metacyclic, supersoluble, monomial, Z-group, 5-hyperelementary
Aliases: C31⋊C5, SmallGroup(155,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C31⋊C5 |
Generators and relations for C31⋊C5
G = < a,b | a31=b5=1, bab-1=a2 >
Character table of C31⋊C5
| class | 1 | 5A | 5B | 5C | 5D | 31A | 31B | 31C | 31D | 31E | 31F | |
| size | 1 | 31 | 31 | 31 | 31 | 5 | 5 | 5 | 5 | 5 | 5 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | ζ53 | ζ54 | ζ5 | ζ52 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ3 | 1 | ζ5 | ζ53 | ζ52 | ζ54 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ4 | 1 | ζ52 | ζ5 | ζ54 | ζ53 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ5 | 1 | ζ54 | ζ52 | ζ53 | ζ5 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ6 | 5 | 0 | 0 | 0 | 0 | ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 | ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 | ζ3116+ζ318+ζ314+ζ312+ζ31 | ζ3124+ζ3117+ζ3112+ζ316+ζ313 | ζ3120+ζ3118+ζ3110+ζ319+ζ315 | ζ3128+ζ3125+ζ3119+ζ3114+ζ317 | complex faithful |
| ρ7 | 5 | 0 | 0 | 0 | 0 | ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 | ζ3124+ζ3117+ζ3112+ζ316+ζ313 | ζ3128+ζ3125+ζ3119+ζ3114+ζ317 | ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 | ζ3116+ζ318+ζ314+ζ312+ζ31 | ζ3120+ζ3118+ζ3110+ζ319+ζ315 | complex faithful |
| ρ8 | 5 | 0 | 0 | 0 | 0 | ζ3128+ζ3125+ζ3119+ζ3114+ζ317 | ζ3120+ζ3118+ζ3110+ζ319+ζ315 | ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 | ζ3116+ζ318+ζ314+ζ312+ζ31 | ζ3124+ζ3117+ζ3112+ζ316+ζ313 | ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 | complex faithful |
| ρ9 | 5 | 0 | 0 | 0 | 0 | ζ3120+ζ3118+ζ3110+ζ319+ζ315 | ζ3116+ζ318+ζ314+ζ312+ζ31 | ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 | ζ3128+ζ3125+ζ3119+ζ3114+ζ317 | ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 | ζ3124+ζ3117+ζ3112+ζ316+ζ313 | complex faithful |
| ρ10 | 5 | 0 | 0 | 0 | 0 | ζ3116+ζ318+ζ314+ζ312+ζ31 | ζ3128+ζ3125+ζ3119+ζ3114+ζ317 | ζ3124+ζ3117+ζ3112+ζ316+ζ313 | ζ3120+ζ3118+ζ3110+ζ319+ζ315 | ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 | ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 | complex faithful |
| ρ11 | 5 | 0 | 0 | 0 | 0 | ζ3124+ζ3117+ζ3112+ζ316+ζ313 | ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 | ζ3120+ζ3118+ζ3110+ζ319+ζ315 | ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 | ζ3128+ζ3125+ζ3119+ζ3114+ζ317 | ζ3116+ζ318+ζ314+ζ312+ζ31 | complex faithful |
►On 31 points: primitive - transitive group 31T4
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 17 9 5 3)(4 18 25 13 7)(6 19 10 21 11)(8 20 26 29 15)(12 22 27 14 23)(16 24 28 30 31)
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), (2,17,9,5,3)(4,18,25,13,7)(6,19,10,21,11)(8,20,26,29,15)(12,22,27,14,23)(16,24,28,30,31)>;
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), (2,17,9,5,3)(4,18,25,13,7)(6,19,10,21,11)(8,20,26,29,15)(12,22,27,14,23)(16,24,28,30,31) );
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)], [(2,17,9,5,3),(4,18,25,13,7),(6,19,10,21,11),(8,20,26,29,15),(12,22,27,14,23),(16,24,28,30,31)]])
G:=TransitiveGroup(31,4);
C31⋊C5 is a maximal subgroup of
C31⋊C10 C31⋊C15
Matrix representation of C31⋊C5 ►in GL5(𝔽2)
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(5,GF(2))| [0,0,0,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,0,0,0,1,1,0,0],[1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,1] >;
C31⋊C5 in GAP, Magma, Sage, TeX
C_{31}\rtimes C_5 % in TeX
G:=Group("C31:C5"); // GroupNames label
G:=SmallGroup(155,1);
// by ID
G=gap.SmallGroup(155,1);
# by ID
G:=PCGroup([2,-5,-31,321]);
// Polycyclic
G:=Group<a,b|a^31=b^5=1,b*a*b^-1=a^2>;
// generators/relations
Export
Subgroup lattice of C31⋊C5 in TeX
Character table of C31⋊C5 in TeX