metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: SD16⋊1F5, C8⋊3(C2×F5), C40⋊3(C2×C4), Q8⋊D5⋊2C4, (Q8×F5)⋊2C2, Q8⋊2(C2×F5), C40⋊C2⋊1C4, D4.D5⋊4C4, D5.D8⋊3C2, C8⋊F5⋊4C2, D4.4(C2×F5), (C2×F5).6D4, (D4×F5).2C2, C2.20(D4×F5), C5⋊(SD16⋊C4), Q8⋊F5⋊2C2, (C5×SD16)⋊1C4, D20.2(C2×C4), C10.19(C4×D4), C4⋊F5.4C22, C4.6(C22×F5), Dic10⋊2(C2×C4), D10.66(C2×D4), D20⋊C4.2C2, C20.6(C22×C4), D5⋊C8.3C22, (D5×SD16).1C2, (D4×D5).8C22, (C4×F5).3C22, (Q8×D5).5C22, D5.3(C8⋊C22), (C8×D5).25C22, (C4×D5).28C23, Dic5.4(C4○D4), D5.2(C8.C22), (C5×Q8)⋊2(C2×C4), C5⋊2C8⋊14(C2×C4), (C5×D4).4(C2×C4), SmallGroup(320,1073)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16⋊F5
G = < a,b,c,d | a8=b2=c5=d4=1, bab=a3, ac=ca, dad-1=a5, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 546 in 120 conjugacy classes, 42 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, D4, Q8, Q8, C23, D5, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, SD16, C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, F5, D10, D10, C2×C10, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C5⋊2C8, C40, C5⋊C8, Dic10, Dic10, C4×D5, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C2×F5, C2×F5, C22×D5, SD16⋊C4, C8×D5, C40⋊C2, D4.D5, Q8⋊D5, C5×SD16, D5⋊C8, C4×F5, C4×F5, C4⋊F5, C4⋊F5, C22⋊F5, D4×D5, Q8×D5, C22×F5, C8⋊F5, D5.D8, D20⋊C4, Q8⋊F5, D5×SD16, D4×F5, Q8×F5, SD16⋊F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, F5, C4×D4, C8⋊C22, C8.C22, C2×F5, SD16⋊C4, C22×F5, D4×F5, SD16⋊F5
Character table of SD16⋊F5
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5 | 8A | 8B | 8C | 8D | 10A | 10B | 20A | 20B | 40A | 40B | |
| size | 1 | 1 | 4 | 5 | 5 | 20 | 2 | 4 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 4 | 4 | 20 | 20 | 20 | 4 | 16 | 8 | 16 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | -1 | i | i | -i | -i | -1 | i | -i | i | 1 | 1 | -i | i | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ10 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -i | -1 | i | i | -i | i | 1 | -i | i | -i | 1 | 1 | -i | i | -1 | 1 | -1 | 1 | -1 | 1 | 1 | linear of order 4 |
| ρ11 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | i | -1 | -i | -i | i | i | -1 | -i | i | -i | 1 | 1 | i | -i | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ12 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | i | -1 | -i | -i | i | -i | 1 | i | -i | i | 1 | 1 | i | -i | -1 | 1 | -1 | 1 | -1 | 1 | 1 | linear of order 4 |
| ρ13 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | i | -1 | -i | -i | i | i | 1 | i | -i | -i | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ14 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | i | -1 | -i | -i | i | -i | -1 | -i | i | i | 1 | -1 | -i | i | 1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ15 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -i | -1 | i | i | -i | -i | 1 | -i | i | i | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ16 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -i | -1 | i | i | -i | i | -1 | i | -i | -i | 1 | -1 | i | -i | 1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ17 | 2 | 2 | 0 | 2 | 2 | 0 | -2 | 0 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 0 | 2 | 2 | 0 | -2 | 0 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 0 | -2 | -2 | 0 | -2 | 0 | -2i | 2 | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | 2 | 0 | -2 | -2 | 0 | -2 | 0 | 2i | 2 | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 4 | -4 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ22 | 4 | 4 | -4 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | -1 | -1 | orthogonal lifted from C2×F5 |
| ρ23 | 4 | 4 | 4 | 0 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ24 | 4 | 4 | 4 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ25 | 4 | 4 | -4 | 0 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 0 | -1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ26 | 4 | -4 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ27 | 8 | 8 | 0 | 0 | 0 | 0 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | orthogonal lifted from D4×F5 |
| ρ28 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -√-10 | √-10 | complex faithful |
| ρ29 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | √-10 | -√-10 | complex faithful |
►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)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 21)(18 24)(20 22)(25 27)(26 30)(29 31)(33 37)(34 40)(36 38)
(1 14 23 32 35)(2 15 24 25 36)(3 16 17 26 37)(4 9 18 27 38)(5 10 19 28 39)(6 11 20 29 40)(7 12 21 30 33)(8 13 22 31 34)
(2 6)(4 8)(9 22 38 31)(10 19 39 28)(11 24 40 25)(12 21 33 30)(13 18 34 27)(14 23 35 32)(15 20 36 29)(16 17 37 26)
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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,21)(18,24)(20,22)(25,27)(26,30)(29,31)(33,37)(34,40)(36,38), (1,14,23,32,35)(2,15,24,25,36)(3,16,17,26,37)(4,9,18,27,38)(5,10,19,28,39)(6,11,20,29,40)(7,12,21,30,33)(8,13,22,31,34), (2,6)(4,8)(9,22,38,31)(10,19,39,28)(11,24,40,25)(12,21,33,30)(13,18,34,27)(14,23,35,32)(15,20,36,29)(16,17,37,26)>;
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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,21)(18,24)(20,22)(25,27)(26,30)(29,31)(33,37)(34,40)(36,38), (1,14,23,32,35)(2,15,24,25,36)(3,16,17,26,37)(4,9,18,27,38)(5,10,19,28,39)(6,11,20,29,40)(7,12,21,30,33)(8,13,22,31,34), (2,6)(4,8)(9,22,38,31)(10,19,39,28)(11,24,40,25)(12,21,33,30)(13,18,34,27)(14,23,35,32)(15,20,36,29)(16,17,37,26) );
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)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16),(17,21),(18,24),(20,22),(25,27),(26,30),(29,31),(33,37),(34,40),(36,38)], [(1,14,23,32,35),(2,15,24,25,36),(3,16,17,26,37),(4,9,18,27,38),(5,10,19,28,39),(6,11,20,29,40),(7,12,21,30,33),(8,13,22,31,34)], [(2,6),(4,8),(9,22,38,31),(10,19,39,28),(11,24,40,25),(12,21,33,30),(13,18,34,27),(14,23,35,32),(15,20,36,29),(16,17,37,26)]])
Matrix representation of SD16⋊F5 ►in GL8(ℤ)
| -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
G:=sub<GL(8,Integers())| [-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1] >;
SD16⋊F5 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\rtimes F_5 % in TeX
G:=Group("SD16:F5"); // GroupNames label
G:=SmallGroup(320,1073);
// by ID
G=gap.SmallGroup(320,1073);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,184,851,438,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^5=d^4=1,b*a*b=a^3,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations
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