Back

〈a, b | abba=b, aaabbb=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 = a²ba⁵

Complete rewriting system

a⁶ ⇒ 1
a³b ⇒ ba³
b² ⇒ a²ba²
bab ⇒ aba
ba²b ⇒ a

Cayley table

	1	a	b	a²	ab	ba	a³	a²b	aba	ba²	a⁴	a²ba	aba²	ba³	a⁵	a²ba²	aba³	ba⁴	a²ba³	aba⁴	ba⁵	a²ba⁴	aba⁵	a²ba⁵
1	1	a	b	a²	ab	ba	a³	a²b	aba	ba²	a⁴	a²ba	aba²	ba³	a⁵	a²ba²	aba³	ba⁴	a²ba³	aba⁴	ba⁵	a²ba⁴	aba⁵	a²ba⁵
a	a	a²	ab	a³	a²b	aba	a⁴	ba³	a²ba	aba²	a⁵	ba⁴	a²ba²	aba³	1	ba⁵	a²ba³	aba⁴	b	a²ba⁴	aba⁵	ba	a²ba⁵	ba²
b	b	ba	a²ba²	ba²	aba	a²ba³	ba³	a	aba²	a²ba⁴	ba⁴	a²	aba³	a²ba⁵	ba⁵	a³	aba⁴	a²b	a⁴	aba⁵	a²ba	a⁵	ab	1
a²	a²	a³	a²b	a⁴	ba³	a²ba	a⁵	aba³	ba⁴	a²ba²	1	aba⁴	ba⁵	a²ba³	a	aba⁵	b	a²ba⁴	ab	ba	a²ba⁵	aba	ba²	aba²
ab	ab	aba	ba⁵	aba²	a²ba	b	aba³	a²	a²ba²	ba	aba⁴	a³	a²ba³	ba²	aba⁵	a⁴	a²ba⁴	ba³	a⁵	a²ba⁵	ba⁴	1	a²b	a
ba	ba	ba²	aba	ba³	a	aba²	ba⁴	a²ba⁵	a²	aba³	ba⁵	a²b	a³	aba⁴	b	a²ba	a⁴	aba⁵	a²ba²	a⁵	ab	a²ba³	1	a²ba⁴
a³	a³	a⁴	ba³	a⁵	aba³	ba⁴	1	a²ba³	aba⁴	ba⁵	a	a²ba⁴	aba⁵	b	a²	a²ba⁵	ab	ba	a²b	aba	ba²	a²ba	aba²	a²ba²
a²b	a²b	a²ba	aba⁵	a²ba²	ba⁴	ab	a²ba³	a³	ba⁵	aba	a²ba⁴	a⁴	b	aba²	a²ba⁵	a⁵	ba	aba³	1	ba²	aba⁴	a	ba³	a²
aba	aba	aba²	a²ba	aba³	a²	a²ba²	aba⁴	ba²	a³	a²ba³	aba⁵	ba³	a⁴	a²ba⁴	ab	ba⁴	a⁵	a²ba⁵	ba⁵	1	a²b	b	a	ba
ba²	ba²	ba³	a	ba⁴	a²ba⁵	a²	ba⁵	aba⁴	a²b	a³	b	aba⁵	a²ba	a⁴	ba	ab	a²ba²	a⁵	aba	a²ba³	1	aba²	a²ba⁴	aba³
a⁴	a⁴	a⁵	aba³	1	a²ba³	aba⁴	a	b	a²ba⁴	aba⁵	a²	ba	a²ba⁵	ab	a³	ba²	a²b	aba	ba³	a²ba	aba²	ba⁴	a²ba²	ba⁵
a²ba	a²ba	a²ba²	ba⁴	a²ba³	a³	ba⁵	a²ba⁴	aba²	a⁴	b	a²ba⁵	aba³	a⁵	ba	a²b	aba⁴	1	ba²	aba⁵	a	ba³	ab	a²	aba
aba²	aba²	aba³	a²	aba⁴	ba²	a³	aba⁵	a²ba⁴	ba³	a⁴	ab	a²ba⁵	ba⁴	a⁵	aba	a²b	ba⁵	1	a²ba	b	a	a²ba²	ba	a²ba³
ba³	ba³	ba⁴	a²ba⁵	ba⁵	aba⁴	a²b	b	a⁴	aba⁵	a²ba	ba	a⁵	ab	a²ba²	ba²	1	aba	a²ba³	a	aba²	a²ba⁴	a²	aba³	a³
a⁵	a⁵	1	a²ba³	a	b	a²ba⁴	a²	ab	ba	a²ba⁵	a³	aba	ba²	a²b	a⁴	aba²	ba³	a²ba	aba³	ba⁴	a²ba²	aba⁴	ba⁵	aba⁵
a²ba²	a²ba²	a²ba³	a³	a²ba⁴	aba²	a⁴	a²ba⁵	ba	aba³	a⁵	a²b	ba²	aba⁴	1	a²ba	ba³	aba⁵	a	ba⁴	ab	a²	ba⁵	aba	b
aba³	aba³	aba⁴	ba²	aba⁵	a²ba⁴	ba³	ab	a⁵	a²ba⁵	ba⁴	aba	1	a²b	ba⁵	aba²	a	a²ba	b	a²	a²ba²	ba	a³	a²ba³	a⁴
ba⁴	ba⁴	ba⁵	aba⁴	b	a⁴	aba⁵	ba	a²ba²	a⁵	ab	ba²	a²ba³	1	aba	ba³	a²ba⁴	a	aba²	a²ba⁵	a²	aba³	a²b	a³	a²ba
a²ba³	a²ba³	a²ba⁴	aba²	a²ba⁵	ba	aba³	a²b	1	ba²	aba⁴	a²ba	a	ba³	aba⁵	a²ba²	a²	ba⁴	ab	a³	ba⁵	aba	a⁴	b	a⁵
aba⁴	aba⁴	aba⁵	a²ba⁴	ab	a⁵	a²ba⁵	aba	ba⁵	1	a²b	aba²	b	a	a²ba	aba³	ba	a²	a²ba²	ba²	a³	a²ba³	ba³	a⁴	ba⁴
ba⁵	ba⁵	b	a⁴	ba	a²ba²	a⁵	ba²	aba	a²ba³	1	ba³	aba²	a²ba⁴	a	ba⁴	aba³	a²ba⁵	a²	aba⁴	a²b	a³	aba⁵	a²ba	ab
a²ba⁴	a²ba⁴	a²ba⁵	ba	a²b	1	ba²	a²ba	aba⁵	a	ba³	a²ba²	ab	a²	ba⁴	a²ba³	aba	a³	ba⁵	aba²	a⁴	b	aba³	a⁵	aba⁴
aba⁵	aba⁵	ab	a⁵	aba	ba⁵	1	aba²	a²ba	b	a	aba³	a²ba²	ba	a²	aba⁴	a²ba³	ba²	a³	a²ba⁴	ba³	a⁴	a²ba⁵	ba⁴	a²b
a²ba⁵	a²ba⁵	a²b	1	a²ba	aba⁵	a	a²ba²	ba⁴	ab	a²	a²ba³	ba⁵	aba	a³	a²ba⁴	b	aba²	a⁴	ba	aba³	a⁵	ba²	aba⁴	ba³

Right Cayley graph

Left Cayley graph

Others with same cardinality

14 unique, 75 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, ababab=1〉	Finite non-Abelian group with 24 elements
11	〈a, b \| aba=b, aaabbbb=1〉	Finite non-Abelian group with 24 elements	7 isomorphic
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

2 total

Length:	Presentation:
11	〈a, b \| abba=b, aabbba=1〉
11	〈a, b \| abba=b, baaabb=1〉