metabelian, supersoluble, monomial
Aliases: C20⋊2F5, (C5×C20)⋊4C4, C5⋊3(C4⋊F5), C4⋊(C52⋊C4), C5⋊D5.6D4, C5⋊D5.4Q8, C52⋊8(C4⋊C4), C52⋊6C4⋊9C4, C10.24(C2×F5), (C4×C5⋊D5).9C2, C2.5(C2×C52⋊C4), (C5×C10).37(C2×C4), (C2×C52⋊C4).5C2, (C2×C5⋊D5).25C22, SmallGroup(400,159)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊2F5
G = < a,b,c | a20=b5=c4=1, ab=ba, cac-1=a7, cbc-1=b3 >
Subgroups: 556 in 68 conjugacy classes, 20 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, D5, C10, C10, C4⋊C4, Dic5, C20, C20, F5, D10, C52, C4×D5, C2×F5, C5⋊D5, C5×C10, C4⋊F5, C52⋊6C4, C5×C20, C52⋊C4, C2×C5⋊D5, C4×C5⋊D5, C2×C52⋊C4, C20⋊2F5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C4⋊C4, F5, C2×F5, C4⋊F5, C52⋊C4, C2×C52⋊C4, C20⋊2F5
►On 40 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)
(1 5 9 13 17)(2 6 10 14 18)(3 7 11 15 19)(4 8 12 16 20)(21 37 33 29 25)(22 38 34 30 26)(23 39 35 31 27)(24 40 36 32 28)
(1 29 11 39)(2 32 20 26)(3 35 9 33)(4 38 18 40)(5 21 7 27)(6 24 16 34)(8 30 14 28)(10 36 12 22)(13 25 19 23)(15 31 17 37)
G:=sub<Sym(40)| (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), (1,5,9,13,17)(2,6,10,14,18)(3,7,11,15,19)(4,8,12,16,20)(21,37,33,29,25)(22,38,34,30,26)(23,39,35,31,27)(24,40,36,32,28), (1,29,11,39)(2,32,20,26)(3,35,9,33)(4,38,18,40)(5,21,7,27)(6,24,16,34)(8,30,14,28)(10,36,12,22)(13,25,19,23)(15,31,17,37)>;
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), (1,5,9,13,17)(2,6,10,14,18)(3,7,11,15,19)(4,8,12,16,20)(21,37,33,29,25)(22,38,34,30,26)(23,39,35,31,27)(24,40,36,32,28), (1,29,11,39)(2,32,20,26)(3,35,9,33)(4,38,18,40)(5,21,7,27)(6,24,16,34)(8,30,14,28)(10,36,12,22)(13,25,19,23)(15,31,17,37) );
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)], [(1,5,9,13,17),(2,6,10,14,18),(3,7,11,15,19),(4,8,12,16,20),(21,37,33,29,25),(22,38,34,30,26),(23,39,35,31,27),(24,40,36,32,28)], [(1,29,11,39),(2,32,20,26),(3,35,9,33),(4,38,18,40),(5,21,7,27),(6,24,16,34),(8,30,14,28),(10,36,12,22),(13,25,19,23),(15,31,17,37)]])
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | ··· | 4F | 5A | ··· | 5F | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | ··· | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 25 | 25 | 2 | 50 | ··· | 50 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | F5 | C2×F5 | C4⋊F5 | C52⋊C4 | C2×C52⋊C4 | C20⋊2F5 |
| kernel | C20⋊2F5 | C4×C5⋊D5 | C2×C52⋊C4 | C52⋊6C4 | C5×C20 | C5⋊D5 | C5⋊D5 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C20⋊2F5 ►in GL4(𝔽41) generated by
| 0 | 9 | 0 | 0 |
| 32 | 22 | 0 | 0 |
| 0 | 0 | 32 | 22 |
| 0 | 0 | 19 | 22 |
| 34 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 7 |
| 0 | 0 | 34 | 7 |
| 0 | 0 | 40 | 7 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(41))| [0,32,0,0,9,22,0,0,0,0,32,19,0,0,22,22],[34,40,0,0,1,0,0,0,0,0,40,34,0,0,7,7],[0,0,1,0,0,0,0,1,40,0,0,0,7,1,0,0] >;
C20⋊2F5 in GAP, Magma, Sage, TeX
C_{20}\rtimes_2F_5 % in TeX
G:=Group("C20:2F5"); // GroupNames label
G:=SmallGroup(400,159);
// by ID
G=gap.SmallGroup(400,159);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,55,1444,496,5765,2897]);
// Polycyclic
G:=Group<a,b,c|a^20=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^7,c*b*c^-1=b^3>;
// generators/relations