Conway group:Co0

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

This group, denoted ${\mathrm{C}}_{\mathrm{o}}$ or $2\cdot {\mathrm{C}}_{\mathrm{o}}$, 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.

Its center is cyclic group:Z2 and the inner automorphism group is 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: $2^{22}\cdot 3^9\cdot 5^4\cdot 7^1\cdot 11\cdot 13\cdot 23$.

Arithmetic functions of a counting nature

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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.