Projective symplectic group:PSp(4,3)

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This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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Definition

This group is defined in the following equivalent ways:

  1. It is the projective symplectic group of degree four (i.e., arising from 4 \times 4 matrices) over field:F3. This is denoted PSp(4,3) and the Chevalley notation is C_2(3) -- however, due to collision with another family (see definition (3)) it is denoted B_2(3).
  2. It is the projective special unitary group of degree four (i.e., arising from 4 \times 4 matrices) over field:F2. This is denoted PSU(4,2) and the Chevalley notation is PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE].
  3. It is the Chevalley group of type B B_2(3), i.e., the group \Omega_5(3). This is kernel of the spinor norm on the special orthogonal group SO(5,3).

Arithmetic functions

Basic arithmetic functions

Function Value Similar groups Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set) 25920 groups with same order As PSp(4,q), q = 3: q^4(q^4 - 1)(q^2 - 1)/\operatorname{gcd}(2,q-1) = 3^4(3^4-1)(3^2 - 1)/\operatorname{gcd}(2,3-1) = 25920

Arithmetic functions of a counting nature

Function Value Similar groups Explanation for function value
number of conjugacy classes 20 groups with same order and number of conjugacy classes | groups with same number of conjugacy classes As PSp(4,q), q = 3 (odd): (q^2 + 6q + 13)/2 = (3^2 + (6)(3) + 13)/2 = 20 (more here)
See element structure of projective symplectic group:PSp(4,3)
number of conjugacy classes of subgroups 116 groups with same order and number of conjugacy classes of subgroups | groups with same number of conjugacy classes of subgroups See subgroup structure of projective symplectic group:PSp(4,3)
number of subgroups 45649 groups with same order and number of subgroups | groups with same number of subgroups See subgroup structure of projective symplectic group:PSp(4,3)

Group properties

Property Satisfied? Explanation
abelian group No
nilpotent group No
solvable group No
simple group, simple non-abelian group Yes projective symplectic group is simple

GAP implementation

Description Functions used
PSp(4,3) PSp
NormalSubgroups(SO(5,3))[2] SO, NormalSubgroups