↑

Learn more

The database currently contains 544,831 groups from many different sources, the largest of which is $S_{47}$ of order $47!$. In addition, it contains 275,379,753 of their subgroups and 39,933,457 of their irreducible complex characters. You can browse further statistics.

Browse

By order: 1-64  65-127  128  129-255  256  257-383  384  385-511  513-1000  1001-1500  1501-2000  2001- 

By nilpotency class: 1  2  3  4  5  6  7  8  9  (and not nilpotent)

By property: abelian  nonabelian  solvable  nonsolvable  simple  perfect  rational 

Some interesting groups or a random group

Search for subgroups or complex characters

Search

Order e.g. 4, or a range like 3..5 Exponent e.g. 2, or list of integers like 2, 3, 7
Automorphism group e.g. 4.2 Nilpotency class e.g. 4, or a range like 3..5
Automorphism group order e.g. 4, or a range like 3..5 Commutator e.g. 4 or 4.2 (order or label)
Center e.g. 4 or 4.2 (order or label) Abelianization e.g. 4 or 4.2 (order or label)
Central quotient e.g. 4 or 4.2 (order or label) Frattini subgroup e.g. 4.2
Abelian Direct product
Cyclic Semidirect product
Nilpotent Perfect
Simple Solvable
Transitive degree e.g. 4, or a range like 3..5 Permutation degree e.g. 4, or a range like 3..5
Minimal degree of $\C$-irrep e.g. -1, 4, or a range like 3..5 Minimal degree of $\Q$-irrep e.g. -1, 4, or a range like 3..5
Minimal degree of $\C$-rep e.g. 4, or a range like 3..5 Minimal degree of $\Q$-rep e.g. 4, or a range like 3..5
Almost simple Derived length e.g. 4, or a range like 3..5
Quasisimple Supersolvable
Outer aut. group e.g. 4.2 Metabelian
Outer aut. group order e.g. 4, or a range like 3..5 Metacyclic
A-group Monomial
Z-group Rational
Schur multiplier e.g. [2,4,12] Wreath product
Number of subgroups e.g. 4, or a range like 3..5 Number of normal subgroups e.g. 4, or a range like 3..5
Number of characteristic subgroups e.g. 4, or a range like 3..5 Number of divisions e.g. 4, or a range like 3..5
Number of conjugacy classes e.g. 4, or a range like 3..5 Number of autjugacy classes e.g. 4, or a range like 3..5
Order statistics e.g. 1^1, 2^3, 3^2 Rank e.g. 4, or a range like 3..5
Order factorization e.g. [2,1] or [8] Commutator length e.g. 1 or 2-4
Results to display
Display:

Find

Label or name
e.g. 8.3, GL(2,3), 8T34, C3:C4, C2*A5 or C16.D4