# 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 $T\times T\le G\le T_0 {\rm wr} Z/2Z$ where the socle T of T0 is simple, and T0 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.[1] 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.

A6 = L2(9) = Sp4(2)' = M10' S4 15 = 1+6+8 Pairs of points, or sets of 3 blocks of 2, in the 6-point permutation representation; two classes
A9 L2(8):3 120 = 1+56+63 Projective line P1(8); two classes
A10 (A5×A5):4 126 = 1+25+100 Sets of 2 blocks of 5 in the natural 10-point permutation representation
L2(8) 7:2 = Dih(7) 36 = 1+14+21 Pairs of points in P1(8)
L3(4) A6 56 = 1+10+45 Hyperovals in P2(4); three classes
L4(3) PSp4(3):2 117 = 1+36+80 Symplectic polarities of P3(3); two classes
G2(2)' = U3(3) PSL3(2) 36 = 1+14+21 Suzuki chain
U3(5) A7 50 = 1+7+42 The action on the vertices of the Hoffman-Singleton graph; three classes
U4(3) L3(4) 162 = 1+56+105 Two classes
Sp6(2) G2(2) = U3(3):2 120 = 1+56+63 The Chevalley group of type G2 acting on the octonion algebra over GF(2)
Ω7(3) G2(3) 1080 = 1+351+728 The Chevalley group of type G2 acting on the imaginary octonions of the octonion algebra over GF(3); two classes
U6(2) U4(3):22 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
M11 M9:2 = 32:SD16 55 = 1+18+36 Pairs of points in the 11-point permutation representation
M12 M10:2 = A6.22 = 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
M22 24:A6 77 = 1+16+60 Blocks of S(3,6,22)
J2 PSU3(3) 100 = 1+36+63 Suzuki chain; the action on the vertices of the Hall-Janko graph
Higman-Sims group HS M22 100 = 1+22+77 The action on the vertices of the Higman-Sims graph
M22 A7 176 = 1+70+105 Two classes
M23 M21:2 = L3(4):22 = PΣL(3,4) 253 = 1+42+210 Pairs of points in the 23-point permutation representation
M23 24:A7 253 = 1+112+140 Blocks of S(4,7,23)
McLaughlin group McL U4(3) 275 = 1+112+162 The action on the vertices of the McLaughlin graph
M24 M22:2 276 = 1+44+231 Pairs of points in the 24-point permutation representation
G2(3) U3(3):2 351 = 1+126+244 Two classes
G2(4) J2 416 = 1+100+315 Suzuki chain
M24 M12:2 1288 = 1+495+792 Pairs of complementary dodecads in the 24-point permutation representation
Suzuki group Suz G2(4) 1782 = 1+416+1365 Suzuki chain
G2(4) U3(4):2 2016 = 1+975+1040
Co2 PSU6(2):2 2300 = 1+1891+1408
Rudvalis group Ru ²F₄(2) 4060 = 1+1755+2304
Fi22 2.PSU6(2) 3510 = 1+693+2816 3-transpositions
Fi22 Ω7(3) 14080 = 1+3159+10920 Two classes
Fi23 2.Fi22 31671 = 1+3510+28160 3-transpositions
G2(8).3 SU3(8).6 130816 = 1+32319+98496
Fi23 8+(3).S3 137632 = 1+28431+109200
Fi24' Fi23 306936 = 1+31671+275264 3-transpositions

## Notes

1. ^ The three orbits are: the fixed pair itself; those pairs having one element in common with the fixed pair; and those pairs having no element in common with the fixed pair.