direct product, metabelian, supersoluble, monomial, A-group
Aliases: Dic5xC52, C53:10C4, C52:8C20, C5:2(C5xC20), C10.(C5xC10), C2.(D5xC52), (C5xC10).7C10, C10.10(C5xD5), (C5xC10).11D5, (C52xC10).1C2, SmallGroup(500,37)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — Dic5xC52 |
Generators and relations for Dic5xC52
G = < a,b,c,d | a5=b5=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 176 in 96 conjugacy classes, 40 normal (10 characteristic)
C1, C2, C4, C5, C5, C5, C10, C10, C10, Dic5, C20, C52, C52, C52, C5xC10, C5xC10, C5xC10, C5xDic5, C5xC20, C53, C52xC10, Dic5xC52
Quotients: C1, C2, C4, C5, D5, C10, Dic5, C20, C52, C5xD5, C5xC10, C5xDic5, C5xC20, D5xC52, Dic5xC52
►On 100 points
Generators in S100
(1 37 15 43 21)(2 38 16 44 22)(3 39 17 45 23)(4 40 18 46 24)(5 31 19 47 25)(6 32 20 48 26)(7 33 11 49 27)(8 34 12 50 28)(9 35 13 41 29)(10 36 14 42 30)(51 83 65 97 79)(52 84 66 98 80)(53 85 67 99 71)(54 86 68 100 72)(55 87 69 91 73)(56 88 70 92 74)(57 89 61 93 75)(58 90 62 94 76)(59 81 63 95 77)(60 82 64 96 78)
(1 49 39 29 19)(2 50 40 30 20)(3 41 31 21 11)(4 42 32 22 12)(5 43 33 23 13)(6 44 34 24 14)(7 45 35 25 15)(8 46 36 26 16)(9 47 37 27 17)(10 48 38 28 18)(51 91 81 71 61)(52 92 82 72 62)(53 93 83 73 63)(54 94 84 74 64)(55 95 85 75 65)(56 96 86 76 66)(57 97 87 77 67)(58 98 88 78 68)(59 99 89 79 69)(60 100 90 80 70)
(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 56 6 51)(2 55 7 60)(3 54 8 59)(4 53 9 58)(5 52 10 57)(11 64 16 69)(12 63 17 68)(13 62 18 67)(14 61 19 66)(15 70 20 65)(21 74 26 79)(22 73 27 78)(23 72 28 77)(24 71 29 76)(25 80 30 75)(31 84 36 89)(32 83 37 88)(33 82 38 87)(34 81 39 86)(35 90 40 85)(41 94 46 99)(42 93 47 98)(43 92 48 97)(44 91 49 96)(45 100 50 95)
G:=sub<Sym(100)| (1,37,15,43,21)(2,38,16,44,22)(3,39,17,45,23)(4,40,18,46,24)(5,31,19,47,25)(6,32,20,48,26)(7,33,11,49,27)(8,34,12,50,28)(9,35,13,41,29)(10,36,14,42,30)(51,83,65,97,79)(52,84,66,98,80)(53,85,67,99,71)(54,86,68,100,72)(55,87,69,91,73)(56,88,70,92,74)(57,89,61,93,75)(58,90,62,94,76)(59,81,63,95,77)(60,82,64,96,78), (1,49,39,29,19)(2,50,40,30,20)(3,41,31,21,11)(4,42,32,22,12)(5,43,33,23,13)(6,44,34,24,14)(7,45,35,25,15)(8,46,36,26,16)(9,47,37,27,17)(10,48,38,28,18)(51,91,81,71,61)(52,92,82,72,62)(53,93,83,73,63)(54,94,84,74,64)(55,95,85,75,65)(56,96,86,76,66)(57,97,87,77,67)(58,98,88,78,68)(59,99,89,79,69)(60,100,90,80,70), (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), (1,56,6,51)(2,55,7,60)(3,54,8,59)(4,53,9,58)(5,52,10,57)(11,64,16,69)(12,63,17,68)(13,62,18,67)(14,61,19,66)(15,70,20,65)(21,74,26,79)(22,73,27,78)(23,72,28,77)(24,71,29,76)(25,80,30,75)(31,84,36,89)(32,83,37,88)(33,82,38,87)(34,81,39,86)(35,90,40,85)(41,94,46,99)(42,93,47,98)(43,92,48,97)(44,91,49,96)(45,100,50,95)>;
G:=Group( (1,37,15,43,21)(2,38,16,44,22)(3,39,17,45,23)(4,40,18,46,24)(5,31,19,47,25)(6,32,20,48,26)(7,33,11,49,27)(8,34,12,50,28)(9,35,13,41,29)(10,36,14,42,30)(51,83,65,97,79)(52,84,66,98,80)(53,85,67,99,71)(54,86,68,100,72)(55,87,69,91,73)(56,88,70,92,74)(57,89,61,93,75)(58,90,62,94,76)(59,81,63,95,77)(60,82,64,96,78), (1,49,39,29,19)(2,50,40,30,20)(3,41,31,21,11)(4,42,32,22,12)(5,43,33,23,13)(6,44,34,24,14)(7,45,35,25,15)(8,46,36,26,16)(9,47,37,27,17)(10,48,38,28,18)(51,91,81,71,61)(52,92,82,72,62)(53,93,83,73,63)(54,94,84,74,64)(55,95,85,75,65)(56,96,86,76,66)(57,97,87,77,67)(58,98,88,78,68)(59,99,89,79,69)(60,100,90,80,70), (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), (1,56,6,51)(2,55,7,60)(3,54,8,59)(4,53,9,58)(5,52,10,57)(11,64,16,69)(12,63,17,68)(13,62,18,67)(14,61,19,66)(15,70,20,65)(21,74,26,79)(22,73,27,78)(23,72,28,77)(24,71,29,76)(25,80,30,75)(31,84,36,89)(32,83,37,88)(33,82,38,87)(34,81,39,86)(35,90,40,85)(41,94,46,99)(42,93,47,98)(43,92,48,97)(44,91,49,96)(45,100,50,95) );
G=PermutationGroup([[(1,37,15,43,21),(2,38,16,44,22),(3,39,17,45,23),(4,40,18,46,24),(5,31,19,47,25),(6,32,20,48,26),(7,33,11,49,27),(8,34,12,50,28),(9,35,13,41,29),(10,36,14,42,30),(51,83,65,97,79),(52,84,66,98,80),(53,85,67,99,71),(54,86,68,100,72),(55,87,69,91,73),(56,88,70,92,74),(57,89,61,93,75),(58,90,62,94,76),(59,81,63,95,77),(60,82,64,96,78)], [(1,49,39,29,19),(2,50,40,30,20),(3,41,31,21,11),(4,42,32,22,12),(5,43,33,23,13),(6,44,34,24,14),(7,45,35,25,15),(8,46,36,26,16),(9,47,37,27,17),(10,48,38,28,18),(51,91,81,71,61),(52,92,82,72,62),(53,93,83,73,63),(54,94,84,74,64),(55,95,85,75,65),(56,96,86,76,66),(57,97,87,77,67),(58,98,88,78,68),(59,99,89,79,69),(60,100,90,80,70)], [(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)], [(1,56,6,51),(2,55,7,60),(3,54,8,59),(4,53,9,58),(5,52,10,57),(11,64,16,69),(12,63,17,68),(13,62,18,67),(14,61,19,66),(15,70,20,65),(21,74,26,79),(22,73,27,78),(23,72,28,77),(24,71,29,76),(25,80,30,75),(31,84,36,89),(32,83,37,88),(33,82,38,87),(34,81,39,86),(35,90,40,85),(41,94,46,99),(42,93,47,98),(43,92,48,97),(44,91,49,96),(45,100,50,95)]])
200 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | ··· | 5X | 5Y | ··· | 5BV | 10A | ··· | 10X | 10Y | ··· | 10BV | 20A | ··· | 20AV |
| order | 1 | 2 | 4 | 4 | 5 | ··· | 5 | 5 | ··· | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 5 | 5 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | D5 | Dic5 | C5xD5 | C5xDic5 |
| kernel | Dic5xC52 | C52xC10 | C53 | C5xDic5 | C5xC10 | C52 | C5xC10 | C52 | C10 | C5 |
| # reps | 1 | 1 | 2 | 24 | 24 | 48 | 2 | 2 | 48 | 48 |
Matrix representation of Dic5xC52 ►in GL4(F41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 37 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 10 |
| 9 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,10,0,0,0,0,10],[37,0,0,0,0,16,0,0,0,0,18,0,0,0,0,18],[40,0,0,0,0,1,0,0,0,0,37,0,0,0,0,10],[9,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0] >;
Dic5xC52 in GAP, Magma, Sage, TeX
{\rm Dic}_5\times C_5^2 % in TeX
G:=Group("Dic5xC5^2"); // GroupNames label
G:=SmallGroup(500,37);
// by ID
G=gap.SmallGroup(500,37);
# by ID
G:=PCGroup([5,-2,-5,-5,-2,-5,250,10004]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^10=1,d^2=c^5,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations