metacyclic, supersoluble, monomial, Z-group
Aliases: C37⋊C6, D37⋊C3, C37⋊C3⋊C2, SmallGroup(222,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C37 — C37⋊C6 |
Generators and relations for C37⋊C6
G = < a,b | a37=b6=1, bab-1=a11 >
Character table of C37⋊C6
| class | 1 | 2 | 3A | 3B | 6A | 6B | 37A | 37B | 37C | 37D | 37E | 37F | |
| size | 1 | 37 | 37 | 37 | 37 | 37 | 6 | 6 | 6 | 6 | 6 | 6 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ4 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 6 | 0 | 0 | 0 | 0 | 0 | ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 | ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 | ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 | ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 | ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 | ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 | orthogonal faithful |
| ρ8 | 6 | 0 | 0 | 0 | 0 | 0 | ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 | ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 | ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 | ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 | ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 | ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 | orthogonal faithful |
| ρ9 | 6 | 0 | 0 | 0 | 0 | 0 | ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 | ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 | ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 | ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 | ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 | ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 | orthogonal faithful |
| ρ10 | 6 | 0 | 0 | 0 | 0 | 0 | ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 | ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 | ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 | ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 | ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 | ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 | orthogonal faithful |
| ρ11 | 6 | 0 | 0 | 0 | 0 | 0 | ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 | ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 | ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 | ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 | ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 | ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 | orthogonal faithful |
| ρ12 | 6 | 0 | 0 | 0 | 0 | 0 | ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 | ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 | ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 | ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 | ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 | ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 | 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 28 27 37 11 12)(3 18 16 36 21 23)(4 8 5 35 31 34)(6 25 20 33 14 19)(7 15 9 32 24 30)(10 22 13 29 17 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), (2,28,27,37,11,12)(3,18,16,36,21,23)(4,8,5,35,31,34)(6,25,20,33,14,19)(7,15,9,32,24,30)(10,22,13,29,17,26)>;
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), (2,28,27,37,11,12)(3,18,16,36,21,23)(4,8,5,35,31,34)(6,25,20,33,14,19)(7,15,9,32,24,30)(10,22,13,29,17,26) );
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)], [(2,28,27,37,11,12),(3,18,16,36,21,23),(4,8,5,35,31,34),(6,25,20,33,14,19),(7,15,9,32,24,30),(10,22,13,29,17,26)]])
C37⋊C6 is a maximal subgroup of
C37⋊C12
C37⋊C6 is a maximal quotient of C74.C6
Matrix representation of C37⋊C6 ►in GL6(𝔽223)
| 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 |
| 222 | 125 | 178 | 177 | 178 | 125 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 149 | 54 | 5 | 77 | 191 | 55 |
| 204 | 164 | 54 | 198 | 217 | 162 |
| 53 | 139 | 103 | 161 | 68 | 55 |
| 98 | 14 | 75 | 3 | 127 | 193 |
| 163 | 159 | 34 | 136 | 180 | 49 |
G:=sub<GL(6,GF(223))| [0,0,0,0,0,222,1,0,0,0,0,125,0,1,0,0,0,178,0,0,1,0,0,177,0,0,0,1,0,178,0,0,0,0,1,125],[1,149,204,53,98,163,0,54,164,139,14,159,0,5,54,103,75,34,0,77,198,161,3,136,0,191,217,68,127,180,0,55,162,55,193,49] >;
C37⋊C6 in GAP, Magma, Sage, TeX
C_{37}\rtimes C_6 % in TeX
G:=Group("C37:C6"); // GroupNames label
G:=SmallGroup(222,1);
// by ID
G=gap.SmallGroup(222,1);
# by ID
G:=PCGroup([3,-2,-3,-37,1946,707]);
// Polycyclic
G:=Group<a,b|a^37=b^6=1,b*a*b^-1=a^11>;
// generators/relations
Export
Subgroup lattice of C37⋊C6 in TeX
Character table of C37⋊C6 in TeX