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〈a, b | aba=b, aaabbbb=1〉

Monoid presentation of length 11

Properties

Finite non-Abelian group with 24 elements

Completion parameters

Reduction order: left-to-right a/b

Inverses of generators

a^-1 = a⁵
b^-1 = b³a³

Complete rewriting system

a⁶ ⇒ 1
ab ⇒ ba⁵
b⁴ ⇒ a³

Cayley table

	1	a	b	a²	ba	b²	a³	ba²	b²a	b³	a⁴	ba³	b²a²	b³a	a⁵	ba⁴	b²a³	b³a²	ba⁵	b²a⁴	b³a³	b²a⁵	b³a⁴	b³a⁵
1	1	a	b	a²	ba	b²	a³	ba²	b²a	b³	a⁴	ba³	b²a²	b³a	a⁵	ba⁴	b²a³	b³a²	ba⁵	b²a⁴	b³a³	b²a⁵	b³a⁴	b³a⁵
a	a	a²	ba⁵	a³	b	b²a	a⁴	ba	b²a²	b³a⁵	a⁵	ba²	b²a³	b³	1	ba³	b²a⁴	b³a	ba⁴	b²a⁵	b³a²	b²	b³a³	b³a⁴
b	b	ba	b²	ba²	b²a	b³	ba³	b²a²	b³a	a³	ba⁴	b²a³	b³a²	a⁴	ba⁵	b²a⁴	b³a³	a⁵	b²a⁵	b³a⁴	1	b³a⁵	a	a²
a²	a²	a³	ba⁴	a⁴	ba⁵	b²a²	a⁵	b	b²a³	b³a⁴	1	ba	b²a⁴	b³a⁵	a	ba²	b²a⁵	b³	ba³	b²	b³a	b²a	b³a²	b³a³
ba	ba	ba²	b²a⁵	ba³	b²	b³a	ba⁴	b²a	b³a²	a²	ba⁵	b²a²	b³a³	a³	b	b²a³	b³a⁴	a⁴	b²a⁴	b³a⁵	a⁵	b³	1	a
b²	b²	b²a	b³	b²a²	b³a	a³	b²a³	b³a²	a⁴	ba³	b²a⁴	b³a³	a⁵	ba⁴	b²a⁵	b³a⁴	1	ba⁵	b³a⁵	a	b	a²	ba	ba²
a³	a³	a⁴	ba³	a⁵	ba⁴	b²a³	1	ba⁵	b²a⁴	b³a³	a	b	b²a⁵	b³a⁴	a²	ba	b²	b³a⁵	ba²	b²a	b³	b²a²	b³a	b³a²
ba²	ba²	ba³	b²a⁴	ba⁴	b²a⁵	b³a²	ba⁵	b²	b³a³	a	b	b²a	b³a⁴	a²	ba	b²a²	b³a⁵	a³	b²a³	b³	a⁴	b³a	a⁵	1
b²a	b²a	b²a²	b³a⁵	b²a³	b³	a⁴	b²a⁴	b³a	a⁵	ba²	b²a⁵	b³a²	1	ba³	b²	b³a³	a	ba⁴	b³a⁴	a²	ba⁵	a³	b	ba
b³	b³	b³a	a³	b³a²	a⁴	ba³	b³a³	a⁵	ba⁴	b²a³	b³a⁴	1	ba⁵	b²a⁴	b³a⁵	a	b	b²a⁵	a²	ba	b²	ba²	b²a	b²a²
a⁴	a⁴	a⁵	ba²	1	ba³	b²a⁴	a	ba⁴	b²a⁵	b³a²	a²	ba⁵	b²	b³a³	a³	b	b²a	b³a⁴	ba	b²a²	b³a⁵	b²a³	b³	b³a
ba³	ba³	ba⁴	b²a³	ba⁵	b²a⁴	b³a³	b	b²a⁵	b³a⁴	1	ba	b²	b³a⁵	a	ba²	b²a	b³	a²	b²a²	b³a	a³	b³a²	a⁴	a⁵
b²a²	b²a²	b²a³	b³a⁴	b²a⁴	b³a⁵	a⁵	b²a⁵	b³	1	ba	b²	b³a	a	ba²	b²a	b³a²	a²	ba³	b³a³	a³	ba⁴	a⁴	ba⁵	b
b³a	b³a	b³a²	a²	b³a³	a³	ba⁴	b³a⁴	a⁴	ba⁵	b²a²	b³a⁵	a⁵	b	b²a³	b³	1	ba	b²a⁴	a	ba²	b²a⁵	ba³	b²	b²a
a⁵	a⁵	1	ba	a	ba²	b²a⁵	a²	ba³	b²	b³a	a³	ba⁴	b²a	b³a²	a⁴	ba⁵	b²a²	b³a³	b	b²a³	b³a⁴	b²a⁴	b³a⁵	b³
ba⁴	ba⁴	ba⁵	b²a²	b	b²a³	b³a⁴	ba	b²a⁴	b³a⁵	a⁵	ba²	b²a⁵	b³	1	ba³	b²	b³a	a	b²a	b³a²	a²	b³a³	a³	a⁴
b²a³	b²a³	b²a⁴	b³a³	b²a⁵	b³a⁴	1	b²	b³a⁵	a	b	b²a	b³	a²	ba	b²a²	b³a	a³	ba²	b³a²	a⁴	ba³	a⁵	ba⁴	ba⁵
b³a²	b³a²	b³a³	a	b³a⁴	a²	ba⁵	b³a⁵	a³	b	b²a	b³	a⁴	ba	b²a²	b³a	a⁵	ba²	b²a³	1	ba³	b²a⁴	ba⁴	b²a⁵	b²
ba⁵	ba⁵	b	b²a	ba	b²a²	b³a⁵	ba²	b²a³	b³	a⁴	ba³	b²a⁴	b³a	a⁵	ba⁴	b²a⁵	b³a²	1	b²	b³a³	a	b³a⁴	a²	a³
b²a⁴	b²a⁴	b²a⁵	b³a²	b²	b³a³	a	b²a	b³a⁴	a²	ba⁵	b²a²	b³a⁵	a³	b	b²a³	b³	a⁴	ba	b³a	a⁵	ba²	1	ba³	ba⁴
b³a³	b³a³	b³a⁴	1	b³a⁵	a	b	b³	a²	ba	b²	b³a	a³	ba²	b²a	b³a²	a⁴	ba³	b²a²	a⁵	ba⁴	b²a³	ba⁵	b²a⁴	b²a⁵
b²a⁵	b²a⁵	b²	b³a	b²a	b³a²	a²	b²a²	b³a³	a³	ba⁴	b²a³	b³a⁴	a⁴	ba⁵	b²a⁴	b³a⁵	a⁵	b	b³	1	ba	a	ba²	ba³
b³a⁴	b³a⁴	b³a⁵	a⁵	b³	1	ba	b³a	a	ba²	b²a⁵	b³a²	a²	ba³	b²	b³a³	a³	ba⁴	b²a	a⁴	ba⁵	b²a²	b	b²a³	b²a⁴
b³a⁵	b³a⁵	b³	a⁴	b³a	a⁵	ba²	b³a²	1	ba³	b²a⁴	b³a³	a	ba⁴	b²a⁵	b³a⁴	a²	ba⁵	b²	a³	b	b²a	ba	b²a²	b²a³

Right Cayley graph

Left Cayley graph

Others with same cardinality

14 unique, 70 total

Length:	Presentation:	Description:	Related:
8	〈a, b \| aab=ba, bbb=1〉	Finite non-commutative monoid with 24 elements	6 isomorphic
9	〈a, b \| aaa=1, aaba=bb〉	Finite non-commutative monoid with 24 elements	5 isomorphic
9	〈a, b \| aaa=1, abab=ba〉	Finite non-commutative monoid with 24 elements	2 isomorphic, 3 anti-isomorphic
10	〈a, b \| aba=bb, aabbb=1〉	Finite non-Abelian group with 24 elements	22 isomorphic
10	〈a, b \| bb=aa, aaabab=1〉	Finite non-Abelian group with 24 elements	2 isomorphic
10	〈a, b \| aaa=1, abaab=ba〉	Finite non-commutative monoid with 24 elements	1 isomorphic
11	〈a, b \| ababa=b, abbaa=1〉	Finite non-Abelian group with 24 elements	2 isomorphic
11	〈a, b \| aaaab=1, bbbbbb=1〉	Isomorphic to ℤ₂₄	8 isomorphic
11	〈a, b \| aaabb=1, bababa=1〉	Finite non-Abelian group with 24 elements	3 isomorphic
11	〈a, b \| abba=b, aaabbb=1〉	Finite non-Abelian group with 24 elements	2 isomorphic
11	〈a, b \| abba=b, ababab=1〉	Finite non-Abelian group with 24 elements
11	〈a, b \| aba=a, aaaab=bb〉	Finite non-commutative monoid with 24 elements
11	〈a, b \| aba=b, aaaa=bab〉	Finite non-commutative monoid with 24 elements
11	〈a, b \| aaa=1, aabaaba=b〉	Finite non-commutative monoid with 24 elements

Other isomorphic instances

7 total

Length:	Presentation:
11	〈a, b \| aba=b, aabbabb=1〉
11	〈a, b \| aba=b, aabbbba=1〉
11	〈a, b \| aba=b, abbaabb=1〉
11	〈a, b \| aba=b, abbabba=1〉
11	〈a, b \| aba=b, baaabbb=1〉
11	〈a, b \| aba=b, baabbab=1〉
11	〈a, b \| aba=b, bbaaabb=1〉