#10664 ⟨a, b | aaaa=bbb, abab=1⟩

Properties
Finite structure
Right Cayley graph
Left Cayley graph
Rewriting system
Same cardinality

Properties

Sum of relation sides is 11
Finite non-Abelian group with 336 elements
Inverses of generators:
- a^-1 = a⁵⁵
- b^-1 = aba

Finite structure

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

14 element center:
- 1, a⁴, a⁸, a¹², a¹⁶, a²⁰, a²⁴, a²⁸, a³², a³⁶, a⁴⁰, a⁴⁴, a⁴⁸, a⁵²
Order of generators:
- a: order 56
- b: order 42

Histogram:

order 2	13 elements	ab, ba, ba³ba⁶, ba³ba²⁰, a²⁸, ...
order 3	8 elements	a³ba¹⁹, a²ba²⁰, aba²¹, ba²², a³ba³³, ...
order 4	6 elements	a¹⁴, ba³ba¹³, ba³ba²⁷, a⁴², ba³ba⁴¹, ...
order 6	8 elements	a³ba⁵, a²ba⁶, aba⁷, ba⁸, a³ba⁴⁷, ...
order 7	6 elements	a⁸, a¹⁶, a²⁴, a³², a⁴⁰, ...
order 8	12 elements	a⁷, a³ba¹², a²ba¹³, aba¹⁴, ba¹⁵, ...
order 14	78 elements	a⁴, ba³b, a³ba², a²ba³, aba⁴, ...
order 21	48 elements	a³ba, a²ba², aba³, ba⁴, a³ba³, ...
order 28	36 elements	a², a⁶, ba³ba, (ba³)², a¹⁰, ...
order 42	48 elements	b, a²b, aba, ba², a³ba⁷, ...
order 56	72 elements	a, a³, a³b, a²ba, aba², ...

Right Cayley graph

Left Cayley graph

Rewriting system

Format:	Pretty Plain
Word:
	Enter a word above to compute its normal form. Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Strategy:	Leftmost Rightmost
Result:	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.	a⁵⁶ ⇒ 1	[17]
2.	a⁴b ⇒ ba⁴	[3]
3.	b² ⇒ aba⁵	[4]
4.	bab ⇒ a⁵⁵	[18]
5.	ba²b ⇒ a³ba⁴⁷	[13]
6.	aba³b ⇒ ba³ba⁴³	[15]

# ab:aaaa=bbb,abab=1 a/b - -
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa=1
aaaab=baaaa
bb=abaaaaa
bab=aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
baab=aaabaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
abaaab=baaabaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

Same cardinality

1 unique, 1 total

Σ	#	Presentation	Properties
11	22271	⟨a, b \| aaa=1, ababbb=ba⟩	Noc 336

Up:	Monoid enumeration
Prev:	#10660 ⟨a, b \| aaaa=bab, bbbb=1⟩
Next:	#10683 ⟨a, b \| aaab=aaa, bbbb=1⟩

#10664 ⟨a, b | aaaa=bbb, abab=1⟩

Contents

Properties

Finite structure

Right Cayley graph

Left Cayley graph

Rewriting system

Same cardinality