 80 56 72)
(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 56 21 76)(2 55 22 75)(3 54 23 74)(4 53 24 73)(5 52 25 72)(6 51 26 71)(7 50 27 70)(8 49 28 69)(9 48 29 68)(10 47 30 67)(11 46 31 66)(12 45 32 65)(13 44 33 64)(14 43 34 63)(15 42 35 62)(16 41 36 61)(17 80 37 60)(18 79 38 59)(19 78 39 58)(20 77 40 57)
G:=sub<Sym(80)| (1,25,9,33,17)(2,26,10,34,18)(3,27,11,35,19)(4,28,12,36,20)(5,29,13,37,21)(6,30,14,38,22)(7,31,15,39,23)(8,32,16,40,24)(41,57,73,49,65)(42,58,74,50,66)(43,59,75,51,67)(44,60,76,52,68)(45,61,77,53,69)(46,62,78,54,70)(47,63,79,55,71)(48,64,80,56,72), (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), (1,56,21,76)(2,55,22,75)(3,54,23,74)(4,53,24,73)(5,52,25,72)(6,51,26,71)(7,50,27,70)(8,49,28,69)(9,48,29,68)(10,47,30,67)(11,46,31,66)(12,45,32,65)(13,44,33,64)(14,43,34,63)(15,42,35,62)(16,41,36,61)(17,80,37,60)(18,79,38,59)(19,78,39,58)(20,77,40,57)>;
G:=Group( (1,25,9,33,17)(2,26,10,34,18)(3,27,11,35,19)(4,28,12,36,20)(5,29,13,37,21)(6,30,14,38,22)(7,31,15,39,23)(8,32,16,40,24)(41,57,73,49,65)(42,58,74,50,66)(43,59,75,51,67)(44,60,76,52,68)(45,61,77,53,69)(46,62,78,54,70)(47,63,79,55,71)(48,64,80,56,72), (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), (1,56,21,76)(2,55,22,75)(3,54,23,74)(4,53,24,73)(5,52,25,72)(6,51,26,71)(7,50,27,70)(8,49,28,69)(9,48,29,68)(10,47,30,67)(11,46,31,66)(12,45,32,65)(13,44,33,64)(14,43,34,63)(15,42,35,62)(16,41,36,61)(17,80,37,60)(18,79,38,59)(19,78,39,58)(20,77,40,57) );
G=PermutationGroup([[(1,25,9,33,17),(2,26,10,34,18),(3,27,11,35,19),(4,28,12,36,20),(5,29,13,37,21),(6,30,14,38,22),(7,31,15,39,23),(8,32,16,40,24),(41,57,73,49,65),(42,58,74,50,66),(43,59,75,51,67),(44,60,76,52,68),(45,61,77,53,69),(46,62,78,54,70),(47,63,79,55,71),(48,64,80,56,72)], [(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)], [(1,56,21,76),(2,55,22,75),(3,54,23,74),(4,53,24,73),(5,52,25,72),(6,51,26,71),(7,50,27,70),(8,49,28,69),(9,48,29,68),(10,47,30,67),(11,46,31,66),(12,45,32,65),(13,44,33,64),(14,43,34,63),(15,42,35,62),(16,41,36,61),(17,80,37,60),(18,79,38,59),(19,78,39,58),(20,77,40,57)]])
115 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 8A | 8B | 10A | 10B | 10C | 10D | 10E | ··· | 10N | 20A | ··· | 20X | 20Y | ··· | 20AF | 40A | ··· | 40AV |
| order | 1 | 2 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 20 | 20 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 20 | ··· | 20 | 2 | ··· | 2 |
115 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | |||||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | D4 | D5 | Q16 | D10 | D20 | C5×D4 | C5×D5 | Dic20 | C5×Q16 | D5×C10 | C5×D20 | C5×Dic20 |
| kernel | C5×Dic20 | C5×C40 | C5×Dic10 | Dic20 | C40 | Dic10 | C5×C10 | C40 | C52 | C20 | C10 | C10 | C8 | C5 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | 32 |
Matrix representation of C5×Dic20 ►in GL2(𝔽41) generated by
| 18 | 0 |
| 0 | 18 |
| 34 | 0 |
| 0 | 35 |
| 0 | 1 |
| 40 | 0 |
G:=sub<GL(2,GF(41))| [18,0,0,18],[34,0,0,35],[0,40,1,0] >;
C5×Dic20 in GAP, Magma, Sage, TeX
C_5\times {\rm Dic}_{20} % in TeX
G:=Group("C5xDic20"); // GroupNames label
G:=SmallGroup(400,80);
// by ID
G=gap.SmallGroup(400,80);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-5,240,265,367,1443,69,11525]);
// Polycyclic
G:=Group<a,b,c|a^5=b^40=1,c^2=b^20,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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