#21200 ⟨a, b | aaa=1, ababbbbb=1⟩

Properties

Presentation has sum-of-sides 11
Finite non-Abelian group with 432 elements
Inverses of generators:
- a^-1 = bab⁵
- b^-1 = b⁷¹

Elements

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

18 element center:
- 1, b⁴, b⁸, b¹², b¹⁶, b²⁰, b²⁴, b²⁸, b³², b³⁶, b⁴⁰, b⁴⁴, b⁴⁸, b⁵², b⁵⁶, b⁶⁰, b⁶⁴, b⁶⁸
Order of generators:
- a: order 3
- b: order 72

Histogram:

order 2	1 element	b³⁶
order 3	26 elements	a, ab⁶, bab⁵, b²ab⁴, b³ab³, ...
order 4	18 elements	(ab³)², ab³ab¹², b¹⁸, ab²¹, bab²⁰, ...
order 6	26 elements	b¹², ab¹², bab¹¹, b²ab¹⁰, b³ab⁹, ...
order 8	12 elements	ab³, bab², b²ab, b³a, b⁹, ...
order 9	54 elements	b⁸, ab⁸, bab⁷, b²ab⁶, b³ab⁵, ...
order 12	36 elements	ab³a, b⁶, ab⁹, bab⁸, b²ab⁷, ...
order 18	54 elements	ab², bab, b²a, b⁴, ab⁴, ...
order 24	24 elements	b³, b¹⁵, ab¹⁵, bab¹⁴, b²ab¹³, ...
order 36	108 elements	ab, ba, b², ab³ab, ab⁵, ...
order 72	72 elements	b, b⁵, b⁷, ab⁷, bab⁶, ...

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

b⁷² ⇒ 1
b⁴a ⇒ ab⁴
a² ⇒ bab⁵
aba ⇒ b⁶⁷
ab²a ⇒ b³ab⁵⁹
bab³a ⇒ ab³ab⁵⁵

# ab:aaa=1,ababbbbb=1 b/a
bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=1
bbbba=abbbb
aa=babbbbb
aba=bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb
abba=bbbabbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb
babbba=abbbabbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

Right Cayley graph

Left Cayley graph

Other isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

3 total

Σ	#	Presentation	Mapping
11	21217	⟨a, b \| aaa=1, abbbbbab=1⟩	φ(a) = a, φ(b) = b
11	21240	⟨a, b \| aaa=1, bababbbb=1⟩	φ(a) = a, φ(b) = b
11	21250	⟨a, b \| aaa=1, bbababbb=1⟩	φ(a) = a, φ(b) = b

Up:	Monoids with two generators and two relations
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Next:	#21214 ⟨a, b \| aaa=1, abbbabbb=1⟩