metabelian, supersoluble, monomial
Aliases: C20.31D10, C52⋊10M4(2), C4.16D52, C5⋊2C8⋊5D5, C10.9(C4×D5), C5⋊3(C8⋊D5), C52⋊6C4.5C4, (C5×C20).30C22, C2.3(Dic5⋊2D5), (C5×C5⋊2C8)⋊7C2, (C4×C5⋊D5).4C2, (C2×C5⋊D5).5C4, (C5×C10).44(C2×C4), SmallGroup(400,63)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.31D10
G = < a,b,c | a20=c2=1, b10=a5, bab-1=cac=a9, cbc=a10b9 >
Subgroups: 364 in 56 conjugacy classes, 20 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C8, C2×C4, D5, C10, C10, M4(2), Dic5, C20, C20, D10, C52, C5⋊2C8, C40, C4×D5, C5⋊D5, C5×C10, C8⋊D5, C52⋊6C4, C5×C20, C2×C5⋊D5, C5×C5⋊2C8, C4×C5⋊D5, C20.31D10
Quotients: C1, C2, C4, C22, C2×C4, D5, M4(2), D10, C4×D5, C8⋊D5, D52, Dic5⋊2D5, C20.31D10
►On 40 points
(1 35 29 23 17 11 5 39 33 27 21 15 9 3 37 31 25 19 13 7)(2 28 14 40 26 12 38 24 10 36 22 8 34 20 6 32 18 4 30 16)
(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)
(1 5)(2 34)(3 23)(4 12)(6 30)(7 19)(9 37)(10 26)(11 15)(13 33)(14 22)(16 40)(17 29)(20 36)(21 25)(24 32)(27 39)(31 35)
G:=sub<Sym(40)| (1,35,29,23,17,11,5,39,33,27,21,15,9,3,37,31,25,19,13,7)(2,28,14,40,26,12,38,24,10,36,22,8,34,20,6,32,18,4,30,16), (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), (1,5)(2,34)(3,23)(4,12)(6,30)(7,19)(9,37)(10,26)(11,15)(13,33)(14,22)(16,40)(17,29)(20,36)(21,25)(24,32)(27,39)(31,35)>;
G:=Group( (1,35,29,23,17,11,5,39,33,27,21,15,9,3,37,31,25,19,13,7)(2,28,14,40,26,12,38,24,10,36,22,8,34,20,6,32,18,4,30,16), (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), (1,5)(2,34)(3,23)(4,12)(6,30)(7,19)(9,37)(10,26)(11,15)(13,33)(14,22)(16,40)(17,29)(20,36)(21,25)(24,32)(27,39)(31,35) );
G=PermutationGroup([[(1,35,29,23,17,11,5,39,33,27,21,15,9,3,37,31,25,19,13,7),(2,28,14,40,26,12,38,24,10,36,22,8,34,20,6,32,18,4,30,16)], [(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)], [(1,5),(2,34),(3,23),(4,12),(6,30),(7,19),(9,37),(10,26),(11,15),(13,33),(14,22),(16,40),(17,29),(20,36),(21,25),(24,32),(27,39),(31,35)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 50 | 1 | 1 | 50 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 10 | ··· | 10 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | D5 | M4(2) | D10 | C4×D5 | C8⋊D5 | D52 | Dic5⋊2D5 | C20.31D10 |
| kernel | C20.31D10 | C5×C5⋊2C8 | C4×C5⋊D5 | C52⋊6C4 | C2×C5⋊D5 | C5⋊2C8 | C52 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 8 | 16 | 4 | 4 | 8 |
Matrix representation of C20.31D10 ►in GL4(𝔽41) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 34 |
| 31 | 9 | 0 | 0 |
| 32 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 34 | 40 |
| 1 | 7 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 34 | 40 |
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,0,40,0,0,1,34],[31,32,0,0,9,9,0,0,0,0,1,34,0,0,0,40],[1,0,0,0,7,40,0,0,0,0,1,34,0,0,0,40] >;
C20.31D10 in GAP, Magma, Sage, TeX
C_{20}._{31}D_{10} % in TeX
G:=Group("C20.31D10"); // GroupNames label
G:=SmallGroup(400,63);
// by ID
G=gap.SmallGroup(400,63);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,31,50,970,11525]);
// Polycyclic
G:=Group<a,b,c|a^20=c^2=1,b^10=a^5,b*a*b^-1=c*a*c=a^9,c*b*c=a^10*b^9>;
// generators/relations