#14128 ⟨a, b | aaba=b, bbbbbb=1⟩

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Properties
Elements
Right Cayley graph
Left Cayley graph
Rewriting system
Same cardinality

Properties

Presentation has sum-of-sides 11
Finite non-Abelian group with 378 elements
Inverses of generators:
- a^-1 = a⁶²
- b^-1 = b⁵

Elements

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

3 element center:
- 1, a²¹, a⁴²
Order of generators:
- a: order 63
- b: order 6

Histogram:

order 2	7 elements	b³, b³a⁹, b³a¹⁸, b³a²⁷, b³a³⁶, ...
order 3	44 elements	b², b⁴, b²a³, b⁴a³, b²a⁶, ...
order 6	56 elements	b, ba³, b⁵, b³a³, ba⁶, ...
order 7	6 elements	a⁹, a¹⁸, a²⁷, a³⁶, a⁴⁵, ...
order 9	90 elements	b²a, b²a², b⁴a, b²a⁴, b⁴a², ...
order 18	126 elements	ba, ba², b³a, ba⁴, b³a², ...
order 21	12 elements	a³, a⁶, a¹², a¹⁵, a²⁴, ...
order 63	36 elements	a, a², a⁴, a⁵, a⁸, ...

Right Cayley graph

Left Cayley graph

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
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a⁶³ ⇒ 1	[23]
2.	ab ⇒ ba³¹	[22]
3.	b⁶ ⇒ 1	[2]

# ab:aaba=b,bbbbbb=1 a/b
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa=1
ab=baaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
bbbbbb=1

Same cardinality

1 unique, 1 total

Σ	#	Presentation	Description	Related
11	21778	⟨a, b \| aaa=1, babbbbb=a⟩	Finite non-Abelian group with 378 elements