direct product, metacyclic, supersoluble, monomial, A-group
Aliases: D5×C25, C5⋊C50, C52.2C10, (C5×C25)⋊1C2, (C5×D5).C5, C5.4(C5×D5), SmallGroup(250,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C25 |
Generators and relations for D5×C25
G = < a,b,c | a25=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C25
►On 50 points
Generators in S50
(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)
(1 21 16 11 6)(2 22 17 12 7)(3 23 18 13 8)(4 24 19 14 9)(5 25 20 15 10)(26 31 36 41 46)(27 32 37 42 47)(28 33 38 43 48)(29 34 39 44 49)(30 35 40 45 50)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 26)(24 27)(25 28)
G:=sub<Sym(50)| (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), (1,21,16,11,6)(2,22,17,12,7)(3,23,18,13,8)(4,24,19,14,9)(5,25,20,15,10)(26,31,36,41,46)(27,32,37,42,47)(28,33,38,43,48)(29,34,39,44,49)(30,35,40,45,50), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,26)(24,27)(25,28)>;
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), (1,21,16,11,6)(2,22,17,12,7)(3,23,18,13,8)(4,24,19,14,9)(5,25,20,15,10)(26,31,36,41,46)(27,32,37,42,47)(28,33,38,43,48)(29,34,39,44,49)(30,35,40,45,50), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,26)(24,27)(25,28) );
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)], [(1,21,16,11,6),(2,22,17,12,7),(3,23,18,13,8),(4,24,19,14,9),(5,25,20,15,10),(26,31,36,41,46),(27,32,37,42,47),(28,33,38,43,48),(29,34,39,44,49),(30,35,40,45,50)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,26),(24,27),(25,28)]])
D5×C25 is a maximal subgroup of
D5.D25
100 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 10A | 10B | 10C | 10D | 25A | ··· | 25T | 25U | ··· | 25BH | 50A | ··· | 50T |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 25 | ··· | 25 | 25 | ··· | 25 | 50 | ··· | 50 |
| size | 1 | 5 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | ||||||
| image | C1 | C2 | C5 | C10 | C25 | C50 | D5 | C5×D5 | D5×C25 |
| kernel | D5×C25 | C5×C25 | C5×D5 | C52 | D5 | C5 | C25 | C5 | C1 |
| # reps | 1 | 1 | 4 | 4 | 20 | 20 | 2 | 8 | 40 |
Matrix representation of D5×C25 ►in GL2(𝔽101) generated by
| 79 | 0 |
| 0 | 79 |
| 84 | 0 |
| 97 | 95 |
| 25 | 7 |
| 84 | 76 |
G:=sub<GL(2,GF(101))| [79,0,0,79],[84,97,0,95],[25,84,7,76] >;
D5×C25 in GAP, Magma, Sage, TeX
D_5\times C_{25} % in TeX
G:=Group("D5xC25"); // GroupNames label
G:=SmallGroup(250,4);
// by ID
G=gap.SmallGroup(250,4);
# by ID
G:=PCGroup([4,-2,-5,-5,-5,45,3203]);
// Polycyclic
G:=Group<a,b,c|a^25=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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