#17156 ⟨a, b | aaaa=1, bbabbb=a⟩

Properties

Presentation has sum-of-sides 11
Finite non-Abelian group with 260 elements
Inverses of generators:
- a^-1 = a³
- b^-1 = b⁶⁴

Elements

Elements in the center commute with all other elements.
The order of x is the least n such that xⁿ = 1.

5 element center:
- 1, b¹³, b²⁶, b³⁹, b⁵²
Order of generators:
- a: order 4
- b: order 65

Histogram:

order 2	13 elements	a², a²b⁵, a²b¹⁰, a²b¹⁵, a²b²⁰, ...
order 4	26 elements	a, a³, ab⁵, a³b⁵, ab¹⁰, ...
order 5	4 elements	b¹³, b²⁶, b³⁹, b⁵²
order 10	52 elements	a²b, a²b², a²b³, a²b⁴, a²b⁶, ...
order 13	12 elements	b⁵, b¹⁰, b¹⁵, b²⁰, b²⁵, ...
order 20	104 elements	ab, ab², a³b, ab³, a³b², ...
order 65	48 elements	b, b², b³, b⁴, b⁶, ...

Complete rewriting system

Format:	Pretty Plain
Word to reduce:
	Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Reduction strategy:	Leftmost Rightmost
Path to normal form:	1 1

Reduction order:
- Left-to-right recursive path with deg(b) = 0; deg(a) = 1

b⁶⁵ ⇒ 1
ba ⇒ ab³¹
a⁴ ⇒ 1

# ab:aaaa=1,bbabbb=a b/a
bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=1
ba=abbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb
aaaa=1

Right Cayley graph

Left Cayley graph