metacyclic, supersoluble, monomial, Z-group
Aliases: C31⋊C10, D31⋊C5, C31⋊C5⋊C2, SmallGroup(310,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C31⋊C10 |
Generators and relations for C31⋊C10
G = < a,b | a31=b10=1, bab-1=a23 >
Character table of C31⋊C10
| class | 1 | 2 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 31A | 31B | 31C | |
| size | 1 | 31 | 31 | 31 | 31 | 31 | 31 | 31 | 31 | 31 | 10 | 10 | 10 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | ζ52 | ζ54 | ζ5 | ζ53 | ζ5 | ζ54 | ζ53 | ζ52 | 1 | 1 | 1 | linear of order 5 |
| ρ4 | 1 | 1 | ζ5 | ζ52 | ζ53 | ζ54 | ζ53 | ζ52 | ζ54 | ζ5 | 1 | 1 | 1 | linear of order 5 |
| ρ5 | 1 | -1 | ζ54 | ζ53 | ζ52 | ζ5 | -ζ52 | -ζ53 | -ζ5 | -ζ54 | 1 | 1 | 1 | linear of order 10 |
| ρ6 | 1 | -1 | ζ5 | ζ52 | ζ53 | ζ54 | -ζ53 | -ζ52 | -ζ54 | -ζ5 | 1 | 1 | 1 | linear of order 10 |
| ρ7 | 1 | 1 | ζ54 | ζ53 | ζ52 | ζ5 | ζ52 | ζ53 | ζ5 | ζ54 | 1 | 1 | 1 | linear of order 5 |
| ρ8 | 1 | -1 | ζ52 | ζ54 | ζ5 | ζ53 | -ζ5 | -ζ54 | -ζ53 | -ζ52 | 1 | 1 | 1 | linear of order 10 |
| ρ9 | 1 | -1 | ζ53 | ζ5 | ζ54 | ζ52 | -ζ54 | -ζ5 | -ζ52 | -ζ53 | 1 | 1 | 1 | linear of order 10 |
| ρ10 | 1 | 1 | ζ53 | ζ5 | ζ54 | ζ52 | ζ54 | ζ5 | ζ52 | ζ53 | 1 | 1 | 1 | linear of order 5 |
| ρ11 | 10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3128+ζ3125+ζ3124+ζ3119+ζ3117+ζ3114+ζ3112+ζ317+ζ316+ζ313 | ζ3130+ζ3129+ζ3127+ζ3123+ζ3116+ζ3115+ζ318+ζ314+ζ312+ζ31 | ζ3126+ζ3122+ζ3121+ζ3120+ζ3118+ζ3113+ζ3111+ζ3110+ζ319+ζ315 | orthogonal faithful |
| ρ12 | 10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3130+ζ3129+ζ3127+ζ3123+ζ3116+ζ3115+ζ318+ζ314+ζ312+ζ31 | ζ3126+ζ3122+ζ3121+ζ3120+ζ3118+ζ3113+ζ3111+ζ3110+ζ319+ζ315 | ζ3128+ζ3125+ζ3124+ζ3119+ζ3117+ζ3114+ζ3112+ζ317+ζ316+ζ313 | orthogonal faithful |
| ρ13 | 10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3126+ζ3122+ζ3121+ζ3120+ζ3118+ζ3113+ζ3111+ζ3110+ζ319+ζ315 | ζ3128+ζ3125+ζ3124+ζ3119+ζ3117+ζ3114+ζ3112+ζ317+ζ316+ζ313 | ζ3130+ζ3129+ζ3127+ζ3123+ζ3116+ζ3115+ζ318+ζ314+ζ312+ζ31 | orthogonal faithful |
►On 31 points: primitive - transitive group 31T6
Generators in S31
(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)
(2 28 17 30 9 31 5 16 3 24)(4 20 18 26 25 29 13 15 7 8)(6 12 19 22 10 27 21 14 11 23)
G:=sub<Sym(31)| (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), (2,28,17,30,9,31,5,16,3,24)(4,20,18,26,25,29,13,15,7,8)(6,12,19,22,10,27,21,14,11,23)>;
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), (2,28,17,30,9,31,5,16,3,24)(4,20,18,26,25,29,13,15,7,8)(6,12,19,22,10,27,21,14,11,23) );
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)], [(2,28,17,30,9,31,5,16,3,24),(4,20,18,26,25,29,13,15,7,8),(6,12,19,22,10,27,21,14,11,23)]])
G:=TransitiveGroup(31,6);
Matrix representation of C31⋊C10 ►in GL10(𝔽2)
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
| 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 |
| 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 |
| 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
| 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
G:=sub<GL(10,GF(2))| [0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,1,0,1,1,1,0,1,1,0,1,1,1,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,1,1,0,0,1,0,0,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,1,0,1,1,0,1,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,1,0,0,0,0,0,1,1,1,1,1,1,0,0,1,1,1,1,0,1,1,0,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,1,1,0,0] >;
C31⋊C10 in GAP, Magma, Sage, TeX
C_{31}\rtimes C_{10} % in TeX
G:=Group("C31:C10"); // GroupNames label
G:=SmallGroup(310,1);
// by ID
G=gap.SmallGroup(310,1);
# by ID
G:=PCGroup([3,-2,-5,-31,2702,725]);
// Polycyclic
G:=Group<a,b|a^31=b^10=1,b*a*b^-1=a^23>;
// generators/relations
Export
Subgroup lattice of C31⋊C10 in TeX
Character table of C31⋊C10 in TeX