metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C37⋊C3, SmallGroup(111,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C37 — C37⋊C3 |
Generators and relations for C37⋊C3
G = < a,b | a37=b3=1, bab-1=a10 >
Character table of C37⋊C3
| class | 1 | 3A | 3B | 37A | 37B | 37C | 37D | 37E | 37F | 37G | 37H | 37I | 37J | 37K | 37L | |
| size | 1 | 37 | 37 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
| ρ1 | 1 | 1 | 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 | 1 | 1 | linear of order 3 |
| ρ3 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 3 | 0 | 0 | ζ3734+ζ3733+ζ377 | ζ3732+ζ3724+ζ3718 | ζ3731+ζ3729+ζ3714 | ζ3730+ζ374+ζ373 | ζ3728+ζ3725+ζ3721 | ζ3726+ζ3710+ζ37 | ζ3723+ζ378+ζ376 | ζ3720+ζ3715+ζ372 | ζ3719+ζ3713+ζ375 | ζ3716+ζ3712+ζ379 | ζ3736+ζ3727+ζ3711 | ζ3735+ζ3722+ζ3717 | complex faithful |
| ρ5 | 3 | 0 | 0 | ζ3720+ζ3715+ζ372 | ζ3728+ζ3725+ζ3721 | ζ3730+ζ374+ζ373 | ζ3735+ζ3722+ζ3717 | ζ3723+ζ378+ζ376 | ζ3732+ζ3724+ζ3718 | ζ3734+ζ3733+ζ377 | ζ3736+ζ3727+ζ3711 | ζ3716+ζ3712+ζ379 | ζ3731+ζ3729+ζ3714 | ζ3719+ζ3713+ζ375 | ζ3726+ζ3710+ζ37 | complex faithful |
| ρ6 | 3 | 0 | 0 | ζ3736+ζ3727+ζ3711 | ζ3723+ζ378+ζ376 | ζ3735+ζ3722+ζ3717 | ζ3726+ζ3710+ζ37 | ζ3734+ζ3733+ζ377 | ζ3728+ζ3725+ζ3721 | ζ3720+ζ3715+ζ372 | ζ3719+ζ3713+ζ375 | ζ3731+ζ3729+ζ3714 | ζ3730+ζ374+ζ373 | ζ3716+ζ3712+ζ379 | ζ3732+ζ3724+ζ3718 | complex faithful |
| ρ7 | 3 | 0 | 0 | ζ3719+ζ3713+ζ375 | ζ3734+ζ3733+ζ377 | ζ3726+ζ3710+ζ37 | ζ3732+ζ3724+ζ3718 | ζ3720+ζ3715+ζ372 | ζ3723+ζ378+ζ376 | ζ3736+ζ3727+ζ3711 | ζ3716+ζ3712+ζ379 | ζ3730+ζ374+ζ373 | ζ3735+ζ3722+ζ3717 | ζ3731+ζ3729+ζ3714 | ζ3728+ζ3725+ζ3721 | complex faithful |
| ρ8 | 3 | 0 | 0 | ζ3726+ζ3710+ζ37 | ζ3731+ζ3729+ζ3714 | ζ3720+ζ3715+ζ372 | ζ3736+ζ3727+ζ3711 | ζ3730+ζ374+ζ373 | ζ3716+ζ3712+ζ379 | ζ3735+ζ3722+ζ3717 | ζ3732+ζ3724+ζ3718 | ζ3723+ζ378+ζ376 | ζ3734+ζ3733+ζ377 | ζ3728+ζ3725+ζ3721 | ζ3719+ζ3713+ζ375 | complex faithful |
| ρ9 | 3 | 0 | 0 | ζ3730+ζ374+ζ373 | ζ3719+ζ3713+ζ375 | ζ3723+ζ378+ζ376 | ζ3734+ζ3733+ζ377 | ζ3716+ζ3712+ζ379 | ζ3736+ζ3727+ζ3711 | ζ3731+ζ3729+ζ3714 | ζ3735+ζ3722+ζ3717 | ζ3732+ζ3724+ζ3718 | ζ3728+ζ3725+ζ3721 | ζ3726+ζ3710+ζ37 | ζ3720+ζ3715+ζ372 | complex faithful |
| ρ10 | 3 | 0 | 0 | ζ3728+ζ3725+ζ3721 | ζ3735+ζ3722+ζ3717 | ζ3719+ζ3713+ζ375 | ζ3716+ζ3712+ζ379 | ζ3726+ζ3710+ζ37 | ζ3730+ζ374+ζ373 | ζ3732+ζ3724+ζ3718 | ζ3723+ζ378+ζ376 | ζ3720+ζ3715+ζ372 | ζ3736+ζ3727+ζ3711 | ζ3734+ζ3733+ζ377 | ζ3731+ζ3729+ζ3714 | complex faithful |
| ρ11 | 3 | 0 | 0 | ζ3732+ζ3724+ζ3718 | ζ3730+ζ374+ζ373 | ζ3736+ζ3727+ζ3711 | ζ3719+ζ3713+ζ375 | ζ3735+ζ3722+ζ3717 | ζ3731+ζ3729+ζ3714 | ζ3726+ζ3710+ζ37 | ζ3728+ζ3725+ζ3721 | ζ3734+ζ3733+ζ377 | ζ3720+ζ3715+ζ372 | ζ3723+ζ378+ζ376 | ζ3716+ζ3712+ζ379 | complex faithful |
| ρ12 | 3 | 0 | 0 | ζ3716+ζ3712+ζ379 | ζ3720+ζ3715+ζ372 | ζ3732+ζ3724+ζ3718 | ζ3728+ζ3725+ζ3721 | ζ3736+ζ3727+ζ3711 | ζ3734+ζ3733+ζ377 | ζ3719+ζ3713+ζ375 | ζ3731+ζ3729+ζ3714 | ζ3735+ζ3722+ζ3717 | ζ3726+ζ3710+ζ37 | ζ3730+ζ374+ζ373 | ζ3723+ζ378+ζ376 | complex faithful |
| ρ13 | 3 | 0 | 0 | ζ3723+ζ378+ζ376 | ζ3726+ζ3710+ζ37 | ζ3716+ζ3712+ζ379 | ζ3731+ζ3729+ζ3714 | ζ3732+ζ3724+ζ3718 | ζ3735+ζ3722+ζ3717 | ζ3728+ζ3725+ζ3721 | ζ3734+ζ3733+ζ377 | ζ3736+ζ3727+ζ3711 | ζ3719+ζ3713+ζ375 | ζ3720+ζ3715+ζ372 | ζ3730+ζ374+ζ373 | complex faithful |
| ρ14 | 3 | 0 | 0 | ζ3731+ζ3729+ζ3714 | ζ3736+ζ3727+ζ3711 | ζ3728+ζ3725+ζ3721 | ζ3723+ζ378+ζ376 | ζ3719+ζ3713+ζ375 | ζ3720+ζ3715+ζ372 | ζ3716+ζ3712+ζ379 | ζ3730+ζ374+ζ373 | ζ3726+ζ3710+ζ37 | ζ3732+ζ3724+ζ3718 | ζ3735+ζ3722+ζ3717 | ζ3734+ζ3733+ζ377 | complex faithful |
| ρ15 | 3 | 0 | 0 | ζ3735+ζ3722+ζ3717 | ζ3716+ζ3712+ζ379 | ζ3734+ζ3733+ζ377 | ζ3720+ζ3715+ζ372 | ζ3731+ζ3729+ζ3714 | ζ3719+ζ3713+ζ375 | ζ3730+ζ374+ζ373 | ζ3726+ζ3710+ζ37 | ζ3728+ζ3725+ζ3721 | ζ3723+ζ378+ζ376 | ζ3732+ζ3724+ζ3718 | ζ3736+ζ3727+ζ3711 | complex faithful |
►On 37 points: primitive
Generators in S37
(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)
(2 27 11)(3 16 21)(4 5 31)(6 20 14)(7 9 24)(8 35 34)(10 13 17)(12 28 37)(15 32 30)(18 36 23)(19 25 33)(22 29 26)
G:=sub<Sym(37)| (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), (2,27,11)(3,16,21)(4,5,31)(6,20,14)(7,9,24)(8,35,34)(10,13,17)(12,28,37)(15,32,30)(18,36,23)(19,25,33)(22,29,26)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37), (2,27,11)(3,16,21)(4,5,31)(6,20,14)(7,9,24)(8,35,34)(10,13,17)(12,28,37)(15,32,30)(18,36,23)(19,25,33)(22,29,26) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37)], [(2,27,11),(3,16,21),(4,5,31),(6,20,14),(7,9,24),(8,35,34),(10,13,17),(12,28,37),(15,32,30),(18,36,23),(19,25,33),(22,29,26)]])
C37⋊C3 is a maximal subgroup of
C37⋊C6 C37⋊C9 C37⋊A4
C37⋊C3 is a maximal quotient of C37⋊2C9 C37⋊A4
Matrix representation of C37⋊C3 ►in GL3(𝔽223) generated by
| 52 | 1 | 0 |
| 112 | 0 | 1 |
| 184 | 193 | 141 |
| 62 | 179 | 5 |
| 66 | 206 | 159 |
| 55 | 14 | 178 |
G:=sub<GL(3,GF(223))| [52,112,184,1,0,193,0,1,141],[62,66,55,179,206,14,5,159,178] >;
C37⋊C3 in GAP, Magma, Sage, TeX
C_{37}\rtimes C_3 % in TeX
G:=Group("C37:C3"); // GroupNames label
G:=SmallGroup(111,1);
// by ID
G=gap.SmallGroup(111,1);
# by ID
G:=PCGroup([2,-3,-37,313]);
// Polycyclic
G:=Group<a,b|a^37=b^3=1,b*a*b^-1=a^10>;
// generators/relations
Export
Subgroup lattice of C37⋊C3 in TeX
Character table of C37⋊C3 in TeX