metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5⋊M5(2), D5⋊C16⋊4C2, C5⋊C16⋊2C22, (C4×D5).4C8, C5⋊2(C2×M5(2)), C8.33(C2×F5), (C2×C8).20F5, (C2×C40).24C4, C20.15(C2×C8), C40.40(C2×C4), (C8×D5).12C4, C8.F5⋊5C2, C20.C8⋊7C2, C4.10(D5⋊C8), D10.18(C2×C8), (C22×D5).9C8, C4.47(C22×F5), C10.10(C22×C8), C20.87(C22×C4), C5⋊2C8.37C23, C22.5(D5⋊C8), Dic5.18(C2×C8), (C2×Dic5).14C8, (C8×D5).63C22, (C2×C4×D5).39C4, (D5×C2×C8).31C2, C2.11(C2×D5⋊C8), (C2×C10).12(C2×C8), C5⋊2C8.53(C2×C4), (C4×D5).93(C2×C4), (C2×C4).134(C2×F5), (C2×C20).145(C2×C4), (C2×C5⋊2C8).338C22, SmallGroup(320,1053)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5⋊M5(2)
G = < a,b,c,d | a5=b2=c16=d2=1, bab=a-1, cac-1=a3, ad=da, cbc-1=a2b, bd=db, dcd=c9 >
Subgroups: 250 in 90 conjugacy classes, 50 normal (38 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C16, C2×C8, C2×C8, C22×C4, Dic5, C20, D10, D10, C2×C10, C2×C16, M5(2), C22×C8, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×M5(2), C5⋊C16, C8×D5, C2×C5⋊2C8, C2×C40, C2×C4×D5, D5⋊C16, C8.F5, C20.C8, D5×C2×C8, D5⋊M5(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, F5, M5(2), C22×C8, C2×F5, C2×M5(2), D5⋊C8, C22×F5, C2×D5⋊C8, D5⋊M5(2)
►On 80 points
(1 23 53 34 66)(2 35 24 67 54)(3 68 36 55 25)(4 56 69 26 37)(5 27 57 38 70)(6 39 28 71 58)(7 72 40 59 29)(8 60 73 30 41)(9 31 61 42 74)(10 43 32 75 62)(11 76 44 63 17)(12 64 77 18 45)(13 19 49 46 78)(14 47 20 79 50)(15 80 48 51 21)(16 52 65 22 33)
(1 66)(2 54)(3 25)(4 37)(5 70)(6 58)(7 29)(8 41)(9 74)(10 62)(11 17)(12 45)(13 78)(14 50)(15 21)(16 33)(18 64)(19 46)(22 52)(23 34)(26 56)(27 38)(30 60)(31 42)(35 67)(39 71)(43 75)(47 79)(51 80)(55 68)(59 72)(63 76)
(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)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(33 41)(35 43)(37 45)(39 47)(50 58)(52 60)(54 62)(56 64)(65 73)(67 75)(69 77)(71 79)
G:=sub<Sym(80)| (1,23,53,34,66)(2,35,24,67,54)(3,68,36,55,25)(4,56,69,26,37)(5,27,57,38,70)(6,39,28,71,58)(7,72,40,59,29)(8,60,73,30,41)(9,31,61,42,74)(10,43,32,75,62)(11,76,44,63,17)(12,64,77,18,45)(13,19,49,46,78)(14,47,20,79,50)(15,80,48,51,21)(16,52,65,22,33), (1,66)(2,54)(3,25)(4,37)(5,70)(6,58)(7,29)(8,41)(9,74)(10,62)(11,17)(12,45)(13,78)(14,50)(15,21)(16,33)(18,64)(19,46)(22,52)(23,34)(26,56)(27,38)(30,60)(31,42)(35,67)(39,71)(43,75)(47,79)(51,80)(55,68)(59,72)(63,76), (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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(33,41)(35,43)(37,45)(39,47)(50,58)(52,60)(54,62)(56,64)(65,73)(67,75)(69,77)(71,79)>;
G:=Group( (1,23,53,34,66)(2,35,24,67,54)(3,68,36,55,25)(4,56,69,26,37)(5,27,57,38,70)(6,39,28,71,58)(7,72,40,59,29)(8,60,73,30,41)(9,31,61,42,74)(10,43,32,75,62)(11,76,44,63,17)(12,64,77,18,45)(13,19,49,46,78)(14,47,20,79,50)(15,80,48,51,21)(16,52,65,22,33), (1,66)(2,54)(3,25)(4,37)(5,70)(6,58)(7,29)(8,41)(9,74)(10,62)(11,17)(12,45)(13,78)(14,50)(15,21)(16,33)(18,64)(19,46)(22,52)(23,34)(26,56)(27,38)(30,60)(31,42)(35,67)(39,71)(43,75)(47,79)(51,80)(55,68)(59,72)(63,76), (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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(33,41)(35,43)(37,45)(39,47)(50,58)(52,60)(54,62)(56,64)(65,73)(67,75)(69,77)(71,79) );
G=PermutationGroup([[(1,23,53,34,66),(2,35,24,67,54),(3,68,36,55,25),(4,56,69,26,37),(5,27,57,38,70),(6,39,28,71,58),(7,72,40,59,29),(8,60,73,30,41),(9,31,61,42,74),(10,43,32,75,62),(11,76,44,63,17),(12,64,77,18,45),(13,19,49,46,78),(14,47,20,79,50),(15,80,48,51,21),(16,52,65,22,33)], [(1,66),(2,54),(3,25),(4,37),(5,70),(6,58),(7,29),(8,41),(9,74),(10,62),(11,17),(12,45),(13,78),(14,50),(15,21),(16,33),(18,64),(19,46),(22,52),(23,34),(26,56),(27,38),(30,60),(31,42),(35,67),(39,71),(43,75),(47,79),(51,80),(55,68),(59,72),(63,76)], [(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)], [(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32),(33,41),(35,43),(37,45),(39,47),(50,58),(52,60),(54,62),(56,64),(65,73),(67,75),(69,77),(71,79)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 10A | 10B | 10C | 16A | ··· | 16P | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 5 | 5 | 10 | 1 | 1 | 2 | 5 | 5 | 10 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 4 | 4 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | M5(2) | F5 | C2×F5 | C2×F5 | D5⋊C8 | D5⋊C8 | D5⋊M5(2) |
| kernel | D5⋊M5(2) | D5⋊C16 | C8.F5 | C20.C8 | D5×C2×C8 | C8×D5 | C2×C40 | C2×C4×D5 | C4×D5 | C2×Dic5 | C22×D5 | D5 | C2×C8 | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 8 | 4 | 4 | 8 | 1 | 2 | 1 | 2 | 2 | 8 |
Matrix representation of D5⋊M5(2) ►in GL4(𝔽241) generated by
| 190 | 191 | 0 | 0 |
| 240 | 240 | 0 | 0 |
| 0 | 0 | 0 | 190 |
| 0 | 0 | 52 | 51 |
| 240 | 50 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 240 | 240 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 84 | 84 | 0 | 0 |
| 30 | 157 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
G:=sub<GL(4,GF(241))| [190,240,0,0,191,240,0,0,0,0,0,52,0,0,190,51],[240,0,0,0,50,1,0,0,0,0,1,240,0,0,0,240],[0,0,84,30,0,0,84,157,1,0,0,0,0,1,0,0],[1,0,0,0,0,1,0,0,0,0,240,0,0,0,0,240] >;
D5⋊M5(2) in GAP, Magma, Sage, TeX
D_5\rtimes M_5(2)
% in TeX
G:=Group("D5:M5(2)"); // GroupNames label
G:=SmallGroup(320,1053);
// by ID
G=gap.SmallGroup(320,1053);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,100,80,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^16=d^2=1,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^9>;
// generators/relations