direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: SD16xF5, C5:(C4xSD16), C8:5(C2xF5), C40:4(C2xC4), Q8:D5:1C4, (C8xF5):4C2, (Q8xF5):1C2, Q8:1(C2xF5), C40:C2:3C4, D4.D5:3C4, C40:C4:4C2, D4.3(C2xF5), (D4xF5).1C2, C2.19(D4xF5), Q8:F5:1C2, (C5xSD16):3C4, D20.1(C2xC4), C10.18(C4xD4), (C2xF5).11D4, C4:F5.3C22, C4.5(C22xF5), Dic10:1(C2xC4), D5.3(C4oD8), D10.65(C2xD4), D20:C4.1C2, C20.5(C22xC4), D5.2(C2xSD16), (D5xSD16).2C2, (D4xD5).7C22, (Q8xD5).4C22, D5:C8.11C22, (C4xD5).27C23, (C8xD5).27C22, (C4xF5).11C22, Dic5.3(C4oD4), (C5xQ8):1(C2xC4), C5:2C8:13(C2xC4), (C5xD4).3(C2xC4), SmallGroup(320,1072)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16xF5
G = < a,b,c,d | a8=b2=c5=d4=1, bab=a3, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 546 in 122 conjugacy classes, 44 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2xC4, D4, D4, Q8, Q8, C23, D5, D5, C10, C10, C42, C22:C4, C4:C4, C2xC8, SD16, SD16, C22xC4, C2xD4, C2xQ8, Dic5, Dic5, C20, C20, F5, F5, D10, D10, C2xC10, C4xC8, D4:C4, Q8:C4, C4.Q8, C4xD4, C4xQ8, C2xSD16, C5:2C8, C40, C5:C8, Dic10, Dic10, C4xD5, C4xD5, D20, C5:D4, C5xD4, C5xQ8, C2xF5, C2xF5, C22xD5, C4xSD16, C8xD5, C40:C2, D4.D5, Q8:D5, C5xSD16, D5:C8, C4xF5, C4xF5, C4:F5, C4:F5, C22:F5, D4xD5, Q8xD5, C22xF5, C8xF5, C40:C4, D20:C4, Q8:F5, D5xSD16, D4xF5, Q8xF5, SD16xF5
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, SD16, C22xC4, C2xD4, C4oD4, F5, C4xD4, C2xSD16, C4oD8, C2xF5, C4xSD16, C22xF5, D4xF5, SD16xF5
(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 11)(10 14)(13 15)(17 19)(18 22)(21 23)(25 27)(26 30)(29 31)(33 37)(34 40)(36 38)
(1 24 32 12 35)(2 17 25 13 36)(3 18 26 14 37)(4 19 27 15 38)(5 20 28 16 39)(6 21 29 9 40)(7 22 30 10 33)(8 23 31 11 34)
(1 5)(2 6)(3 7)(4 8)(9 17 29 36)(10 18 30 37)(11 19 31 38)(12 20 32 39)(13 21 25 40)(14 22 26 33)(15 23 27 34)(16 24 28 35)
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)(9,11)(10,14)(13,15)(17,19)(18,22)(21,23)(25,27)(26,30)(29,31)(33,37)(34,40)(36,38), (1,24,32,12,35)(2,17,25,13,36)(3,18,26,14,37)(4,19,27,15,38)(5,20,28,16,39)(6,21,29,9,40)(7,22,30,10,33)(8,23,31,11,34), (1,5)(2,6)(3,7)(4,8)(9,17,29,36)(10,18,30,37)(11,19,31,38)(12,20,32,39)(13,21,25,40)(14,22,26,33)(15,23,27,34)(16,24,28,35)>;
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)(9,11)(10,14)(13,15)(17,19)(18,22)(21,23)(25,27)(26,30)(29,31)(33,37)(34,40)(36,38), (1,24,32,12,35)(2,17,25,13,36)(3,18,26,14,37)(4,19,27,15,38)(5,20,28,16,39)(6,21,29,9,40)(7,22,30,10,33)(8,23,31,11,34), (1,5)(2,6)(3,7)(4,8)(9,17,29,36)(10,18,30,37)(11,19,31,38)(12,20,32,39)(13,21,25,40)(14,22,26,33)(15,23,27,34)(16,24,28,35) );
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),(9,11),(10,14),(13,15),(17,19),(18,22),(21,23),(25,27),(26,30),(29,31),(33,37),(34,40),(36,38)], [(1,24,32,12,35),(2,17,25,13,36),(3,18,26,14,37),(4,19,27,15,38),(5,20,28,16,39),(6,21,29,9,40),(7,22,30,10,33),(8,23,31,11,34)], [(1,5),(2,6),(3,7),(4,8),(9,17,29,36),(10,18,30,37),(11,19,31,38),(12,20,32,39),(13,21,25,40),(14,22,26,33),(15,23,27,34),(16,24,28,35)]])
35 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | ··· | 4N | 5 | 8A | 8B | 8C | ··· | 8H | 10A | 10B | 20A | 20B | 40A | 40B |
order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 20 | 20 | 40 | 40 |
size | 1 | 1 | 4 | 5 | 5 | 20 | 2 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 20 | ··· | 20 | 4 | 2 | 2 | 10 | ··· | 10 | 4 | 16 | 8 | 16 | 8 | 8 |
35 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | C4oD4 | SD16 | C4oD8 | F5 | C2xF5 | C2xF5 | C2xF5 | D4xF5 | SD16xF5 |
kernel | SD16xF5 | C8xF5 | C40:C4 | D20:C4 | Q8:F5 | D5xSD16 | D4xF5 | Q8xF5 | C40:C2 | D4.D5 | Q8:D5 | C5xSD16 | C2xF5 | Dic5 | F5 | D5 | SD16 | C8 | D4 | Q8 | C2 | C1 |
# reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 2 |
Matrix representation of SD16xF5 ►in GL6(F41)
15 | 15 | 0 | 0 | 0 | 0 |
26 | 15 | 0 | 0 | 0 | 0 |
0 | 0 | 40 | 0 | 0 | 0 |
0 | 0 | 0 | 40 | 0 | 0 |
0 | 0 | 0 | 0 | 40 | 0 |
0 | 0 | 0 | 0 | 0 | 40 |
40 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 40 | 0 | 0 | 0 |
0 | 0 | 0 | 40 | 0 | 0 |
0 | 0 | 0 | 0 | 40 | 0 |
0 | 0 | 0 | 0 | 0 | 40 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 40 |
0 | 0 | 1 | 0 | 0 | 40 |
0 | 0 | 0 | 1 | 0 | 40 |
0 | 0 | 0 | 0 | 1 | 40 |
32 | 0 | 0 | 0 | 0 | 0 |
0 | 32 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(41))| [15,26,0,0,0,0,15,15,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;
SD16xF5 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\times F_5
% in TeX
G:=Group("SD16xF5");
// GroupNames label
G:=SmallGroup(320,1072);
// by ID
G=gap.SmallGroup(320,1072);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,184,851,438,102,6278,1595]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations