metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C37⋊C9, C37⋊C3.C3, SmallGroup(333,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C37 — C37⋊C9 |
Generators and relations for C37⋊C9
G = < a,b | a37=b9=1, bab-1=a7 >
Character table of C37⋊C9
| class | 1 | 3A | 3B | 9A | 9B | 9C | 9D | 9E | 9F | 37A | 37B | 37C | 37D | |
| size | 1 | 37 | 37 | 37 | 37 | 37 | 37 | 37 | 37 | 9 | 9 | 9 | 9 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | ζ32 | ζ3 | ζ98 | ζ9 | ζ94 | ζ97 | ζ92 | ζ95 | 1 | 1 | 1 | 1 | linear of order 9 |
| ρ5 | 1 | ζ32 | ζ3 | ζ92 | ζ97 | ζ9 | ζ94 | ζ95 | ζ98 | 1 | 1 | 1 | 1 | linear of order 9 |
| ρ6 | 1 | ζ3 | ζ32 | ζ9 | ζ98 | ζ95 | ζ92 | ζ97 | ζ94 | 1 | 1 | 1 | 1 | linear of order 9 |
| ρ7 | 1 | ζ3 | ζ32 | ζ94 | ζ95 | ζ92 | ζ98 | ζ9 | ζ97 | 1 | 1 | 1 | 1 | linear of order 9 |
| ρ8 | 1 | ζ3 | ζ32 | ζ97 | ζ92 | ζ98 | ζ95 | ζ94 | ζ9 | 1 | 1 | 1 | 1 | linear of order 9 |
| ρ9 | 1 | ζ32 | ζ3 | ζ95 | ζ94 | ζ97 | ζ9 | ζ98 | ζ92 | 1 | 1 | 1 | 1 | linear of order 9 |
| ρ10 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3732+ζ3731+ζ3729+ζ3724+ζ3720+ζ3718+ζ3715+ζ3714+ζ372 | ζ3735+ζ3723+ζ3722+ζ3719+ζ3717+ζ3713+ζ378+ζ376+ζ375 | ζ3734+ζ3733+ζ3726+ζ3716+ζ3712+ζ3710+ζ379+ζ377+ζ37 | ζ3736+ζ3730+ζ3728+ζ3727+ζ3725+ζ3721+ζ3711+ζ374+ζ373 | complex faithful |
| ρ11 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3734+ζ3733+ζ3726+ζ3716+ζ3712+ζ3710+ζ379+ζ377+ζ37 | ζ3736+ζ3730+ζ3728+ζ3727+ζ3725+ζ3721+ζ3711+ζ374+ζ373 | ζ3735+ζ3723+ζ3722+ζ3719+ζ3717+ζ3713+ζ378+ζ376+ζ375 | ζ3732+ζ3731+ζ3729+ζ3724+ζ3720+ζ3718+ζ3715+ζ3714+ζ372 | complex faithful |
| ρ12 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3735+ζ3723+ζ3722+ζ3719+ζ3717+ζ3713+ζ378+ζ376+ζ375 | ζ3732+ζ3731+ζ3729+ζ3724+ζ3720+ζ3718+ζ3715+ζ3714+ζ372 | ζ3736+ζ3730+ζ3728+ζ3727+ζ3725+ζ3721+ζ3711+ζ374+ζ373 | ζ3734+ζ3733+ζ3726+ζ3716+ζ3712+ζ3710+ζ379+ζ377+ζ37 | complex faithful |
| ρ13 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3736+ζ3730+ζ3728+ζ3727+ζ3725+ζ3721+ζ3711+ζ374+ζ373 | ζ3734+ζ3733+ζ3726+ζ3716+ζ3712+ζ3710+ζ379+ζ377+ζ37 | ζ3732+ζ3731+ζ3729+ζ3724+ζ3720+ζ3718+ζ3715+ζ3714+ζ372 | ζ3735+ζ3723+ζ3722+ζ3719+ζ3717+ζ3713+ζ378+ζ376+ζ375 | 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 17 35 27 10 34 11 13 8)(3 33 32 16 19 30 21 25 15)(4 12 29 5 28 26 31 37 22)(6 7 23 20 9 18 14 24 36)
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), (2,17,35,27,10,34,11,13,8)(3,33,32,16,19,30,21,25,15)(4,12,29,5,28,26,31,37,22)(6,7,23,20,9,18,14,24,36)>;
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), (2,17,35,27,10,34,11,13,8)(3,33,32,16,19,30,21,25,15)(4,12,29,5,28,26,31,37,22)(6,7,23,20,9,18,14,24,36) );
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)], [(2,17,35,27,10,34,11,13,8),(3,33,32,16,19,30,21,25,15),(4,12,29,5,28,26,31,37,22),(6,7,23,20,9,18,14,24,36)]])
Matrix representation of C37⋊C9 ►in GL9(𝔽1999)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 544 | 1415 | 698 | 705 | 712 | 467 | 1417 | 621 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 497 | 1963 | 368 | 754 | 1203 | 1495 | 1052 | 581 | 463 |
| 81 | 881 | 540 | 1146 | 1583 | 546 | 1519 | 371 | 495 |
| 252 | 1744 | 1917 | 1895 | 1547 | 1842 | 1878 | 1585 | 702 |
| 668 | 1446 | 1196 | 999 | 1054 | 757 | 612 | 1993 | 1455 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 124 | 741 | 1053 | 1639 | 797 | 1818 | 1816 | 1383 | 1460 |
| 1171 | 1092 | 1262 | 90 | 1350 | 86 | 1090 | 139 | 795 |
G:=sub<GL(9,GF(1999))| [0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,544,0,1,0,0,0,0,0,0,1415,0,0,1,0,0,0,0,0,698,0,0,0,1,0,0,0,0,705,0,0,0,0,1,0,0,0,712,0,0,0,0,0,1,0,0,467,0,0,0,0,0,0,1,0,1417,0,0,0,0,0,0,0,1,621],[1,0,497,81,252,668,0,124,1171,0,0,1963,881,1744,1446,0,741,1092,0,0,368,540,1917,1196,0,1053,1262,0,0,754,1146,1895,999,0,1639,90,0,0,1203,1583,1547,1054,0,797,1350,0,0,1495,546,1842,757,1,1818,86,0,0,1052,1519,1878,612,0,1816,1090,0,1,581,371,1585,1993,0,1383,139,0,0,463,495,702,1455,0,1460,795] >;
C37⋊C9 in GAP, Magma, Sage, TeX
C_{37}\rtimes C_9 % in TeX
G:=Group("C37:C9"); // GroupNames label
G:=SmallGroup(333,3);
// by ID
G=gap.SmallGroup(333,3);
# by ID
G:=PCGroup([3,-3,-3,-37,9,1298,707]);
// Polycyclic
G:=Group<a,b|a^37=b^9=1,b*a*b^-1=a^7>;
// generators/relations
Export
Subgroup lattice of C37⋊C9 in TeX
Character table of C37⋊C9 in TeX