Elements of the group are displayed as permutations of degree 14.
| Group |
Label |
Order |
Size |
Centralizer |
Powers |
Representative |
| 2P |
3P |
5P |
7P |
| $S_7\times \GL(3,2)$ |
1A |
$1$ |
$1$ |
$S_7\times \GL(3,2)$ |
1A |
1A |
1A |
1A |
$()$ |
| $S_7\times \GL(3,2)$ |
2A |
$2$ |
$21$ |
$C_2\times S_5\times \GL(3,2)$ |
1A |
2A |
2A |
2A |
$(9,10)$ |
| $S_7\times \GL(3,2)$ |
2B |
$2$ |
$21$ |
$D_4\times S_7$ |
1A |
2B |
2B |
2B |
$(1,3)(6,7)$ |
| $S_7\times \GL(3,2)$ |
2C |
$2$ |
$105$ |
$S_3\times D_4\times \GL(3,2)$ |
1A |
2C |
2C |
2C |
$(9,13)(10,12)$ |
| $S_7\times \GL(3,2)$ |
2D |
$2$ |
$105$ |
$C_2\times S_4\times \GL(3,2)$ |
1A |
2D |
2D |
2D |
$(8,11)(10,14)(12,13)$ |
| $S_7\times \GL(3,2)$ |
2E |
$2$ |
$441$ |
$C_2\times D_4\times S_5$ |
1A |
2E |
2E |
2E |
$(2,3)(5,6)(12,13)$ |
| $S_7\times \GL(3,2)$ |
2F |
$2$ |
$2205$ |
$S_3\times D_4^2$ |
1A |
2F |
2F |
2F |
$(1,3)(6,7)(8,12)(13,14)$ |
| $S_7\times \GL(3,2)$ |
2G |
$2$ |
$2205$ |
$\GL(2,\mathbb{Z}/4):C_2^2$ |
1A |
2G |
2G |
2G |
$(1,4)(5,6)(8,14)(9,11)(12,13)$ |
| $S_7\times \GL(3,2)$ |
3A |
$3$ |
$56$ |
$C_3\times S_7$ |
3A |
1A |
3A |
3A |
$(1,2,3)(5,6,7)$ |
| $S_7\times \GL(3,2)$ |
3B |
$3$ |
$70$ |
$C_3\times S_4\times \GL(3,2)$ |
3B |
1A |
3B |
3B |
$(8,13,12)$ |
| $S_7\times \GL(3,2)$ |
3C |
$3$ |
$280$ |
$C_3\times S_3\times \GL(3,2)$ |
3C |
1A |
3C |
3C |
$(8,10,13)(11,14,12)$ |
| $S_7\times \GL(3,2)$ |
3D |
$3$ |
$3920$ |
$C_3^2\times S_4$ |
3D |
1A |
3D |
3D |
$(1,2,3)(5,6,7)(11,12,13)$ |
| $S_7\times \GL(3,2)$ |
3E |
$3$ |
$15680$ |
$S_3\times C_3^2$ |
3E |
1A |
3E |
3E |
$(2,5,7)(3,4,6)(8,11,9)(10,14,12)$ |
| $S_7\times \GL(3,2)$ |
4A |
$4$ |
$42$ |
$C_4\times S_7$ |
2B |
4A |
4A |
4A |
$(1,4,5,6)(2,7)$ |
| $S_7\times \GL(3,2)$ |
4B |
$4$ |
$210$ |
$C_4\times S_3\times \GL(3,2)$ |
2C |
4B |
4B |
4B |
$(8,14,10,9)$ |
| $S_7\times \GL(3,2)$ |
4C |
$4$ |
$630$ |
$C_2\times C_4\times \GL(3,2)$ |
2C |
4C |
4C |
4C |
$(8,11,12,14)(9,10)$ |
| $S_7\times \GL(3,2)$ |
4D |
$4$ |
$882$ |
$C_2\times C_4\times S_5$ |
2B |
4D |
4D |
4D |
$(1,7,3,6)(2,4)(13,14)$ |
| $S_7\times \GL(3,2)$ |
4E |
$4$ |
$4410$ |
$C_4^2:D_6$ |
2B |
4E |
4E |
4E |
$(1,4)(2,6,3,5)(8,9)(10,12)$ |
| $S_7\times \GL(3,2)$ |
4F |
$4$ |
$4410$ |
$C_2^4.D_6$ |
2B |
4F |
4F |
4F |
$(1,5,4,6)(2,3)(8,11)(9,12)(10,13)$ |
| $S_7\times \GL(3,2)$ |
4G |
$4$ |
$4410$ |
$C_4^2:D_6$ |
2C |
4G |
4G |
4G |
$(1,7)(2,5)(9,14,13,12)$ |
| $S_7\times \GL(3,2)$ |
4H |
$4$ |
$8820$ |
$S_3\times C_4^2$ |
2F |
4H |
4H |
4H |
$(1,6,3,7)(2,4)(8,14,12,13)$ |
| $S_7\times \GL(3,2)$ |
4I |
$4$ |
$13230$ |
$C_4^2:C_2^2$ |
2C |
4I |
4I |
4I |
$(3,5)(4,7)(8,11)(9,10,13,12)$ |
| $S_7\times \GL(3,2)$ |
4J |
$4$ |
$26460$ |
$C_2\times C_4^2$ |
2F |
4J |
4J |
4J |
$(1,2,3,4)(6,7)(8,12)(9,14,10,11)$ |
| $S_7\times \GL(3,2)$ |
5A |
$5$ |
$504$ |
$C_{10}\times \GL(3,2)$ |
5A |
5A |
1A |
5A |
$(8,10,13,14,11)$ |
| $S_7\times \GL(3,2)$ |
6A |
$6$ |
$210$ |
$C_3\times D_4\times \GL(3,2)$ |
3B |
2C |
6A |
6A |
$(8,10)(9,14)(11,13,12)$ |
| $S_7\times \GL(3,2)$ |
6B |
$6$ |
$420$ |
$C_2\times C_6\times \GL(3,2)$ |
3B |
2A |
6B |
6B |
$(8,12,13)(9,10)$ |
| $S_7\times \GL(3,2)$ |
6C |
$6$ |
$840$ |
$C_6\times \GL(3,2)$ |
3C |
2D |
6C |
6C |
$(8,12,10,11,13,14)$ |
| $S_7\times \GL(3,2)$ |
6D |
$6$ |
$1176$ |
$C_6\times S_5$ |
3A |
2A |
6D |
6D |
$(1,6,2)(3,7,4)(12,14)$ |
| $S_7\times \GL(3,2)$ |
6E |
$6$ |
$1470$ |
$\GL(2,\mathbb{Z}/4):C_6$ |
3B |
2B |
6E |
6E |
$(1,3)(6,7)(8,12,9)$ |
| $S_7\times \GL(3,2)$ |
6F |
$6$ |
$4410$ |
$C_3\times D_4^2$ |
3B |
2F |
6F |
6F |
$(1,3)(6,7)(8,12)(9,11,10)(13,14)$ |
| $S_7\times \GL(3,2)$ |
6G |
$6$ |
$5880$ |
$C_{12}:D_6$ |
3A |
2C |
6G |
6G |
$(1,3,2)(5,7,6)(8,10)(9,13)$ |
| $S_7\times \GL(3,2)$ |
6H |
$6$ |
$5880$ |
$C_{12}:D_6$ |
3C |
2B |
6H |
6H |
$(1,4)(5,6)(8,9,10)(11,12,13)$ |
| $S_7\times \GL(3,2)$ |
6I |
$6$ |
$5880$ |
$C_6\times S_4$ |
3A |
2D |
6I |
6I |
$(1,6,2)(3,7,4)(8,11)(10,12)(13,14)$ |
| $S_7\times \GL(3,2)$ |
6J |
$6$ |
$8820$ |
$C_{12}:C_2^3$ |
3B |
2E |
6J |
6J |
$(3,7)(4,5)(8,10,11)(9,12)$ |
| $S_7\times \GL(3,2)$ |
6K |
$6$ |
$11760$ |
$D_4\times C_3^2$ |
3D |
2C |
6K |
6K |
$(1,3,2)(5,7,6)(8,10)(9,14)(11,13,12)$ |
| $S_7\times \GL(3,2)$ |
6L |
$6$ |
$17640$ |
$C_6\times D_4$ |
3C |
2G |
6L |
6L |
$(1,4)(5,6)(8,9,13,14,11,12)$ |
| $S_7\times \GL(3,2)$ |
6M |
$6$ |
$23520$ |
$C_6^2$ |
3D |
2A |
6M |
6M |
$(1,3,4)(5,6,7)(8,11)(9,10,14)$ |
| $S_7\times \GL(3,2)$ |
6N |
$6$ |
$47040$ |
$C_3\times C_6$ |
3E |
2D |
6N |
6N |
$(2,7,5)(3,6,4)(8,12,11,10,9,14)$ |
| $S_7\times \GL(3,2)$ |
7A1 |
$7$ |
$24$ |
$C_7\times S_7$ |
7A1 |
7A-1 |
7A-1 |
1A |
$(1,7,6,2,3,4,5)$ |
| $S_7\times \GL(3,2)$ |
7A-1 |
$7$ |
$24$ |
$C_7\times S_7$ |
7A-1 |
7A1 |
7A1 |
1A |
$(1,5,4,3,2,6,7)$ |
| $S_7\times \GL(3,2)$ |
7B |
$7$ |
$720$ |
$C_7\times \PSL(2,7)$ |
7B |
7B |
7B |
1A |
$(8,13,11,14,10,12,9)$ |
| $S_7\times \GL(3,2)$ |
7C1 |
$7$ |
$17280$ |
$C_7^2$ |
7C1 |
7C-1 |
7C-1 |
1A |
$(1,2,4,6,5,3,7)(8,12,13,10,9,11,14)$ |
| $S_7\times \GL(3,2)$ |
7C-1 |
$7$ |
$17280$ |
$C_7^2$ |
7C-1 |
7C1 |
7C1 |
1A |
$(1,7,3,5,6,4,2)(8,14,11,9,10,13,12)$ |
| $S_7\times \GL(3,2)$ |
10A |
$10$ |
$504$ |
$C_{10}\times \GL(3,2)$ |
5A |
10A |
2A |
10A |
$(8,14,10,11,13)(9,12)$ |
| $S_7\times \GL(3,2)$ |
10B |
$10$ |
$10584$ |
$D_4\times C_{10}$ |
5A |
10B |
2B |
10B |
$(3,5)(4,7)(8,9,10,14,12)$ |
| $S_7\times \GL(3,2)$ |
10C |
$10$ |
$10584$ |
$D_4\times C_{10}$ |
5A |
10C |
2E |
10C |
$(2,3)(5,6)(8,10,14,11,9)(12,13)$ |
| $S_7\times \GL(3,2)$ |
12A |
$12$ |
$420$ |
$C_{12}\times \GL(3,2)$ |
6A |
4B |
12A |
12A |
$(8,14,10,9)(11,12,13)$ |
| $S_7\times \GL(3,2)$ |
12B |
$12$ |
$2940$ |
$C_{12}\times S_4$ |
6E |
4A |
12B |
12B |
$(1,3)(2,7,4,6)(9,13,11)$ |
| $S_7\times \GL(3,2)$ |
12C |
$12$ |
$8820$ |
$D_4\times C_{12}$ |
6E |
4E |
12C |
12C |
$(1,4)(2,5,3,6)(8,9)(10,12)(11,14,13)$ |
| $S_7\times \GL(3,2)$ |
12D |
$12$ |
$8820$ |
$D_4\times C_{12}$ |
6A |
4G |
12D |
12D |
$(1,7)(2,5)(8,11,10)(9,12,13,14)$ |
