#6935 ⟨a, b | bb=aa, aaabab=1⟩

Properties

Presentation has sum-of-sides 10
Finite non-Abelian group with 24 elements
Inverses of generators:
- a^-1 = a¹¹
- b^-1 = ba¹⁰

Element profile

6 element center:
- 1, a², a⁴, a⁶, a⁸, a¹⁰

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

a¹² ⇒ 1
ab ⇒ ba⁷
b² ⇒ a²

# ab:bb=aa,aaabab=1 a/b
aaaaaaaaaaaa=1
ab=baaaaaaa
bb=aa

Cayley table

	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹	ba¹⁰	ba¹¹
1	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹	ba¹⁰	ba¹¹
a	a	a²	ba⁷	a³	ba⁸	a⁴	ba⁹	a⁵	ba¹⁰	a⁶	ba¹¹	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a¹¹	ba⁴	1	ba⁵	ba⁶
b	b	ba	a²	ba²	a³	ba³	a⁴	ba⁴	a⁵	ba⁵	a⁶	ba⁶	a⁷	ba⁷	a⁸	ba⁸	a⁹	ba⁹	a¹⁰	ba¹⁰	a¹¹	ba¹¹	1	a
a²	a²	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹	ba¹⁰	1	ba¹¹	a	b	ba
ba	ba	ba²	a⁹	ba³	a¹⁰	ba⁴	a¹¹	ba⁵	1	ba⁶	a	ba⁷	a²	ba⁸	a³	ba⁹	a⁴	ba¹⁰	a⁵	ba¹¹	a⁶	b	a⁷	a⁸
a³	a³	a⁴	ba⁹	a⁵	ba¹⁰	a⁶	ba¹¹	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a¹¹	ba⁴	1	ba⁵	a	ba⁶	a²	ba⁷	ba⁸
ba²	ba²	ba³	a⁴	ba⁴	a⁵	ba⁵	a⁶	ba⁶	a⁷	ba⁷	a⁸	ba⁸	a⁹	ba⁹	a¹⁰	ba¹⁰	a¹¹	ba¹¹	1	b	a	ba	a²	a³
a⁴	a⁴	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹	ba¹⁰	1	ba¹¹	a	b	a²	ba	a³	ba²	ba³
ba³	ba³	ba⁴	a¹¹	ba⁵	1	ba⁶	a	ba⁷	a²	ba⁸	a³	ba⁹	a⁴	ba¹⁰	a⁵	ba¹¹	a⁶	b	a⁷	ba	a⁸	ba²	a⁹	a¹⁰
a⁵	a⁵	a⁶	ba¹¹	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a¹¹	ba⁴	1	ba⁵	a	ba⁶	a²	ba⁷	a³	ba⁸	a⁴	ba⁹	ba¹⁰
ba⁴	ba⁴	ba⁵	a⁶	ba⁶	a⁷	ba⁷	a⁸	ba⁸	a⁹	ba⁹	a¹⁰	ba¹⁰	a¹¹	ba¹¹	1	b	a	ba	a²	ba²	a³	ba³	a⁴	a⁵
a⁶	a⁶	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹	ba¹⁰	1	ba¹¹	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	ba⁵
ba⁵	ba⁵	ba⁶	a	ba⁷	a²	ba⁸	a³	ba⁹	a⁴	ba¹⁰	a⁵	ba¹¹	a⁶	b	a⁷	ba	a⁸	ba²	a⁹	ba³	a¹⁰	ba⁴	a¹¹	1
a⁷	a⁷	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a¹¹	ba⁴	1	ba⁵	a	ba⁶	a²	ba⁷	a³	ba⁸	a⁴	ba⁹	a⁵	ba¹⁰	a⁶	ba¹¹	b
ba⁶	ba⁶	ba⁷	a⁸	ba⁸	a⁹	ba⁹	a¹⁰	ba¹⁰	a¹¹	ba¹¹	1	b	a	ba	a²	ba²	a³	ba³	a⁴	ba⁴	a⁵	ba⁵	a⁶	a⁷
a⁸	a⁸	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹	ba¹⁰	1	ba¹¹	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	ba⁷
ba⁷	ba⁷	ba⁸	a³	ba⁹	a⁴	ba¹⁰	a⁵	ba¹¹	a⁶	b	a⁷	ba	a⁸	ba²	a⁹	ba³	a¹⁰	ba⁴	a¹¹	ba⁵	1	ba⁶	a	a²
a⁹	a⁹	a¹⁰	ba³	a¹¹	ba⁴	1	ba⁵	a	ba⁶	a²	ba⁷	a³	ba⁸	a⁴	ba⁹	a⁵	ba¹⁰	a⁶	ba¹¹	a⁷	b	a⁸	ba	ba²
ba⁸	ba⁸	ba⁹	a¹⁰	ba¹⁰	a¹¹	ba¹¹	1	b	a	ba	a²	ba²	a³	ba³	a⁴	ba⁴	a⁵	ba⁵	a⁶	ba⁶	a⁷	ba⁷	a⁸	a⁹
a¹⁰	a¹⁰	a¹¹	ba¹⁰	1	ba¹¹	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	ba⁹
ba⁹	ba⁹	ba¹⁰	a⁵	ba¹¹	a⁶	b	a⁷	ba	a⁸	ba²	a⁹	ba³	a¹⁰	ba⁴	a¹¹	ba⁵	1	ba⁶	a	ba⁷	a²	ba⁸	a³	a⁴
a¹¹	a¹¹	1	ba⁵	a	ba⁶	a²	ba⁷	a³	ba⁸	a⁴	ba⁹	a⁵	ba¹⁰	a⁶	ba¹¹	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	ba⁴
ba¹⁰	ba¹⁰	ba¹¹	1	b	a	ba	a²	ba²	a³	ba³	a⁴	ba⁴	a⁵	ba⁵	a⁶	ba⁶	a⁷	ba⁷	a⁸	ba⁸	a⁹	ba⁹	a¹⁰	a¹¹
ba¹¹	ba¹¹	b	a⁷	ba	a⁸	ba²	a⁹	ba³	a¹⁰	ba⁴	a¹¹	ba⁵	1	ba⁶	a	ba⁷	a²	ba⁸	a³	ba⁹	a⁴	ba¹⁰	a⁵	a⁶

Right Cayley graph

Left Cayley graph

Others with same cardinality

6 unique, 75 total

Σ	#	Presentation	Description	Related
8	425	⟨a, b \| aab=ba, bbb=1⟩	Finite non-commutative monoid with 24 elements	18 iso, 3 anti-iso
10	5009	⟨a, b \| aba=bb, aabbb=1⟩	Finite non-Abelian group with 24 elements	30 iso
11	11515	⟨a, b \| ababa=b, abbaa=1⟩	Finite non-Abelian group with 24 elements	10 iso
11	13372	⟨a, b \| aaaab=1, bbbbbb=1⟩	Isomorphic to ℤ₂₄	8 iso
11	19638	⟨a, b \| aba=a, aaaab=bb⟩	Finite non-commutative monoid with 24 elements
11	20252	⟨a, b \| aba=b, aaaa=bab⟩	Finite non-commutative monoid with 24 elements

Other isomorphic instances

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

2 total

Σ	#	Presentation	Mapping
10	6939	⟨a, b \| bb=aa, aababa=1⟩	φ(a) = a, φ(b) = b
10	6944	⟨a, b \| bb=aa, abaaab=1⟩	φ(a) = a, φ(b) = b

Up:	Monoids with two generators and two relations
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Next:	#6950 ⟨a, b \| ab=aa, aaaaa=a⟩