metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C37⋊C4, D37.C2, SmallGroup(148,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C37 — C37⋊C4 |
Generators and relations for C37⋊C4
G = < a,b | a37=b4=1, bab-1=a6 >
Character table of C37⋊C4
| class | 1 | 2 | 4A | 4B | 37A | 37B | 37C | 37D | 37E | 37F | 37G | 37H | 37I | |
| size | 1 | 37 | 37 | 37 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 4 | 0 | 0 | 0 | ζ3734+ζ3719+ζ3718+ζ373 | ζ3733+ζ3724+ζ3713+ζ374 | ζ3736+ζ3731+ζ376+ζ37 | ζ3722+ζ3721+ζ3716+ζ3715 | ζ3729+ζ3726+ζ3711+ζ378 | ζ3735+ζ3725+ζ3712+ζ372 | ζ3727+ζ3723+ζ3714+ζ3710 | ζ3728+ζ3720+ζ3717+ζ379 | ζ3732+ζ3730+ζ377+ζ375 | orthogonal faithful |
| ρ6 | 4 | 0 | 0 | 0 | ζ3733+ζ3724+ζ3713+ζ374 | ζ3732+ζ3730+ζ377+ζ375 | ζ3729+ζ3726+ζ3711+ζ378 | ζ3728+ζ3720+ζ3717+ζ379 | ζ3727+ζ3723+ζ3714+ζ3710 | ζ3722+ζ3721+ζ3716+ζ3715 | ζ3736+ζ3731+ζ376+ζ37 | ζ3735+ζ3725+ζ3712+ζ372 | ζ3734+ζ3719+ζ3718+ζ373 | orthogonal faithful |
| ρ7 | 4 | 0 | 0 | 0 | ζ3727+ζ3723+ζ3714+ζ3710 | ζ3736+ζ3731+ζ376+ζ37 | ζ3728+ζ3720+ζ3717+ζ379 | ζ3733+ζ3724+ζ3713+ζ374 | ζ3735+ζ3725+ζ3712+ζ372 | ζ3734+ζ3719+ζ3718+ζ373 | ζ3722+ζ3721+ζ3716+ζ3715 | ζ3732+ζ3730+ζ377+ζ375 | ζ3729+ζ3726+ζ3711+ζ378 | orthogonal faithful |
| ρ8 | 4 | 0 | 0 | 0 | ζ3722+ζ3721+ζ3716+ζ3715 | ζ3728+ζ3720+ζ3717+ζ379 | ζ3732+ζ3730+ζ377+ζ375 | ζ3736+ζ3731+ζ376+ζ37 | ζ3734+ζ3719+ζ3718+ζ373 | ζ3727+ζ3723+ζ3714+ζ3710 | ζ3733+ζ3724+ζ3713+ζ374 | ζ3729+ζ3726+ζ3711+ζ378 | ζ3735+ζ3725+ζ3712+ζ372 | orthogonal faithful |
| ρ9 | 4 | 0 | 0 | 0 | ζ3728+ζ3720+ζ3717+ζ379 | ζ3735+ζ3725+ζ3712+ζ372 | ζ3734+ζ3719+ζ3718+ζ373 | ζ3729+ζ3726+ζ3711+ζ378 | ζ3733+ζ3724+ζ3713+ζ374 | ζ3736+ζ3731+ζ376+ζ37 | ζ3732+ζ3730+ζ377+ζ375 | ζ3727+ζ3723+ζ3714+ζ3710 | ζ3722+ζ3721+ζ3716+ζ3715 | orthogonal faithful |
| ρ10 | 4 | 0 | 0 | 0 | ζ3735+ζ3725+ζ3712+ζ372 | ζ3722+ζ3721+ζ3716+ζ3715 | ζ3733+ζ3724+ζ3713+ζ374 | ζ3727+ζ3723+ζ3714+ζ3710 | ζ3732+ζ3730+ζ377+ζ375 | ζ3729+ζ3726+ζ3711+ζ378 | ζ3734+ζ3719+ζ3718+ζ373 | ζ3736+ζ3731+ζ376+ζ37 | ζ3728+ζ3720+ζ3717+ζ379 | orthogonal faithful |
| ρ11 | 4 | 0 | 0 | 0 | ζ3736+ζ3731+ζ376+ζ37 | ζ3729+ζ3726+ζ3711+ζ378 | ζ3735+ζ3725+ζ3712+ζ372 | ζ3732+ζ3730+ζ377+ζ375 | ζ3722+ζ3721+ζ3716+ζ3715 | ζ3733+ζ3724+ζ3713+ζ374 | ζ3728+ζ3720+ζ3717+ζ379 | ζ3734+ζ3719+ζ3718+ζ373 | ζ3727+ζ3723+ζ3714+ζ3710 | orthogonal faithful |
| ρ12 | 4 | 0 | 0 | 0 | ζ3732+ζ3730+ζ377+ζ375 | ζ3734+ζ3719+ζ3718+ζ373 | ζ3727+ζ3723+ζ3714+ζ3710 | ζ3735+ζ3725+ζ3712+ζ372 | ζ3736+ζ3731+ζ376+ζ37 | ζ3728+ζ3720+ζ3717+ζ379 | ζ3729+ζ3726+ζ3711+ζ378 | ζ3722+ζ3721+ζ3716+ζ3715 | ζ3733+ζ3724+ζ3713+ζ374 | orthogonal faithful |
| ρ13 | 4 | 0 | 0 | 0 | ζ3729+ζ3726+ζ3711+ζ378 | ζ3727+ζ3723+ζ3714+ζ3710 | ζ3722+ζ3721+ζ3716+ζ3715 | ζ3734+ζ3719+ζ3718+ζ373 | ζ3728+ζ3720+ζ3717+ζ379 | ζ3732+ζ3730+ζ377+ζ375 | ζ3735+ζ3725+ζ3712+ζ372 | ζ3733+ζ3724+ζ3713+ζ374 | ζ3736+ζ3731+ζ376+ζ37 | orthogonal 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 32 37 7)(3 26 36 13)(4 20 35 19)(5 14 34 25)(6 8 33 31)(9 27 30 12)(10 21 29 18)(11 15 28 24)(16 22 23 17)
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), (2,32,37,7)(3,26,36,13)(4,20,35,19)(5,14,34,25)(6,8,33,31)(9,27,30,12)(10,21,29,18)(11,15,28,24)(16,22,23,17)>;
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), (2,32,37,7)(3,26,36,13)(4,20,35,19)(5,14,34,25)(6,8,33,31)(9,27,30,12)(10,21,29,18)(11,15,28,24)(16,22,23,17) );
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)], [(2,32,37,7),(3,26,36,13),(4,20,35,19),(5,14,34,25),(6,8,33,31),(9,27,30,12),(10,21,29,18),(11,15,28,24),(16,22,23,17)]])
C37⋊C4 is a maximal subgroup of
C37⋊C12 C37⋊Dic3
C37⋊C4 is a maximal quotient of C37⋊C8 C37⋊Dic3
Matrix representation of C37⋊C4 ►in GL4(𝔽149) generated by
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 148 | 134 | 3 | 134 |
| 1 | 0 | 0 | 0 |
| 70 | 22 | 14 | 96 |
| 73 | 71 | 136 | 56 |
| 64 | 70 | 87 | 139 |
G:=sub<GL(4,GF(149))| [0,0,0,148,1,0,0,134,0,1,0,3,0,0,1,134],[1,70,73,64,0,22,71,70,0,14,136,87,0,96,56,139] >;
C37⋊C4 in GAP, Magma, Sage, TeX
C_{37}\rtimes C_4 % in TeX
G:=Group("C37:C4"); // GroupNames label
G:=SmallGroup(148,3);
// by ID
G=gap.SmallGroup(148,3);
# by ID
G:=PCGroup([3,-2,-2,-37,6,1118,653]);
// Polycyclic
G:=Group<a,b|a^37=b^4=1,b*a*b^-1=a^6>;
// generators/relations
Export
Subgroup lattice of C37⋊C4 in TeX
Character table of C37⋊C4 in TeX