#17152 ⟨a, b | aaaa=1, babbbb=a⟩

Properties

Presentation has sum-of-sides 11
Finite non-Abelian group with 1020 elements
Inverses of generators:
- a^-1 = a³
- b^-1 = b²⁵⁴

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, b⁵¹, b¹⁰², b¹⁵³, b²⁰⁴
Order of generators:
- a: order 4
- b: order 255

Histogram:

order 2	17 elements	a², a²b¹⁵, a²b³⁰, a²b⁴⁵, a²b⁶⁰, ...
order 3	2 elements	b⁸⁵, b¹⁷⁰
order 4	102 elements	a, a³, ab⁵, a³b⁵, ab¹⁰, ...
order 5	4 elements	b⁵¹, b¹⁰², b¹⁵³, b²⁰⁴
order 6	34 elements	a²b⁵, a²b¹⁰, a²b²⁰, a²b²⁵, a²b³⁵, ...
order 10	68 elements	a²b³, a²b⁶, a²b⁹, a²b¹², a²b¹⁸, ...
order 15	8 elements	b¹⁷, b³⁴, b⁶⁸, b¹¹⁹, b¹³⁶, ...
order 17	16 elements	b¹⁵, b³⁰, b⁴⁵, b⁶⁰, b⁷⁵, ...
order 20	408 elements	ab, ab², a³b, ab³, a³b², ...
order 30	136 elements	a²b, a²b², a²b⁴, a²b⁷, a²b⁸, ...
order 51	32 elements	b⁵, b¹⁰, b²⁰, b²⁵, b³⁵, ...
order 85	64 elements	b³, b⁶, b⁹, b¹², b¹⁸, ...
order 255	128 elements	b, b², b⁴, b⁷, b⁸, ...

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

b²⁵⁵ ⇒ 1
ba ⇒ ab²⁵¹
a⁴ ⇒ 1

# ab:aaaa=1,babbbb=a b/a
bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=1
ba=abbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb
aaaa=1

Up:	Monoids with two generators and two relations
Prev:	#17151 ⟨a, b \| aaaa=1, babbab=b⟩
Next:	#17153 ⟨a, b \| aaaa=1, babbbb=b⟩

#17152 ⟨a, b | aaaa=1, babbbb=a⟩

Properties

Elements

Complete rewriting system

Right Cayley graph

Left Cayley graph