In this paper, we define the Tribonacci-type balancing numbers via a Diophantine equation with a complex variable and then give their miscellaneous properties. Also, we study the Tribonacci-type balancing sequence modulo $m$ and then obtain some interesting results concerning the periods of the Tribonacci-type balancing sequences for any $m$. Furthermore, we produce the cyclic groups using the multiplicative orders of the generating matrices of the Tribonacci-type balancing numbers when read modulo $m$. Then give the connections between the periods of the Tribonacci-type balancing sequences modulo $m$ and the orders of the cyclic groups produced. Finally, we expand the Tribonacci-type balancing sequences to groups and give the definition of the Tribonacci-type balancing sequences in the $3$-generator groups and also, investigate these sequences in the non-abelian finite groups in detail. In addition, we obtain the periods of the Tribonacci-type balancing sequences in the polyhedral groups $(2,2,n)$, $(2,n,2)$, $(n,2,2)$, $(2,3,3)$, $(2,3,4)$, $(2,3,5)$.