metacyclic, supersoluble, monomial, Z-group
Aliases: C31⋊C6, D31⋊C3, C31⋊C3⋊C2, SmallGroup(186,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C31⋊C6 |
Generators and relations for C31⋊C6
G = < a,b | a31=b6=1, bab-1=a6 >
Character table of C31⋊C6
| class | 1 | 2 | 3A | 3B | 6A | 6B | 31A | 31B | 31C | 31D | 31E | |
| size | 1 | 31 | 31 | 31 | 31 | 31 | 6 | 6 | 6 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 6 | 0 | 0 | 0 | 0 | 0 | ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 | ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 | ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 | ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 | ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 | orthogonal faithful |
| ρ8 | 6 | 0 | 0 | 0 | 0 | 0 | ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 | ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 | ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 | ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 | ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 | orthogonal faithful |
| ρ9 | 6 | 0 | 0 | 0 | 0 | 0 | ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 | ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 | ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 | ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 | ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 | orthogonal faithful |
| ρ10 | 6 | 0 | 0 | 0 | 0 | 0 | ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 | ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 | ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 | ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 | ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 | orthogonal faithful |
| ρ11 | 6 | 0 | 0 | 0 | 0 | 0 | ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 | ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 | ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 | ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 | ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 | orthogonal faithful |
►On 31 points: primitive - transitive group 31T5
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 27 26 31 6 7)(3 22 20 30 11 13)(4 17 14 29 16 19)(5 12 8 28 21 25)(9 23 15 24 10 18)
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), (2,27,26,31,6,7)(3,22,20,30,11,13)(4,17,14,29,16,19)(5,12,8,28,21,25)(9,23,15,24,10,18)>;
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), (2,27,26,31,6,7)(3,22,20,30,11,13)(4,17,14,29,16,19)(5,12,8,28,21,25)(9,23,15,24,10,18) );
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)], [(2,27,26,31,6,7),(3,22,20,30,11,13),(4,17,14,29,16,19),(5,12,8,28,21,25),(9,23,15,24,10,18)]])
G:=TransitiveGroup(31,5);
C31⋊C6 is a maximal quotient of C31⋊C12
Matrix representation of C31⋊C6 ►in GL6(𝔽373)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 372 | 57 | 291 | 217 | 291 | 57 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 372 | 57 | 291 | 217 | 291 | 57 |
| 195 | 252 | 72 | 78 | 328 | 276 |
| 220 | 4 | 365 | 365 | 4 | 220 |
| 276 | 328 | 78 | 72 | 252 | 195 |
| 57 | 291 | 217 | 291 | 57 | 372 |
G:=sub<GL(6,GF(373))| [0,0,0,0,0,372,1,0,0,0,0,57,0,1,0,0,0,291,0,0,1,0,0,217,0,0,0,1,0,291,0,0,0,0,1,57],[1,372,195,220,276,57,0,57,252,4,328,291,0,291,72,365,78,217,0,217,78,365,72,291,0,291,328,4,252,57,0,57,276,220,195,372] >;
C31⋊C6 in GAP, Magma, Sage, TeX
C_{31}\rtimes C_6 % in TeX
G:=Group("C31:C6"); // GroupNames label
G:=SmallGroup(186,1);
// by ID
G=gap.SmallGroup(186,1);
# by ID
G:=PCGroup([3,-2,-3,-31,1622,680]);
// Polycyclic
G:=Group<a,b|a^31=b^6=1,b*a*b^-1=a^6>;
// generators/relations
Export
Subgroup lattice of C31⋊C6 in TeX
Character table of C31⋊C6 in TeX