direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×D8, D4⋊C10, C8⋊1C10, C40⋊5C2, C10.14D4, C20.17C22, (C5×D4)⋊4C2, C2.3(C5×D4), C4.1(C2×C10), SmallGroup(80,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D8
G = < a,b,c | a5=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D8
►On 40 points
Generators in S40
(1 23 13 39 25)(2 24 14 40 26)(3 17 15 33 27)(4 18 16 34 28)(5 19 9 35 29)(6 20 10 36 30)(7 21 11 37 31)(8 22 12 38 32)
(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 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 20)(18 19)(21 24)(22 23)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 40)(38 39)
G:=sub<Sym(40)| (1,23,13,39,25)(2,24,14,40,26)(3,17,15,33,27)(4,18,16,34,28)(5,19,9,35,29)(6,20,10,36,30)(7,21,11,37,31)(8,22,12,38,32), (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), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,20)(18,19)(21,24)(22,23)(25,32)(26,31)(27,30)(28,29)(33,36)(34,35)(37,40)(38,39)>;
G:=Group( (1,23,13,39,25)(2,24,14,40,26)(3,17,15,33,27)(4,18,16,34,28)(5,19,9,35,29)(6,20,10,36,30)(7,21,11,37,31)(8,22,12,38,32), (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), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,20)(18,19)(21,24)(22,23)(25,32)(26,31)(27,30)(28,29)(33,36)(34,35)(37,40)(38,39) );
G=PermutationGroup([[(1,23,13,39,25),(2,24,14,40,26),(3,17,15,33,27),(4,18,16,34,28),(5,19,9,35,29),(6,20,10,36,30),(7,21,11,37,31),(8,22,12,38,32)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,16),(10,15),(11,14),(12,13),(17,20),(18,19),(21,24),(22,23),(25,32),(26,31),(27,30),(28,29),(33,36),(34,35),(37,40),(38,39)]])
C5×D8 is a maximal subgroup of
C5⋊D16 D8.D5 D8⋊D5 D8⋊3D5
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 5C | 5D | 8A | 8B | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 4 | 4 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | D4 | D8 | C5×D4 | C5×D8 |
| kernel | C5×D8 | C40 | C5×D4 | D8 | C8 | D4 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C5×D8 ►in GL2(𝔽31) generated by
| 16 | 0 |
| 0 | 16 |
| 0 | 30 |
| 1 | 23 |
| 23 | 1 |
| 30 | 8 |
G:=sub<GL(2,GF(31))| [16,0,0,16],[0,1,30,23],[23,30,1,8] >;
C5×D8 in GAP, Magma, Sage, TeX
C_5\times D_8
% in TeX
G:=Group("C5xD8"); // GroupNames label
G:=SmallGroup(80,25);
// by ID
G=gap.SmallGroup(80,25);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-2,221,1203,608,58]);
// 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^-1>;
// generators/relations
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