Chevalley group of type B:B3(3)

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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 is defined as $B_{3}(3)$, the Chevalley group of type B and parameter value 3 for field:F3. Explicitly, it is the kernel of the spinor norm map from special orthogonal group:SO(7,3).

It is an example of a simple non-abelian group that is not the only simple non-abelian group of its order. There are at most two finite simple groups of any order. The other group in this case is projective symplectic group:PSp(6,3), which in the Chevalley notation is $C_{3}(3)$. In general, $B_{n}(q)$ and $C_{n}(q)$ have the same order, but are isomorphic only if $n=2$ or $q$ is a prime power.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 4585351680#Arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	4585351680	groups with same order	As $B_{m}(q),m=3,q=3$: $q^{m^{2}}[\prod _{i=1}^{m}(q^{2i}-1)]/\operatorname {gcd} (2,q-1)$ which becomes $3^{9}(3^{2}-1)(3^{4}-1)(3^{6}-1)/\operatorname {gcd} (2,2)=(19683)(8)(80)(728)/2=4585351680$

GAP implementation

Description	Functions used
DerivedSubgroup(SO(7,3))	DerivedSubgroup, SO