# Conway group:Co0

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## Definition

This group, denoted $\operatorname{Co}_0$ or $2 \cdot \operatorname{Co}_1$, is defined in the following equivalent ways:

1. It is the automorphism group of the Leech lattice.
2. It is the Schur covering group (specifically, it is a double cover) of Conway group:Co1.

## Arithmetic functions

### Basic arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 8315553613086720000 groups with same order The order has factorization: $2^{22} \cdot 3^9 \cdot 5^4 \cdot 7^1 \cdot 11 \cdot 13 \cdot 23$.

## Group properties

Property Satisfied? Explanation
simple group No the center is nontrivial
perfect group Yes
quasisimple group Yes
solvable group No

## Linear representation theory

Further information: linear representation theory of Conway group:Co0

Item Value
degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$) too long to list, see Linear representation theory of Conway group:Co0#GAP implementation
number: 167, sum of squares: 8315553613086720000, maximum: 1021620600, quasirandom degree: 24

## GAP implementation

The group itself is too large to be constructed, stored and manipulated in GAP. However, information about its linear representation theory and character table is stored under the symbol "2.Co1" -- for more on this, see Linear representation theory of Conway group:Co0#GAP implementation.