#11251 ⟨a, b | aaaba=b, bbbbb=1⟩

Properties

Presentation has sum-of-sides 11
Finite non-Abelian group with 1220 elements
Inverses of generators:
- a^-1 = a²⁴³
- b^-1 = b⁴

Elements

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

4 element center:
- 1, a⁶¹, a¹²², a¹⁸³
Order of generators:
- a: order 244
- b: order 5

Histogram:

order 2	1 element	a¹²²
order 4	2 elements	a⁶¹, a¹⁸³
order 5	244 elements	b, b², b³, b⁴, ba⁴, ...
order 10	244 elements	ba², b²a², b³a², b⁴a², ba⁶, ...
order 20	488 elements	ba, b²a, ba³, b³a, b²a³, ...
order 61	60 elements	a⁴, a⁸, a¹², a¹⁶, a²⁰, ...
order 122	60 elements	a², a⁶, a¹⁰, a¹⁴, a¹⁸, ...
order 244	120 elements	a, a³, a⁵, a⁷, a⁹, ...

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(a) = 0; deg(b) = 1

a²⁴⁴ ⇒ 1
ab ⇒ ba⁸¹
b⁵ ⇒ 1

# ab:aaaba=b,bbbbb=1 a/b
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa=1
ab=baaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
bbbbb=1

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#11251 ⟨a, b | aaaba=b, bbbbb=1⟩

Properties

Elements

Complete rewriting system

Right Cayley graph

Left Cayley graph