Suzuki sporadic group

In the area of modern algebra known as group theory, the Suzuki group Suz or Sz is a sporadic simple group of order

:   448,345,497,600 = 2¹³ · 3⁷ · 5² · 7 · 11 · 13 ≈ 4.

History

Suz is one of the 26 Sporadic groups and was discovered by as a rank 3 permutation group on 1782 points with point stabilizer G₂(4). It is not related to the Suzuki groups of Lie type. The Schur multiplier has order 6 and the outer automorphism group has order 2.

Complex Leech lattice

The 24-dimensional Leech lattice has a fixed-point-free automorphism of order 3. Identifying this with a complex cube root of 1 makes the Leech lattice into a 12 dimensional lattice over the Eisenstein integers, called the complex Leech lattice. The automorphism group of the complex Leech lattice is the universal cover 6 · Suz of the Suzuki group. This makes the group 6 · Suz · 2 into a maximal subgroup of Conway's group Co₀ = 2 · Co₁ of automorphisms of the Leech lattice, and shows that it has two complex irreducible representations of dimension 12. The group 6 · Suz acting on the complex Leech lattice is analogous to the group 2 · Co₁ acting on the Leech lattice.

Suzuki chain

The Suzuki chain or Suzuki tower is the following tower of rank 3 permutation groups from , each of which is the point stabilizer of the next.

• G₂(2) = U(3, 3) · 2 has a rank 3 action on 36 = 1 + 14 + 21 points with point stabilizer PSL(3, 2) · 2
• J₂ · 2 has a rank 3 action on 100 = 1 + 36 + 63 points with point stabilizer G₂(2)
• G₂(4) · 2 has a rank 3 action on 416 = 1 + 100 + 315 points with point stabilizer J₂ · 2
• Suz · 2 has a rank 3 action on 1782 = 1 + 416 + 1365 points with point stabilizer G₂(4) · 2

Maximal subgroups

found the 17 conjugacy classes of maximal subgroups of Suz as follows:

{| class="wikitable"

|+ Maximal subgroups of Suz

|-

! No. !! Structure !! Order !! Index !! Comments

|-

| 1 ||G₂(4) ||align=right|251,596,800<br />=&nbsp;2¹²·3³·5²·7·13||align=right|1,782 <br />=&nbsp;2·3⁴·11 ||

|-

| 2 ||3₂^·&thinsp;U(4, 3) : 2'₃ ||align=right|19,595,520 <br />=&nbsp;2⁸·3⁷·5·7 ||align=right|22,880 <br />=&nbsp;2⁵·5·11·13 ||normalizer of a subgroup of order 3 (class 3A)

|-

| 3 ||U(5, 2) ||align=right|13,685,760 <br />=&nbsp;2¹⁰·3⁵·5·11 ||align=right|32,760 <br />=&nbsp;2³·3²·5·7·13 ||

|-

| 4 ||2^{&thinsp;·&thinsp;}U(4, 2) ||align=right|3,317,760 <br />=&nbsp;2¹³·3⁴·5 ||align=right|135,135 <br />=&nbsp;3³·5·7·11·13 ||centralizer of an involution of class 2A

|-

| 5 ||3⁵ : M₁₁ ||align=right|1,924,560 <br />=&nbsp;2⁴·3⁷·5·11 ||align=right|232,960 <br />=&nbsp;2⁹·5·7·13 ||

|-

| 6 ||J₂ : 2 ||align=right|1,209,600 <br />=&nbsp;2⁸·3³·5²·7 ||align=right|370,656 <br />=&nbsp;2⁵·3^4·11·13 ||the subgroup fixed by an outer involution of class 2C

|-

| 7 ||2⁴⁺⁶ : 3A₆ ||align=right|1,105,920 <br />=&nbsp;2¹³·3³·5 ||align=right|405,405 <br />=&nbsp;3⁴·5·7·11·13 ||

|-

| 8 ||(A₄ × L₃(4)) : 2 ||align=right|483,840 <br />=&nbsp;2⁹·3³·5·7 ||align=right|926,640 <br />=&nbsp;2⁴·3⁴·5·11·13 ||

|-

| 9 ||2²⁺⁸ : (A₅ × S₃) ||align=right|368,640 <br />=&nbsp;2¹³·3²·5 ||align=right|1,216,215 <br />=&nbsp;3⁵·5·7·11·13 ||

|-

|10 ||M₁₂ : 2 ||align=right|190,080 <br />=&nbsp;2⁷·3³·5·11 ||align=right|2,358,720 <br />=&nbsp;2⁶·3⁴·5·7·13 ||the subgroup fixed by an outer involution of class 2D

|-

|11 ||3²⁺⁴ : 2(A₄ × 2²).2 ||align=right|139,968 <br />=&nbsp;2⁶·3⁷ ||align=right|3,203,200 <br />=&nbsp;2⁷·5²·7·11·13 ||

|-

|12 ||(A₆ × A₅) · 2 ||align=right|43,200 <br />=&nbsp;2⁶·3³·5² ||align=right|10,378,368 <br />=&nbsp;2⁷·3^4·7·11·13 ||

|-

|13 ||(A₆ × 3² : 4)^{&thinsp;·&thinsp;}2 ||align=right|25,920 <br />=&nbsp;2⁶·3⁴·5 ||align=right|17,297,280 <br />=&nbsp;2⁷·3³·5·7·11·13||

|-

|14,15||L₃(3) : 2 ||align=right|11,232 <br />=&nbsp;2⁵·3³·13 ||align=right|39,916,800 <br />=&nbsp;2⁸·3⁴·5^2·7·11 ||two classes, fused by an outer automorphism

|-

|16 ||L₂(25) ||align=right|7,800 <br />=&nbsp;2³·3·5²·13 ||align=right|57,480,192 <br />=&nbsp;2¹⁰·3⁶·7·11 ||

|-

|17 ||A₇ ||align=right|2,520 <br />=&nbsp;2³·3²·5·7 ||align=right|177,914,880<br />=&nbsp;2¹⁰·3⁵·5·11·13 ||

|-

|}

Bibliography

• Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A.: "Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups." Oxford, England 1985.

External links

• MathWorld: Suzuki group
• Atlas of Finite Group Representations: Suzuki group