direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8xD5, C40:3C2, C8oDic5, D10.4C4, C4.12D10, Dic5.4C4, C20.12C22, C5:3(C2xC8), C8o(C5:2C8), C5:2C8:6C2, C2.1(C4xD5), C10.8(C2xC4), (C4xD5).7C2, SmallGroup(80,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C8xD5 |
Generators and relations for C8xD5
G = < a,b,c | a8=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Quotients: C1, C2, C4, C22, C8, C2xC4, D5, C2xC8, D10, C4xD5, C8xD5
Smallest permutation representation of C8xD5
►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 35 9 22 26)(2 36 10 23 27)(3 37 11 24 28)(4 38 12 17 29)(5 39 13 18 30)(6 40 14 19 31)(7 33 15 20 32)(8 34 16 21 25)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 13)(10 14)(11 15)(12 16)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 33)
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), (1,35,9,22,26)(2,36,10,23,27)(3,37,11,24,28)(4,38,12,17,29)(5,39,13,18,30)(6,40,14,19,31)(7,33,15,20,32)(8,34,16,21,25), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,13)(10,14)(11,15)(12,16)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,33)>;
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), (1,35,9,22,26)(2,36,10,23,27)(3,37,11,24,28)(4,38,12,17,29)(5,39,13,18,30)(6,40,14,19,31)(7,33,15,20,32)(8,34,16,21,25), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,13)(10,14)(11,15)(12,16)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,33) );
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)], [(1,35,9,22,26),(2,36,10,23,27),(3,37,11,24,28),(4,38,12,17,29),(5,39,13,18,30),(6,40,14,19,31),(7,33,15,20,32),(8,34,16,21,25)], [(1,30),(2,31),(3,32),(4,25),(5,26),(6,27),(7,28),(8,29),(9,13),(10,14),(11,15),(12,16),(17,34),(18,35),(19,36),(20,37),(21,38),(22,39),(23,40),(24,33)]])
C8xD5 is a maximal subgroup of
C80:C2 D5:C16 C8.F5 C8:F5 C40:C4 D5.D8 C40.C4 D10.Q8 D20.3C4 D20.2C4 D8:3D5 SD16:3D5 Q8.D10 D15:2C8 C20.29D10 Dic5.4F5 C8.A5
C8xD5 is a maximal quotient of
C80:C2 C20.8Q8 D10:1C8 D15:2C8 C20.29D10 Dic5.4F5
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 5 | 5 | 2 | 2 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D5 | D10 | C4xD5 | C8xD5 |
| kernel | C8xD5 | C5:2C8 | C40 | C4xD5 | Dic5 | D10 | D5 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 4 | 8 |
Matrix representation of C8xD5 ►in GL2(F41) generated by
| 14 | 0 |
| 0 | 14 |
| 0 | 38 |
| 14 | 34 |
| 7 | 20 |
| 14 | 34 |
G:=sub<GL(2,GF(41))| [14,0,0,14],[0,14,38,34],[7,14,20,34] >;
C8xD5 in GAP, Magma, Sage, TeX
C_8\times D_5
% in TeX
G:=Group("C8xD5"); // GroupNames label
G:=SmallGroup(80,4);
// by ID
G=gap.SmallGroup(80,4);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,26,42,1604]);
// Polycyclic
G:=Group<a,b,c|a^8=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export