# 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