,"GAP ID","Arithmetic function value"
"Additive group of p-adic integers",,
"Baer-Specker group",,"Frattini length; 1,Fitting length; 1"
"Circle group",,"Dimension of a real Lie group; 1,Dimension of an algebraic group; 1"
"Cyclic group:Z16","16; 1","Underlying prime of p-group; 2,Order of a group; 16,Prime-base logarithm of order; 4,Max-length of a group; 4,Chief length; 4,Composition length; 4,Exponent of a group; 16,Prime-base logarithm of exponent; 4,Frattini length; 4,Minimum size of generating set; 1,Subgroup rank of a group; 1,Rank of a p-group; 1,Normal rank of a p-group; 1,Characteristic rank of a p-group; 1,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Cyclic group:Z2","2; 1","Underlying prime of p-group; 2,Order of a group; 2,Prime-base logarithm of order; 1,Max-length of a group; 1,Chief length; 1,Composition length; 1,Exponent of a group; 2,Prime-base logarithm of exponent; 1,Frattini length; 1,Minimum size of generating set; 1,Subgroup rank of a group; 1,Rank of a p-group; 1,Normal rank of a p-group; 1,Characteristic rank of a p-group; 1,Nilpotency class; 1,Derived length; 1,Fitting length; 1,Number of conjugacy classes; 2,Number of subgroups; 2,Number of conjugacy classes of subgroups; 2,Number of equivalence classes under real conjugacy; 2,Number of equivalence classes under rational conjugacy; 2"
"Cyclic group:Z20","20; 2","Order of a group; 20,Exponent of a group; 20,Frattini length; 2,Fitting length; 1,Derived length; 1"
"Cyclic group:Z27","27; 1","Underlying prime of p-group; 3,Order of a group; 27,Prime-base logarithm of order; 3,Max-length of a group; 3,Chief length; 3,Composition length; 3,Exponent of a group; 27,Prime-base logarithm of exponent; 3,Frattini length; 3,Minimum size of generating set; 1,Subgroup rank of a group; 1,Rank of a p-group; 1,Normal rank of a p-group; 1,Characteristic rank of a p-group; 1,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Cyclic group:Z3","3; 1","Underlying prime of p-group; 3,Order of a group; 3,Prime-base logarithm of order; 1,Max-length of a group; 1,Chief length; 1,Composition length; 1,Exponent of a group; 3,Prime-base logarithm of exponent; 1,Frattini length; 1,Minimum size of generating set; 1,Subgroup rank of a group; 1,Rank of a p-group; 1,Normal rank of a p-group; 1,Characteristic rank of a p-group; 1,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Cyclic group:Z32","32; 1","Underlying prime of p-group; 2,Order of a group; 32,Prime-base logarithm of order; 5,Max-length of a group; 5,Chief length; 5,Composition length; 5,Exponent of a group; 32,Prime-base logarithm of exponent; 5,Frattini length; 5,Minimum size of generating set; 1,Subgroup rank of a group; 1,Rank of a p-group; 1,Normal rank of a p-group; 1,Characteristic rank of a p-group; 1,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Cyclic group:Z4","4; 1","Underlying prime of p-group; 2,Order of a group; 4,Prime-base logarithm of order; 2,Max-length of a group; 2,Chief length; 2,Composition length; 2,Exponent of a group; 4,Prime-base logarithm of exponent; 2,Frattini length; 2,Minimum size of generating set; 1,Subgroup rank of a group; 1,Rank of a p-group; 1,Normal rank of a p-group; 1,Characteristic rank of a p-group; 1,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Cyclic group:Z5","5; 1","Order of a group; 5,Exponent of a group; 5,Derived length; 1,Frattini length; 1,Fitting length; 1"
"Cyclic group:Z64","64; 1","Underlying prime of p-group; 2,Order of a group; 64,Prime-base logarithm of order; 6,Max-length of a group; 6,Chief length; 6,Composition length; 6,Exponent of a group; 64,Prime-base logarithm of exponent; 6,Frattini length; 6,Minimum size of generating set; 1,Subgroup rank of a group; 1,Rank of a p-group; 1,Normal rank of a p-group; 1,Characteristic rank of a p-group; 1,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Cyclic group:Z7","7; 1","Order of a group; 7,Exponent of a group; 7,Frattini length; 1,Fitting length; 1,Subgroup rank of a group; 1,Rank of a p-group; 1"
"Cyclic group:Z8","8; 1","Underlying prime of p-group; 2,Order of a group; 8,Prime-base logarithm of order; 3,Max-length of a group; 3,Chief length; 3,Composition length; 3,Exponent of a group; 8,Prime-base logarithm of exponent; 3,Frattini length; 3,Minimum size of generating set; 1,Subgroup rank of a group; 1,Rank of a p-group; 1,Normal rank of a p-group; 1,Characteristic rank of a p-group; 1,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Cyclic group:Z81","81; 1","Underlying prime of p-group; 3,Order of a group; 81,Prime-base logarithm of order; 4,Max-length of a group; 4,Chief length; 4,Composition length; 4,Exponent of a group; 81,Prime-base logarithm of exponent; 4,Frattini length; 4,Minimum size of generating set; 1,Subgroup rank of a group; 1,Rank of a p-group; 1,Normal rank of a p-group; 1,Characteristic rank of a p-group; 1,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Cyclic group:Z9","9; 1","Order of a group; 9,Exponent of a group; 9,Derived length; 1,Frattini length; 2,Fitting length; 1,Subgroup rank of a group; 1"
"Direct product of E8 and Z4","32; 45","Underlying prime of p-group; 2,Order of a group; 32,Prime-base logarithm of order; 5,Max-length of a group; 5,Chief length; 5,Composition length; 5,Exponent of a group; 4,Prime-base logarithm of exponent; 2,Frattini length; 2,Minimum size of generating set; 4,Subgroup rank of a group; 4,Rank of a p-group; 4,Normal rank of a p-group; 4,Characteristic rank of a p-group; 4,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Direct product of Z16 and V4","64; 183","Underlying prime of p-group; 2,Order of a group; 64,Prime-base logarithm of order; 6,Max-length of a group; 6,Chief length; 6,Composition length; 6,Exponent of a group; 16,Prime-base logarithm of exponent; 4,Frattini length; 4,Minimum size of generating set; 3,Subgroup rank of a group; 3,Rank of a p-group; 3,Normal rank of a p-group; 3,Characteristic rank of a p-group; 3,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Direct product of Z27 and Z3","81; 5","Underlying prime of p-group; 3,Order of a group; 81,Prime-base logarithm of order; 4,Max-length of a group; 4,Chief length; 4,Composition length; 4,Exponent of a group; 27,Prime-base logarithm of exponent; 3,Frattini length; 3,Minimum size of generating set; 2,Subgroup rank of a group; 2,Rank of a p-group; 2,Normal rank of a p-group; 2,Characteristic rank of a p-group; 2,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Direct product of Z4 and Z2","8; 2","Underlying prime of p-group; 2,Order of a group; 8,Prime-base logarithm of order; 3,Exponent of a group; 4,Prime-base logarithm of exponent; 2,Frattini length; 2,Minimum size of generating set; 2,Subgroup rank of a group; 2,Rank of a p-group; 2,Normal rank of a p-group; 2,Characteristic rank of a p-group; 2,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Direct product of Z4 and Z4","16; 2","Underlying prime of p-group; 2,Order of a group; 16,Prime-base logarithm of order; 4,Max-length of a group; 4,Chief length; 4,Composition length; 4,Exponent of a group; 4,Prime-base logarithm of exponent; 2,Frattini length; 2,Minimum size of generating set; 2,Subgroup rank of a group; 2,Rank of a p-group; 2,Normal rank of a p-group; 2,Characteristic rank of a p-group; 2,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Direct product of Z8 and V4","32; 36","Underlying prime of p-group; 2,Order of a group; 32,Prime-base logarithm of order; 5,Max-length of a group; 5,Chief length; 5,Composition length; 5,Exponent of a group; 8,Prime-base logarithm of exponent; 3,Frattini length; 3,Minimum size of generating set; 3,Subgroup rank of a group; 3,Rank of a p-group; 3,Normal rank of a p-group; 3,Characteristic rank of a p-group; 3,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Direct product of Z8 and Z2","16; 5","Underlying prime of p-group; 2,Order of a group; 16,Prime-base logarithm of order; 4,Max-length of a group; 4,Chief length; 4,Composition length; 4,Exponent of a group; 8,Prime-base logarithm of exponent; 3,Frattini length; 3,Minimum size of generating set; 2,Subgroup rank of a group; 2,Rank of a p-group; 2,Normal rank of a p-group; 2,Characteristic rank of a p-group; 2,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Direct product of Z8 and Z4","32; 3","Underlying prime of p-group; 2,Order of a group; 32,Prime-base logarithm of order; 5,Max-length of a group; 5,Chief length; 5,Composition length; 5,Exponent of a group; 8,Prime-base logarithm of exponent; 3,Frattini length; 3,Minimum size of generating set; 2,Subgroup rank of a group; 2,Rank of a p-group; 2,Normal rank of a p-group; 2,Characteristic rank of a p-group; 2,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Direct product of Z8 and Z8","64; 2","Underlying prime of p-group; 2,Order of a group; 64,Prime-base logarithm of order; 6,Max-length of a group; 6,Chief length; 6,Composition length; 6,Exponent of a group; 8,Prime-base logarithm of exponent; 3,Frattini length; 3,Minimum size of generating set; 2,Subgroup rank of a group; 2,Rank of a p-group; 2,Normal rank of a p-group; 2,Characteristic rank of a p-group; 2,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Direct product of Z9 and E9","81; 11","Underlying prime of p-group; 3,Order of a group; 81,Prime-base logarithm of order; 4,Max-length of a group; 4,Chief length; 4,Composition length; 4,Exponent of a group; 9,Prime-base logarithm of exponent; 2,Frattini length; 2,Minimum size of generating set; 3,Subgroup rank of a group; 3,Rank of a p-group; 3,Normal rank of a p-group; 3,Characteristic rank of a p-group; 3,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Direct product of Z9 and Z9","81; 2","Underlying prime of p-group; 3,Order of a group; 81,Prime-base logarithm of order; 4,Exponent of a group; 9,Prime-base logarithm of exponent; 2,Frattini length; 2,Minimum size of generating set; 2,Subgroup rank of a group; 2,Rank of a p-group; 2,Normal rank of a p-group; 2,Characteristic rank of a p-group; 2,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Elementary abelian group:E64","64; 267","Underlying prime of p-group; 2,Order of a group; 64,Prime-base logarithm of order; 6,Max-length of a group; 6,Chief length; 6,Composition length; 6,Exponent of a group; 2,Prime-base logarithm of exponent; 1,Frattini length; 1,Minimum size of generating set; 6,Subgroup rank of a group; 6,Rank of a p-group; 6,Normal rank of a p-group; 6,Characteristic rank of a p-group; 6,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Elementary abelian group:E8","8; 5","Underlying prime of p-group; 2,Order of a group; 8,Prime-base logarithm of order; 3,Max-length of a group; 3,Chief length; 3,Composition length; 3,Exponent of a group; 2,Prime-base logarithm of exponent; 1,Frattini length; 1,Minimum size of generating set; 3,Subgroup rank of a group; 3,Rank of a p-group; 3,Normal rank of a p-group; 3,Characteristic rank of a p-group; 3,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Elementary abelian group:E81","81; 15","Underlying prime of p-group; 3,Order of a group; 81,Prime-base logarithm of order; 4,Max-length of a group; 4,Chief length; 4,Composition length; 4,Exponent of a group; 3,Prime-base logarithm of exponent; 1,Frattini length; 1,Minimum size of generating set; 4,Subgroup rank of a group; 4,Rank of a p-group; 4,Normal rank of a p-group; 4,Characteristic rank of a p-group; 4,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Elementary abelian group:E9","9; 2","Underlying prime of p-group; 3,Order of a group; 9,Prime-base logarithm of order; 2,Max-length of a group; 2,Chief length; 2,Composition length; 2,Exponent of a group; 3,Prime-base logarithm of exponent; 1,Frattini length; 1,Minimum size of generating set; 2,Subgroup rank of a group; 2,Rank of a p-group; 2,Normal rank of a p-group; 2,Characteristic rank of a p-group; 2,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Free abelian group of rank two",,
"Group of integers",,"Derived length; 1,Nilpotency class; 1,Fitting length; 1,Frattini length; 1,Subgroup rank of a group; 1"
"Group of rational numbers modulo integers",,
"Group of rational numbers with square-free denominators",,
"Klein four-group","4; 2","Underlying prime of p-group; 2,Order of a group; 4,Prime-base logarithm of order; 2,Max-length of a group; 2,Chief length; 2,Composition length; 2,Exponent of a group; 2,Prime-base logarithm of exponent; 1,Frattini length; 1,Minimum size of generating set; 2,Subgroup rank of a group; 2,Rank of a p-group; 2,Normal rank of a p-group; 2,Characteristic rank of a p-group; 2,Nilpotency class; 1,Derived length; 1,Fitting length; 1"
"Profinite completion of the integers",,
"Quasicyclic group",,
"Quotient of profinite completion of the integers by the integers",,
"Trivial group","1; 1","Order of a group; 1,Exponent of a group; 1,Nilpotency class; 0,Derived length; 0,Frattini length; 0,Minimum size of generating set; 0,Subgroup rank of a group; 0"