# Rank 3 permutation group

In mathematical finite group theory, a rank 3 permutation group acts transitively on a set such that the stabilizer of a point has 3 orbits. The study of these groups was started by Higman (1964, 1971). Several of the sporadic simple groups were discovered as rank 3 permutation groups.

## Classification

The primitive rank 3 permutation groups are all in one of the following classes:
* Cameron (1981) classified the ones such that $T\times T\leq G\leq T_{0}\operatorname {wr} Z/2Z$ where the socle T of T$_0$ is simple, and T$_0$ is a 2-transitive group of degree √n.
* Liebeck (1987) classified the ones with a regular elementary abelian normal subgroup
* Bannai (1971–72) classified the ones whose socle is a simple alternating group
* Kantor & Liebler (1982) classified the ones whose socle is a simple classical group
* Liebeck & Saxl (1986) classified the ones whose socle is a simple exceptional or sporadic group.

## Examples

If G is any 4-transitive group acting on a set S, then its action on pairs of elements of S is a rank 3 permutation group. In particular most of the alternating groups, symmetric groups, and Mathieu groups have 4-transitive actions, and so can be made into rank 3 permutation groups.
The projective general linear group acting on lines in a projective space of dimension at least 3 is a rank-3 permutation group.
Several 3-transposition groups are rank-3 permutation groups (in the action on transpositions).
It is common for the point-stabilizer of a rank-3 permutation group acting on one of the orbits to be a rank-3 permutation group. This gives several "chains" of rank-3 permutation groups, such as the Suzuki chain and the chain ending with the Fischer groups.
Some unusual rank-3 permutation groups (many from (Liebeck & Saxl 1986)) are listed below.
For each row in the table below, in the grid in the column marked "size", the number to the left of the equal sign is the degree of the permutation group for the permutation group mentioned in the row. In the grid, the sum to the right of the equal sign shows the lengths of the three orbits of the stabilizer of a point of the permutation group. For example, the expression 15 = 1+6+8 in the first row of the table under the heading means that the permutation group for the first row has degree 15, and the lengths of three orbits of the stabilizer of a point of the permutation group are 1, 6 and 8 respectively.
| Group | Point stabilizer | size | Comments |
| A$_6$ = L$_2$(9) = Sp$_4$(2)' = M$_{10}$' | S$_4$ | 15 = 1+6+8 | Pairs of points, or sets of 3 blocks of 2, in the 6-point permutation representation; two classes |
| A$_9$ | L$_2$(8):3 | 120 = 1+56+63 | Projective line P$_1$(8); two classes |
| A$_{10}$ | (A$_5$×A$_5$):4 | 126 = 1+25+100 | Sets of 2 blocks of 5 in the natural 10-point permutation representation |
| L$_2$(8) | 7:2 = Dih(7) | 36 = 1+14+21 | Pairs of points in P$_1$(8) |
| L$_3$(4) | A$_6$ | 56 = 1+10+45 | Hyperovals in P$_2$(4); three classes |
| L$_4$(3) | PSp$_4$(3):2 | 117 = 1+36+80 | Symplectic polarities of P$_3$(3); two classes |
| G$_2$(2)' = U$_3$(3) | PSL$_3$(2) | 36 = 1+14+21 | Suzuki chain |
| U$_3$(5) | A$_7$ | 50 = 1+7+42 | The action on the vertices of the Hoffman-Singleton graph; three classes |
| U$_4$(3) | L$_3$(4) | 162 = 1+56+105 | Two classes |
| Sp$_6$(2) | G$_2$(2) = U$_3$(3):2 | 120 = 1+56+63 | The Chevalley group of type G$_2$ acting on the octonion algebra over GF(2) |
| Ω$_7$(3) | G$_2$(3) | 1080 = 1+351+728 | The Chevalley group of type G$_2$ acting on the imaginary octonions of the octonion algebra over GF(3); two classes |
| U$_6$(2) | U$_4$(3):2$_2$ | 1408 = 1+567+840 | The point stabilizer is the image of the linear representation resulting from "bringing down" the complex representation of Mitchell's group (a complex reflection group) modulo 2; three classes |
| M$_{11}$ | M$_9$:2 = 3$^2$:SD$_{16}$ | 55 = 1+18+36 | Pairs of points in the 11-point permutation representation |
| M$_{12}$ | M$_{10}$:2 = A$_6$.2$^2$ = PΓL(2,9) | 66 = 1+20+45 | Pairs of points, or pairs of complementary blocks of S(5,6,12), in the 12-point permutation representation; two classes |
| M$_{22}$ | 2$^4$:A$_6$ | 77 = 1+16+60 | Blocks of S(3,6,22) |
| J$_2$ | U$_3$(3) | 100 = 1+36+63 | Suzuki chain; the action on the vertices of the Hall-Janko graph |
| Higman-Sims group HS | M$_{22}$ | 100 = 1+22+77 | The action on the vertices of the Higman-Sims graph |
| M$_{22}$ | A$_7$ | 176 = 1+70+105 | Two classes |
| M$_{23}$ | M$_{21}$:2 = L$_3$(4):2$_2$ = PΣL(3,4) | 253 = 1+42+210 | Pairs of points in the 23-point permutation representation |
| M$_{23}$ | 2$^4$:A$_7$ | 253 = 1+112+140 | Blocks of S(4,7,23) |
| McLaughlin group McL | U$_4$(3) | 275 = 1+112+162 | The action on the vertices of the McLaughlin graph |
| M$_{24}$ | M$_{22}$:2 | 276 = 1+44+231 | Pairs of points in the 24-point permutation representation |
| G$_2$(3) | U$_3$(3):2 | 351 = 1+126+244 | Two classes |
| G$_2$(4) | J$_2$ | 416 = 1+100+315 | Suzuki chain |
| M$_{24}$ | M$_{12}$:2 | 1288 = 1+495+792 | Pairs of complementary dodecads in the 24-point permutation representation |
| Suzuki group Suz | G$_2$(4) | 1782 = 1+416+1365 | Suzuki chain |
| G$_2$(4) | U$_3$(4):2 | 2016 = 1+975+1040 | |
| Co$_2$ | PSU$_6$(2):2 | 2300 = 1+891+1408 | |
| Rudvalis group Ru | $^2$F$_4$(2) | 4060 = 1+1755+2304 | |
| Fi$_{22}$ | 2.PSU$_6$(2) | 3510 = 1+693+2816 | 3-transpositions |
| Fi$_{22}$ | Ω$_7$(3) | 14080 = 1+3159+10920 | Two classes |
| Fi$_{23}$ | 2.Fi$_{22}$ | 31671 = 1+3510+28160 | 3-transpositions |
| G$_2$(8).3 | SU$_3$(8).6 | 130816 = 1+32319+98496 | |
| Fi$_{23}$ | PΩ$_8$$^+$(3).S$_3$ | 137632 = 1+28431+109200 | |
| Fi$_{24}$' | Fi$_{23}$ | 306936 = 1+31671+275264 | 3-transpositions |