direct product, metabelian, supersoluble, monomial, A-group
Aliases: D5×C52, C53⋊1C2, C52⋊3C10, C5⋊(C5×C10), SmallGroup(250,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C52 |
Generators and relations for D5×C52
G = < a,b,c,d | a5=b5=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 112 in 56 conjugacy classes, 24 normal (6 characteristic)
C1, C2, C5, C5, C5, D5, C10, C52, C52, C52, C5×D5, C5×C10, C53, D5×C52
Quotients: C1, C2, C5, D5, C10, C52, C5×D5, C5×C10, D5×C52
►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 35 40 45 26)(2 31 36 41 27)(3 32 37 42 28)(4 33 38 43 29)(5 34 39 44 30)(6 11 16 21 50)(7 12 17 22 46)(8 13 18 23 47)(9 14 19 24 48)(10 15 20 25 49)
(1 26 45 40 35)(2 27 41 36 31)(3 28 42 37 32)(4 29 43 38 33)(5 30 44 39 34)(6 11 16 21 50)(7 12 17 22 46)(8 13 18 23 47)(9 14 19 24 48)(10 15 20 25 49)
(1 50)(2 46)(3 47)(4 48)(5 49)(6 35)(7 31)(8 32)(9 33)(10 34)(11 40)(12 36)(13 37)(14 38)(15 39)(16 45)(17 41)(18 42)(19 43)(20 44)(21 26)(22 27)(23 28)(24 29)(25 30)
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), (1,35,40,45,26)(2,31,36,41,27)(3,32,37,42,28)(4,33,38,43,29)(5,34,39,44,30)(6,11,16,21,50)(7,12,17,22,46)(8,13,18,23,47)(9,14,19,24,48)(10,15,20,25,49), (1,26,45,40,35)(2,27,41,36,31)(3,28,42,37,32)(4,29,43,38,33)(5,30,44,39,34)(6,11,16,21,50)(7,12,17,22,46)(8,13,18,23,47)(9,14,19,24,48)(10,15,20,25,49), (1,50)(2,46)(3,47)(4,48)(5,49)(6,35)(7,31)(8,32)(9,33)(10,34)(11,40)(12,36)(13,37)(14,38)(15,39)(16,45)(17,41)(18,42)(19,43)(20,44)(21,26)(22,27)(23,28)(24,29)(25,30)>;
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), (1,35,40,45,26)(2,31,36,41,27)(3,32,37,42,28)(4,33,38,43,29)(5,34,39,44,30)(6,11,16,21,50)(7,12,17,22,46)(8,13,18,23,47)(9,14,19,24,48)(10,15,20,25,49), (1,26,45,40,35)(2,27,41,36,31)(3,28,42,37,32)(4,29,43,38,33)(5,30,44,39,34)(6,11,16,21,50)(7,12,17,22,46)(8,13,18,23,47)(9,14,19,24,48)(10,15,20,25,49), (1,50)(2,46)(3,47)(4,48)(5,49)(6,35)(7,31)(8,32)(9,33)(10,34)(11,40)(12,36)(13,37)(14,38)(15,39)(16,45)(17,41)(18,42)(19,43)(20,44)(21,26)(22,27)(23,28)(24,29)(25,30) );
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)], [(1,35,40,45,26),(2,31,36,41,27),(3,32,37,42,28),(4,33,38,43,29),(5,34,39,44,30),(6,11,16,21,50),(7,12,17,22,46),(8,13,18,23,47),(9,14,19,24,48),(10,15,20,25,49)], [(1,26,45,40,35),(2,27,41,36,31),(3,28,42,37,32),(4,29,43,38,33),(5,30,44,39,34),(6,11,16,21,50),(7,12,17,22,46),(8,13,18,23,47),(9,14,19,24,48),(10,15,20,25,49)], [(1,50),(2,46),(3,47),(4,48),(5,49),(6,35),(7,31),(8,32),(9,33),(10,34),(11,40),(12,36),(13,37),(14,38),(15,39),(16,45),(17,41),(18,42),(19,43),(20,44),(21,26),(22,27),(23,28),(24,29),(25,30)]])
D5×C52 is a maximal subgroup of
C53⋊C4
100 conjugacy classes
| class | 1 | 2 | 5A | ··· | 5X | 5Y | ··· | 5BV | 10A | ··· | 10X |
| order | 1 | 2 | 5 | ··· | 5 | 5 | ··· | 5 | 10 | ··· | 10 |
| size | 1 | 5 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C5 | C10 | D5 | C5×D5 |
| kernel | D5×C52 | C53 | C5×D5 | C52 | C52 | C5 |
| # reps | 1 | 1 | 24 | 24 | 2 | 48 |
Matrix representation of D5×C52 ►in GL3(𝔽11) generated by
| 1 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 9 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 1 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 4 |
| 10 | 0 | 0 |
| 0 | 0 | 4 |
| 0 | 3 | 0 |
G:=sub<GL(3,GF(11))| [1,0,0,0,9,0,0,0,9],[9,0,0,0,4,0,0,0,4],[1,0,0,0,3,0,0,0,4],[10,0,0,0,0,3,0,4,0] >;
D5×C52 in GAP, Magma, Sage, TeX
D_5\times C_5^2
% in TeX
G:=Group("D5xC5^2"); // GroupNames label
G:=SmallGroup(250,12);
// by ID
G=gap.SmallGroup(250,12);
# by ID
G:=PCGroup([4,-2,-5,-5,-5,3203]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations