direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×Dic5, C10⋊2C4, C22.D5, C2.2D10, C10.4C22, C5⋊3(C2×C4), (C2×C10).C2, SmallGroup(40,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C2×Dic5 |
Generators and relations for C2×Dic5
G = < a,b,c | a2=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >
Character table of C2×Dic5
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 10A | 10B | 10C | 10D | 10E | 10F | |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | 1 | i | -i | -i | i | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | i | i | -i | -i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 4 |
| ρ7 | 1 | -1 | -1 | 1 | -i | i | i | -i | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | -i | -i | i | i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ10 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1+√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1-√5/2 | orthogonal lifted from D10 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1-√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1+√5/2 | orthogonal lifted from D10 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | -1-√5/2 | 1+√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ14 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | -1+√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | -1+√5/2 | 1-√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | -1-√5/2 | symplectic lifted from Dic5, Schur index 2 |
►Regular action on 40 points
Generators in S40
(1 19)(2 20)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 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 26 6 21)(2 25 7 30)(3 24 8 29)(4 23 9 28)(5 22 10 27)(11 34 16 39)(12 33 17 38)(13 32 18 37)(14 31 19 36)(15 40 20 35)
G:=sub<Sym(40)| (1,19)(2,20)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (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), (1,26,6,21)(2,25,7,30)(3,24,8,29)(4,23,9,28)(5,22,10,27)(11,34,16,39)(12,33,17,38)(13,32,18,37)(14,31,19,36)(15,40,20,35)>;
G:=Group( (1,19)(2,20)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (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), (1,26,6,21)(2,25,7,30)(3,24,8,29)(4,23,9,28)(5,22,10,27)(11,34,16,39)(12,33,17,38)(13,32,18,37)(14,31,19,36)(15,40,20,35) );
G=PermutationGroup([[(1,19),(2,20),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)], [(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)], [(1,26,6,21),(2,25,7,30),(3,24,8,29),(4,23,9,28),(5,22,10,27),(11,34,16,39),(12,33,17,38),(13,32,18,37),(14,31,19,36),(15,40,20,35)]])
C2×Dic5 is a maximal subgroup of
C10.D4 C4⋊Dic5 D10⋊C4 C23.D5 C22.F5 C2×C4×D5 D4⋊2D5
C2×Dic5 is a maximal quotient of C4.Dic5 C4⋊Dic5 C23.D5
Matrix representation of C2×Dic5 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 1 |
| 0 | 0 | 33 | 7 |
| 9 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 35 |
| 0 | 0 | 34 | 0 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,1,0,0,0,0,40,33,0,0,1,7],[9,0,0,0,0,40,0,0,0,0,0,34,0,0,35,0] >;
C2×Dic5 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_5 % in TeX
G:=Group("C2xDic5"); // GroupNames label
G:=SmallGroup(40,7);
// by ID
G=gap.SmallGroup(40,7);
# by ID
G:=PCGroup([4,-2,-2,-2,-5,16,515]);
// Polycyclic
G:=Group<a,b,c|a^2=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C2×Dic5 in TeX
Character table of C2×Dic5 in TeX