#1470 ⟨a, b | aaa=bb, abab=1⟩

Properties

Presentation has sum-of-sides 9
Finite non-Abelian group with 30 elements
Inverses of generators:
- a^-1 = a¹⁴
- b^-1 = ba¹²

Element profile

5 element center:
- 1, a³, a⁶, a⁹, a¹²

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

a¹⁵ ⇒ 1
ab ⇒ ba¹¹
b² ⇒ a³

# ab:aaa=bb,abab=1 a/b
aaaaaaaaaaaaaaa=1
ab=baaaaaaaaaaa
bb=aaa

Right Cayley graph

Left Cayley graph

Others with same cardinality

8 unique, 50 total

Σ	#	Presentation	Description	Related
9	1907	⟨a, b \| aab=a, bbbbb=1⟩	Finite non-commutative monoid with 30 elements	11 iso, 4 anti-iso
9	2443	⟨a, b \| aaa=1, abbbb=b⟩	Finite non-commutative monoid with 30 elements	14 iso, 5 anti-iso
10	6560	⟨a, b \| aaa=a, bbbb=ab⟩	Finite non-commutative monoid with 30 elements
10	7013	⟨a, b \| ab=aa, bbbbb=b⟩	Finite non-commutative monoid with 30 elements
11	11058	⟨a, b \| abba=bbb, baba=1⟩	Finite non-Abelian group with 30 elements	8 iso
11	13115	⟨a, b \| bbb=aaa, aaba=b⟩	Finite non-commutative monoid with 30 elements
11	15979	⟨a, b \| aaa=aa, bbbb=ab⟩	Finite non-commutative monoid with 30 elements
11	20847	⟨a, b \| ab=aa, bbbbb=bb⟩	Finite non-commutative monoid with 30 elements

Other isomorphic instances

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

73 total

Σ	#	Presentation	Mapping
9	3241	⟨a, b \| aa=1, ababbbb=1⟩	φ(a) = ba, φ(b) = a
9	3249	⟨a, b \| aa=1, abbbbab=1⟩	φ(a) = ba, φ(b) = a
9	3262	⟨a, b \| aa=1, bababbb=1⟩	φ(a) = ba, φ(b) = a
9	3268	⟨a, b \| aa=1, bbababb=1⟩	φ(a) = ba, φ(b) = a
9	3400	⟨a, b \| aa=1, babbbb=a⟩	φ(a) = ba, φ(b) = a
10	9458	⟨a, b \| aa=1, abababbb=1⟩	φ(a) = ba, φ(b) = b
10	9462	⟨a, b \| aa=1, ababbbab=1⟩	φ(a) = ba, φ(b) = b
10	9477	⟨a, b \| aa=1, abbbabab=1⟩	φ(a) = ba, φ(b) = b
10	9502	⟨a, b \| aa=1, babababb=1⟩	φ(a) = ba, φ(b) = b
10	9780	⟨a, b \| aa=1, bababbb=a⟩	φ(a) = ba, φ(b) = b
10	9784	⟨a, b \| aa=1, babbbab=a⟩	φ(a) = ba, φ(b) = b
11	10641	⟨a, b \| aaaa=abb, abab=1⟩	φ(a) = a, φ(b) = b
11	10646	⟨a, b \| aaaa=abb, baba=1⟩	φ(a) = a, φ(b) = b
11	10771	⟨a, b \| aaab=bbb, abab=1⟩	φ(a) = a, φ(b) = b
11	10776	⟨a, b \| aaab=bbb, baba=1⟩	φ(a) = a, φ(b) = b
11	12847	⟨a, b \| bab=aaa, aabbb=1⟩	φ(a) = b, φ(b) = aaaa
11	12852	⟨a, b \| bab=aaa, abbba=1⟩	φ(a) = b, φ(b) = aaaa
11	12855	⟨a, b \| bab=aaa, baabb=1⟩	φ(a) = b, φ(b) = aaaa
11	15387	⟨a, b \| aba=bb, abbabb=1⟩	φ(a) = b, φ(b) = aaaaaaaa
11	15396	⟨a, b \| aba=bb, babbab=1⟩	φ(a) = b, φ(b) = aaaaaaaa
11	17485	⟨a, b \| abab=1, aaaaba=b⟩	φ(a) = a, φ(b) = b
11	17509	⟨a, b \| abab=1, abaaaa=b⟩	φ(a) = a, φ(b) = b
11	18019	⟨a, b \| abab=1, aaaba=bb⟩	φ(a) = b, φ(b) = a
11	18039	⟨a, b \| abab=1, abaaa=bb⟩	φ(a) = b, φ(b) = a
11	18185	⟨a, b \| aaa=a, ababbbb=1⟩	φ(a) = ba, φ(b) = a
11	18193	⟨a, b \| aaa=a, abbbbab=1⟩	φ(a) = ba, φ(b) = a
11	18206	⟨a, b \| aaa=a, bababbb=1⟩	φ(a) = ba, φ(b) = a
11	18212	⟨a, b \| aaa=a, bbababb=1⟩	φ(a) = ba, φ(b) = a
11	18463	⟨a, b \| aab=b, ababbbb=1⟩	φ(a) = ba, φ(b) = a
11	18477	⟨a, b \| aab=b, abbbbab=1⟩	φ(a) = ba, φ(b) = a
11	18503	⟨a, b \| aab=b, bababbb=1⟩	φ(a) = ba, φ(b) = a
11	18510	⟨a, b \| aab=b, babbbba=1⟩	φ(a) = ba, φ(b) = a
11	18523	⟨a, b \| aab=b, bbababb=1⟩	φ(a) = ba, φ(b) = a
11	18533	⟨a, b \| aab=b, bbbabab=1⟩	φ(a) = ba, φ(b) = a
11	18538	⟨a, b \| aab=b, bbbbaba=1⟩	φ(a) = ba, φ(b) = a
11	25958	⟨a, b \| aa=1, aaababbbb=1⟩	φ(a) = ba, φ(b) = a
11	25971	⟨a, b \| aa=1, aaabbbbab=1⟩	φ(a) = ba, φ(b) = a
11	25994	⟨a, b \| aa=1, aabababbb=1⟩	φ(a) = ba, φ(b) = a
11	26000	⟨a, b \| aa=1, aababbbba=1⟩	φ(a) = ba, φ(b) = a
11	26010	⟨a, b \| aa=1, aabbababb=1⟩	φ(a) = ba, φ(b) = a
11	26018	⟨a, b \| aa=1, aabbbabab=1⟩	φ(a) = ba, φ(b) = a
11	26022	⟨a, b \| aa=1, aabbbbaba=1⟩	φ(a) = ba, φ(b) = a
11	26039	⟨a, b \| aa=1, abaaabbbb=1⟩	φ(a) = ba, φ(b) = a
11	26049	⟨a, b \| aa=1, abaabbbab=1⟩	φ(a) = ba, φ(b) = a
11	26056	⟨a, b \| aa=1, ababaabbb=1⟩	φ(a) = ba, φ(b) = a
11	26061	⟨a, b \| aa=1, abababbba=1⟩	φ(a) = ba, φ(b) = a
11	26064	⟨a, b \| aa=1, ababbaabb=1⟩	φ(a) = ba, φ(b) = a
11	26068	⟨a, b \| aa=1, ababbbaab=1⟩	φ(a) = ba, φ(b) = a
11	26081	⟨a, b \| aa=1, abbaabbab=1⟩	φ(a) = ba, φ(b) = a
11	26087	⟨a, b \| aa=1, abbababba=1⟩	φ(a) = ba, φ(b) = a
11	26096	⟨a, b \| aa=1, abbbaabab=1⟩	φ(a) = ba, φ(b) = a
11	26103	⟨a, b \| aa=1, abbbbaaab=1⟩	φ(a) = ba, φ(b) = a
11	26123	⟨a, b \| aa=1, baaababbb=1⟩	φ(a) = ba, φ(b) = a
11	26132	⟨a, b \| aa=1, baabababb=1⟩	φ(a) = ba, φ(b) = a
11	26136	⟨a, b \| aa=1, baabbabab=1⟩	φ(a) = ba, φ(b) = a
11	26144	⟨a, b \| aa=1, babaaabbb=1⟩	φ(a) = ba, φ(b) = a
11	26148	⟨a, b \| aa=1, bababaabb=1⟩	φ(a) = ba, φ(b) = a
11	26166	⟨a, b \| aa=1, bbaaababb=1⟩	φ(a) = ba, φ(b) = a
11	26528	⟨a, b \| aa=1, aababbbb=a⟩	φ(a) = ba, φ(b) = a
11	26550	⟨a, b \| aa=1, aabbbbab=a⟩	φ(a) = ba, φ(b) = a
11	26588	⟨a, b \| aa=1, abababbb=a⟩	φ(a) = ba, φ(b) = a
11	26598	⟨a, b \| aa=1, ababbbba=a⟩	φ(a) = ba, φ(b) = a
11	26614	⟨a, b \| aa=1, abbababb=a⟩	φ(a) = ba, φ(b) = a
11	26626	⟨a, b \| aa=1, abbbabab=a⟩	φ(a) = ba, φ(b) = a
11	26654	⟨a, b \| aa=1, baaabbbb=a⟩	φ(a) = ba, φ(b) = a
11	26666	⟨a, b \| aa=1, baabbbab=a⟩	φ(a) = ba, φ(b) = a
11	26674	⟨a, b \| aa=1, babaabbb=a⟩	φ(a) = ba, φ(b) = a
11	26682	⟨a, b \| aa=1, babbaabb=a⟩	φ(a) = ba, φ(b) = a
11	27126	⟨a, b \| aa=1, ababbbb=aa⟩	φ(a) = ba, φ(b) = a
11	27157	⟨a, b \| aa=1, abbbbab=aa⟩	φ(a) = ba, φ(b) = a
11	27205	⟨a, b \| aa=1, bababbb=aa⟩	φ(a) = ba, φ(b) = a
11	27227	⟨a, b \| aa=1, bbababb=aa⟩	φ(a) = ba, φ(b) = a
11	27740	⟨a, b \| aa=1, babbbb=aaa⟩	φ(a) = ba, φ(b) = a

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Next:	#1472 ⟨a, b \| aaa=bb, abbb=1⟩