Conway group:Co1

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Definition

The group, denoted $\operatorname {Co} _{1}$, is defined as the inner automorphism group of $\operatorname {Co} _{0}$ (see Conway group:Co0), which in turn is defined as the automorphism group of the Leech lattice.

Note that $\operatorname {Co} _{0}$ has a center which is isomorphic to cyclic group:Z2, with the non-identity element corresponding to the automorphism given by sending every vector to its negative. Thus, $\operatorname {Co} _{1}$ is the quotient group of $\operatorname {Co} _{0}$ by a subgroup of order two.

The group is a finite simple non-abelian group, and in fact, it is one of the 26 sporadic simple groups. It is also one of the three Conway groups.

Arithmetic functions

Basic arithmetic functions

Function	Value	Similar groups	Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set)	4157776806543360000	groups with same order

Arithmetic functions of a counting nature

Function	Value	Similar groups	Explanation for function value
number of conjugacy classes	101	groups with same order and number of conjugacy classes | groups with same number of conjugacy classes	equals number of irreducible representations, see also linear representation theory of Conway group:Co1.

Linear representation theory

Further information: linear representation theory of Conway group:Co1

External links

• Co1 on the Atlas of Finite Groups