direct product, metacyclic, supersoluble, monomial, A-group
Aliases: Dic5×C20, C20⋊4C20, C52⋊9C42, C102.16C22, C5⋊2(C4×C20), (C5×C20)⋊11C4, C2.2(D5×C20), (C2×C20).9C10, (C2×C20).22D5, C10.30(C4×D5), C10.14(C2×C20), (C10×C20).11C2, (C2×C10).40D10, C2.2(C10×Dic5), C22.3(D5×C10), (C2×Dic5).6C10, C10.26(C2×Dic5), (C10×Dic5).12C2, (C2×C4).6(C5×D5), (C2×C10).5(C2×C10), (C5×C10).52(C2×C4), SmallGroup(400,83)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — Dic5×C20 |
Generators and relations for Dic5×C20
G = < a,b,c | a20=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 148 in 76 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, C2×C4, C10, C10, C10, C42, Dic5, C20, C20, C2×C10, C2×C10, C52, C2×Dic5, C2×C20, C2×C20, C5×C10, C5×C10, C4×Dic5, C4×C20, C5×Dic5, C5×C20, C102, C10×Dic5, C10×C20, Dic5×C20
Quotients: C1, C2, C4, C22, C5, C2×C4, D5, C10, C42, Dic5, C20, D10, C2×C10, C4×D5, C2×Dic5, C2×C20, C5×D5, C4×Dic5, C4×C20, C5×Dic5, D5×C10, D5×C20, C10×Dic5, Dic5×C20
►On 80 points
(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 32 13 24 5 36 17 28 9 40)(2 33 14 25 6 37 18 29 10 21)(3 34 15 26 7 38 19 30 11 22)(4 35 16 27 8 39 20 31 12 23)(41 80 49 68 57 76 45 64 53 72)(42 61 50 69 58 77 46 65 54 73)(43 62 51 70 59 78 47 66 55 74)(44 63 52 71 60 79 48 67 56 75)
(1 53 36 68)(2 54 37 69)(3 55 38 70)(4 56 39 71)(5 57 40 72)(6 58 21 73)(7 59 22 74)(8 60 23 75)(9 41 24 76)(10 42 25 77)(11 43 26 78)(12 44 27 79)(13 45 28 80)(14 46 29 61)(15 47 30 62)(16 48 31 63)(17 49 32 64)(18 50 33 65)(19 51 34 66)(20 52 35 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), (1,32,13,24,5,36,17,28,9,40)(2,33,14,25,6,37,18,29,10,21)(3,34,15,26,7,38,19,30,11,22)(4,35,16,27,8,39,20,31,12,23)(41,80,49,68,57,76,45,64,53,72)(42,61,50,69,58,77,46,65,54,73)(43,62,51,70,59,78,47,66,55,74)(44,63,52,71,60,79,48,67,56,75), (1,53,36,68)(2,54,37,69)(3,55,38,70)(4,56,39,71)(5,57,40,72)(6,58,21,73)(7,59,22,74)(8,60,23,75)(9,41,24,76)(10,42,25,77)(11,43,26,78)(12,44,27,79)(13,45,28,80)(14,46,29,61)(15,47,30,62)(16,48,31,63)(17,49,32,64)(18,50,33,65)(19,51,34,66)(20,52,35,67)>;
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), (1,32,13,24,5,36,17,28,9,40)(2,33,14,25,6,37,18,29,10,21)(3,34,15,26,7,38,19,30,11,22)(4,35,16,27,8,39,20,31,12,23)(41,80,49,68,57,76,45,64,53,72)(42,61,50,69,58,77,46,65,54,73)(43,62,51,70,59,78,47,66,55,74)(44,63,52,71,60,79,48,67,56,75), (1,53,36,68)(2,54,37,69)(3,55,38,70)(4,56,39,71)(5,57,40,72)(6,58,21,73)(7,59,22,74)(8,60,23,75)(9,41,24,76)(10,42,25,77)(11,43,26,78)(12,44,27,79)(13,45,28,80)(14,46,29,61)(15,47,30,62)(16,48,31,63)(17,49,32,64)(18,50,33,65)(19,51,34,66)(20,52,35,67) );
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)], [(1,32,13,24,5,36,17,28,9,40),(2,33,14,25,6,37,18,29,10,21),(3,34,15,26,7,38,19,30,11,22),(4,35,16,27,8,39,20,31,12,23),(41,80,49,68,57,76,45,64,53,72),(42,61,50,69,58,77,46,65,54,73),(43,62,51,70,59,78,47,66,55,74),(44,63,52,71,60,79,48,67,56,75)], [(1,53,36,68),(2,54,37,69),(3,55,38,70),(4,56,39,71),(5,57,40,72),(6,58,21,73),(7,59,22,74),(8,60,23,75),(9,41,24,76),(10,42,25,77),(11,43,26,78),(12,44,27,79),(13,45,28,80),(14,46,29,61),(15,47,30,62),(16,48,31,63),(17,49,32,64),(18,50,33,65),(19,51,34,66),(20,52,35,67)]])
160 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 10A | ··· | 10L | 10M | ··· | 10AP | 20A | ··· | 20P | 20Q | ··· | 20BD | 20BE | ··· | 20CJ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | ··· | 5 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 |
160 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C20 | C20 | D5 | Dic5 | D10 | C4×D5 | C5×D5 | C5×Dic5 | D5×C10 | D5×C20 |
| kernel | Dic5×C20 | C10×Dic5 | C10×C20 | C5×Dic5 | C5×C20 | C4×Dic5 | C2×Dic5 | C2×C20 | Dic5 | C20 | C2×C20 | C20 | C2×C10 | C10 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 4 | 8 | 4 | 32 | 16 | 2 | 4 | 2 | 8 | 8 | 16 | 8 | 32 |
Matrix representation of Dic5×C20 ►in GL3(𝔽41) generated by
| 18 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 40 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 37 |
| 9 | 0 | 0 |
| 0 | 0 | 9 |
| 0 | 32 | 0 |
G:=sub<GL(3,GF(41))| [18,0,0,0,8,0,0,0,8],[40,0,0,0,10,0,0,0,37],[9,0,0,0,0,32,0,9,0] >;
Dic5×C20 in GAP, Magma, Sage, TeX
{\rm Dic}_5\times C_{20} % in TeX
G:=Group("Dic5xC20"); // GroupNames label
G:=SmallGroup(400,83);
// by ID
G=gap.SmallGroup(400,83);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-5,120,247,11525]);
// Polycyclic
G:=Group<a,b,c|a^20=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations