metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C31⋊C3, SmallGroup(93,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C31⋊C3 |
Generators and relations for C31⋊C3
G = < a,b | a31=b3=1, bab-1=a5 >
Character table of C31⋊C3
| class | 1 | 3A | 3B | 31A | 31B | 31C | 31D | 31E | 31F | 31G | 31H | 31I | 31J | |
| size | 1 | 31 | 31 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ3 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 3 | 0 | 0 | ζ3129+ζ3121+ζ3112 | ζ3128+ζ3118+ζ3116 | ζ3127+ζ3124+ζ3111 | ζ3125+ζ315+ζ31 | ζ3115+ζ3113+ζ313 | ζ3123+ζ3122+ζ3117 | ζ3119+ζ3110+ζ312 | ζ3130+ζ3126+ζ316 | ζ3120+ζ317+ζ314 | ζ3114+ζ319+ζ318 | complex faithful |
| ρ5 | 3 | 0 | 0 | ζ3130+ζ3126+ζ316 | ζ3114+ζ319+ζ318 | ζ3129+ζ3121+ζ3112 | ζ3128+ζ3118+ζ3116 | ζ3123+ζ3122+ζ3117 | ζ3127+ζ3124+ζ3111 | ζ3125+ζ315+ζ31 | ζ3115+ζ3113+ζ313 | ζ3119+ζ3110+ζ312 | ζ3120+ζ317+ζ314 | complex faithful |
| ρ6 | 3 | 0 | 0 | ζ3114+ζ319+ζ318 | ζ3129+ζ3121+ζ3112 | ζ3128+ζ3118+ζ3116 | ζ3127+ζ3124+ζ3111 | ζ3119+ζ3110+ζ312 | ζ3125+ζ315+ζ31 | ζ3123+ζ3122+ζ3117 | ζ3120+ζ317+ζ314 | ζ3115+ζ3113+ζ313 | ζ3130+ζ3126+ζ316 | complex faithful |
| ρ7 | 3 | 0 | 0 | ζ3120+ζ317+ζ314 | ζ3130+ζ3126+ζ316 | ζ3114+ζ319+ζ318 | ζ3129+ζ3121+ζ3112 | ζ3125+ζ315+ζ31 | ζ3128+ζ3118+ζ3116 | ζ3127+ζ3124+ζ3111 | ζ3119+ζ3110+ζ312 | ζ3123+ζ3122+ζ3117 | ζ3115+ζ3113+ζ313 | complex faithful |
| ρ8 | 3 | 0 | 0 | ζ3125+ζ315+ζ31 | ζ3123+ζ3122+ζ3117 | ζ3119+ζ3110+ζ312 | ζ3115+ζ3113+ζ313 | ζ3114+ζ319+ζ318 | ζ3120+ζ317+ζ314 | ζ3130+ζ3126+ζ316 | ζ3128+ζ3118+ζ3116 | ζ3129+ζ3121+ζ3112 | ζ3127+ζ3124+ζ3111 | complex faithful |
| ρ9 | 3 | 0 | 0 | ζ3123+ζ3122+ζ3117 | ζ3119+ζ3110+ζ312 | ζ3115+ζ3113+ζ313 | ζ3120+ζ317+ζ314 | ζ3129+ζ3121+ζ3112 | ζ3130+ζ3126+ζ316 | ζ3114+ζ319+ζ318 | ζ3127+ζ3124+ζ3111 | ζ3128+ζ3118+ζ3116 | ζ3125+ζ315+ζ31 | complex faithful |
| ρ10 | 3 | 0 | 0 | ζ3119+ζ3110+ζ312 | ζ3115+ζ3113+ζ313 | ζ3120+ζ317+ζ314 | ζ3130+ζ3126+ζ316 | ζ3128+ζ3118+ζ3116 | ζ3114+ζ319+ζ318 | ζ3129+ζ3121+ζ3112 | ζ3125+ζ315+ζ31 | ζ3127+ζ3124+ζ3111 | ζ3123+ζ3122+ζ3117 | complex faithful |
| ρ11 | 3 | 0 | 0 | ζ3115+ζ3113+ζ313 | ζ3120+ζ317+ζ314 | ζ3130+ζ3126+ζ316 | ζ3114+ζ319+ζ318 | ζ3127+ζ3124+ζ3111 | ζ3129+ζ3121+ζ3112 | ζ3128+ζ3118+ζ3116 | ζ3123+ζ3122+ζ3117 | ζ3125+ζ315+ζ31 | ζ3119+ζ3110+ζ312 | complex faithful |
| ρ12 | 3 | 0 | 0 | ζ3128+ζ3118+ζ3116 | ζ3127+ζ3124+ζ3111 | ζ3125+ζ315+ζ31 | ζ3123+ζ3122+ζ3117 | ζ3120+ζ317+ζ314 | ζ3119+ζ3110+ζ312 | ζ3115+ζ3113+ζ313 | ζ3114+ζ319+ζ318 | ζ3130+ζ3126+ζ316 | ζ3129+ζ3121+ζ3112 | complex faithful |
| ρ13 | 3 | 0 | 0 | ζ3127+ζ3124+ζ3111 | ζ3125+ζ315+ζ31 | ζ3123+ζ3122+ζ3117 | ζ3119+ζ3110+ζ312 | ζ3130+ζ3126+ζ316 | ζ3115+ζ3113+ζ313 | ζ3120+ζ317+ζ314 | ζ3129+ζ3121+ζ3112 | ζ3114+ζ319+ζ318 | ζ3128+ζ3118+ζ3116 | complex faithful |
►On 31 points: primitive - transitive group 31T3
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 26 6)(3 20 11)(4 14 16)(5 8 21)(7 27 31)(9 15 10)(12 28 25)(13 22 30)(17 29 19)(18 23 24)
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), (2,26,6)(3,20,11)(4,14,16)(5,8,21)(7,27,31)(9,15,10)(12,28,25)(13,22,30)(17,29,19)(18,23,24)>;
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), (2,26,6)(3,20,11)(4,14,16)(5,8,21)(7,27,31)(9,15,10)(12,28,25)(13,22,30)(17,29,19)(18,23,24) );
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)], [(2,26,6),(3,20,11),(4,14,16),(5,8,21),(7,27,31),(9,15,10),(12,28,25),(13,22,30),(17,29,19),(18,23,24)]])
G:=TransitiveGroup(31,3);
C31⋊C3 is a maximal subgroup of
C31⋊C6 C31⋊A4 C31⋊C15
C31⋊C3 is a maximal quotient of C31⋊C9 C31⋊A4
Matrix representation of C31⋊C3 ►in GL3(𝔽5) generated by
| 3 | 0 | 2 |
| 4 | 0 | 4 |
| 3 | 1 | 1 |
| 1 | 0 | 0 |
| 0 | 0 | 4 |
| 0 | 1 | 4 |
G:=sub<GL(3,GF(5))| [3,4,3,0,0,1,2,4,1],[1,0,0,0,0,1,0,4,4] >;
C31⋊C3 in GAP, Magma, Sage, TeX
C_{31}\rtimes C_3 % in TeX
G:=Group("C31:C3"); // GroupNames label
G:=SmallGroup(93,1);
// by ID
G=gap.SmallGroup(93,1);
# by ID
G:=PCGroup([2,-3,-31,301]);
// Polycyclic
G:=Group<a,b|a^31=b^3=1,b*a*b^-1=a^5>;
// generators/relations
Export
Subgroup lattice of C31⋊C3 in TeX
Character table of C31⋊C3 in TeX