# Category:Particular groups

This is a category of articles about particular objects or specific concrete examples in a certain collection of objects

This page lists particular groups, viz groups, each unique up to isomorphism.

See also number of groups of given order to get an idea of how many groups there are of particular orders, along with links to pages that compare and contrast groups of a particular order.

If you want to search for a group given its group ID as per the SmallGroup library for GAP or Magma, type in SmallGroup(order,ID) into the search bar at the top right of the page. For instance, if the ID is (32,33), type in SmallGroup(32,33) in the search bar. If the ID is (12,3), type in SmallGroup(12,3) in the search bar.

## Particular groups of importance

Extremely important (importance rank 1):

GAP ID
Cyclic group:Z22 (1)
Cyclic group:Z33 (1)
Cyclic group:Z44 (1)
Group of integers
Klein four-group4 (2)
Symmetric group:S36 (1)
Trivial group1 (1)

Very important (importance rank 2):

GAP ID
Alternating group:A412 (3)
Alternating group:A560 (5)
Alternating group:A6360 (118)
Dihedral group:D88 (3)
Direct product of Z4 and Z28 (2)
Free group:F2
Projective special linear group:PSL(3,2)168 (42)
Quaternion group8 (4)
Special linear group:SL(2,3)24 (3)
Special linear group:SL(2,5)120 (5)
Symmetric group:S424 (12)
Symmetric group:S5120 (34)

Somewhat important (importance rank 3):

GAP ID
Alternating group:A7
Alternating group:A8
Binary octahedral group48 (28)
Dihedral group:D1616 (7)
Direct product of A4 and Z224 (13)
Direct product of D8 and Z216 (11)
Direct product of S4 and Z248 (48)
Double cover of symmetric group:S5 of minus type240 (89)
Double cover of symmetric group:S5 of plus type240 (90)
General linear group:GL(2,3)48 (29)
Generalized quaternion group:Q1616 (9)
M1616 (6)
Mathieu group:M10720 (765)
Mathieu group:M972 (41)
Projective general linear group:PGL(2,7)336 (208)
Projective general linear group:PGL(2,9)720 (764)
Projective special linear group:PSL(2,11)660 (13)
Projective special linear group:PSL(2,8)504 (156)
Semidihedral group:SD1616 (8)
Special linear group:SL(2,7)336 (114)
Special linear group:SL(2,9)720 (409)
Symmetric group:S7