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