Sporadic JacquesTits group T
Derived twisted group 2F4(2)'
Order = 17 971 200 = 211.33.52.13.
Mult = 1.
Out = 2.
Robert Wilson's ATLAS page for 2F4(2)' is available here.
2F4(2)': Length 143, 2-generator, 6-relator.
< x, y | x2 = y3 = (xy)13 = [x, y]5 = [x, yxy]4 = (xyxyxyxyxy-1)6 = 1 >
Remark: x and y are R.A.Wilson's standard generators for 2F4(2)'.
The presentation is available in MAGMA code here. (This applies to the subgroups below too.)
Some subgroups:
- M1 = < y, (xy)3(xy-1)4xy(xy-1)2xyxyx > = L3(3):2 of order 11 232 and index 1 600.
- M2 = < y, xyxyxy-1xy-1xyxyx > = L3(3):2 of order 11 232 and index 1 600.
- M3 = < x, y-1xyxyxy-1xyxyxyxy-1xy-1 > = 2.[28]:5:4 of order 10 240 and index 1 755.
- M4 = < y, xyxyxyxy-1xyxyxy-1xyxyxyx > = L2(25) of order 7 800 and index 2 304.
- M5 = < y, xyxyxyxyxy-1xy-1xyxyxy-1xyxy-1x > = 22.[28]:S3 of order 6 144 and index 2 925.
- M6 = < x, yxyxyxy-1xy > = A6.22 of order 1 440 and index 12 480.
- M7 = < x, yxy-1xyxy-1xy-1xy > = A6.22 of order 1 440 and index 12 480.
- M8 = < x, yxy > = 52:4A4 of order 1 200 and index 14 976.
- H1 = < y, xyxyxy-1x > = L3(3) of order 5 616 and index 3 200.
- H2 = < y, xyxyxyxyxy-1xyxyxyxyx > = L3(3) of order 5 616 and index 3 200.
- H3 = < x, yxy-1xyxy-1xy, [x, yxy]2 > = 2.[28]:D10 of order 5 120 and index 3 510.
- H4 = < x, yxy-1xyxy-1xy, [x, yxy]2, yxyxyxyxyxy-1xyxy[x, yxyxyxyxyxy-1xyxy]2 > = 2.[28]:5:4 of order 10 240 and index 1 755.
- H5 = < y, xyxyxy-1xyxyxy-1xyxyx > = 22.[28]:3 of order 3 072 and index 5 850.
- H6 = < y, xyx > = 52:4A4 of order 1 200 and index 14 976.
NB: The following pairs are non-conjugate in G: M1 and M2, M6 and M7 and H1 and H2. H1 is conjugate to M1' and H2 is conjugate to M2'.
We have now added more subgroups to the MAGMA file, namely M6a, M6b and M6c of index 2 in M6 and M7a, M7b and M7c of index 2 in M7. We have the isomorphisms: M6a = M7a = S6, M6b = M7b = PGL2(9) and M6c = M7c = M10. In the action of G on the cosets of various subgroups, we found that xyxyxyxy-1 of order 8 had the cycle-type shown in the table below.
|
M1 |
M2 |
H1 |
H2 |
M6 |
M6a |
M6b |
M6c |
M7 |
M7a |
M7b |
M7c |
| 1^ | 2 | 2 | 4 | 0 |
2 | 0 | 0 | 4 |
2 | 0 | 4 | 0 |
| 2^ | 1 | 1 | 2 | 4 |
5 | 4 | 4 | 10 |
5 | 4 | 2 | 12 |
| 4^ | 11 | 11 | 6 | 6 |
17 | 38 | 22 | 18 |
17 | 38 | 22 | 18 |
| 8^ | 194 | 194 | 396 | 396 |
1550 | 3100 | 3108 | 3108 |
1550 | 3100 | 3108 | 3108 |
Realisation:
You can do coset enumeration over one of the subgroups above. Alternatively, you can visit Rob Wilson's ATLAS page and find some (representations) there.
2F4(2): Length ??, ?-generator, ?-relator.
Hmmmm.
Last updated 5th August, 1997
John N. Bray