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