direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×SD16, Q8⋊C10, C40⋊6C2, C8⋊2C10, D4.C10, C10.15D4, C20.18C22, (C5×Q8)⋊4C2, C2.4(C5×D4), C4.2(C2×C10), (C5×D4).2C2, SmallGroup(80,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×SD16
G = < a,b,c | a5=b8=c2=1, ab=ba, ac=ca, cbc=b3 >
Smallest permutation representation of C5×SD16
►On 40 points
Generators in S40
(1 18 29 35 10)(2 19 30 36 11)(3 20 31 37 12)(4 21 32 38 13)(5 22 25 39 14)(6 23 26 40 15)(7 24 27 33 16)(8 17 28 34 9)
(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 23)(19 21)(20 24)(26 28)(27 31)(30 32)(33 37)(34 40)(36 38)
G:=sub<Sym(40)| (1,18,29,35,10)(2,19,30,36,11)(3,20,31,37,12)(4,21,32,38,13)(5,22,25,39,14)(6,23,26,40,15)(7,24,27,33,16)(8,17,28,34,9), (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)(17,23)(19,21)(20,24)(26,28)(27,31)(30,32)(33,37)(34,40)(36,38)>;
G:=Group( (1,18,29,35,10)(2,19,30,36,11)(3,20,31,37,12)(4,21,32,38,13)(5,22,25,39,14)(6,23,26,40,15)(7,24,27,33,16)(8,17,28,34,9), (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)(17,23)(19,21)(20,24)(26,28)(27,31)(30,32)(33,37)(34,40)(36,38) );
G=PermutationGroup([[(1,18,29,35,10),(2,19,30,36,11),(3,20,31,37,12),(4,21,32,38,13),(5,22,25,39,14),(6,23,26,40,15),(7,24,27,33,16),(8,17,28,34,9)], [(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),(17,23),(19,21),(20,24),(26,28),(27,31),(30,32),(33,37),(34,40),(36,38)]])
C5×SD16 is a maximal subgroup of
D40⋊C2 SD16⋊D5 SD16⋊3D5
35 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 5A | 5B | 5C | 5D | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 4 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | D4 | SD16 | C5×D4 | C5×SD16 |
| kernel | C5×SD16 | C40 | C5×D4 | C5×Q8 | SD16 | C8 | D4 | Q8 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 2 | 4 | 8 |
Matrix representation of C5×SD16 ►in GL2(𝔽11) generated by
| 4 | 0 |
| 0 | 4 |
| 8 | 5 |
| 9 | 0 |
| 10 | 7 |
| 0 | 1 |
G:=sub<GL(2,GF(11))| [4,0,0,4],[8,9,5,0],[10,0,7,1] >;
C5×SD16 in GAP, Magma, Sage, TeX
C_5\times {\rm SD}_{16} % in TeX
G:=Group("C5xSD16"); // GroupNames label
G:=SmallGroup(80,26);
// by ID
G=gap.SmallGroup(80,26);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-2,200,221,1203,608,58]);
// Polycyclic
G:=Group<a,b,c|a^5=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations
Export