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〈a, b | abba=b, ababab=1〉

Monoid presentation of length 11

Properties

Finite non-Abelian group with 24 elements

Completion parameters

Reduction order: left-to-right b/a

Inverses of generators

a^-1 = b²ab⁵
b^-1 = b⁵

Complete rewriting system

b⁶ ⇒ 1
b³a ⇒ ab³
a² ⇒ b²ab⁵
aba ⇒ bab⁴
ab²a ⇒ b

Cayley table

	1	a	b	ab	ba	b²	ab²	bab	b²a	b³	ab³	bab²	b²ab	b⁴	ab⁴	bab³	b²ab²	b⁵	ab⁵	bab⁴	b²ab³	bab⁵	b²ab⁴	b²ab⁵
1	1	a	b	ab	ba	b²	ab²	bab	b²a	b³	ab³	bab²	b²ab	b⁴	ab⁴	bab³	b²ab²	b⁵	ab⁵	bab⁴	b²ab³	bab⁵	b²ab⁴	b²ab⁵
a	a	b²ab⁵	ab	b²a	bab⁴	ab²	b²ab	bab⁵	b	ab³	b²ab²	ba	b²	ab⁴	b²ab³	bab	b³	ab⁵	b²ab⁴	bab²	b⁴	bab³	b⁵	1
b	b	ba	b²	bab	b²a	b³	bab²	b²ab	ab³	b⁴	bab³	b²ab²	ab⁴	b⁵	bab⁴	b²ab³	ab⁵	1	bab⁵	b²ab⁴	a	b²ab⁵	ab	ab²
ab	ab	bab⁴	ab²	bab⁵	b	ab³	ba	b²	b²ab²	ab⁴	bab	b³	b²ab³	ab⁵	bab²	b⁴	b²ab⁴	a	bab³	b⁵	b²ab⁵	1	b²a	b²ab
ba	ba	ab²	bab	ab³	b²ab⁴	bab²	ab⁴	b²ab⁵	b²	bab³	ab⁵	b²a	b³	bab⁴	a	b²ab	b⁴	bab⁵	ab	b²ab²	b⁵	b²ab³	1	b
b²	b²	b²a	b³	b²ab	ab³	b⁴	b²ab²	ab⁴	bab³	b⁵	b²ab³	ab⁵	bab⁴	1	b²ab⁴	a	bab⁵	b	b²ab⁵	ab	ba	ab²	bab	bab²
ab²	ab²	b	ab³	b²	b²ab²	ab⁴	b³	b²ab³	bab	ab⁵	b⁴	b²ab⁴	bab²	a	b⁵	b²ab⁵	bab³	ab	1	b²a	bab⁴	b²ab	bab⁵	ba
bab	bab	b²ab⁴	bab²	b²ab⁵	b²	bab³	b²a	b³	ab⁵	bab⁴	b²ab	b⁴	a	bab⁵	b²ab²	b⁵	ab	ba	b²ab³	1	ab²	b	ab³	ab⁴
b²a	b²a	bab²	b²ab	bab³	ab	b²ab²	bab⁴	ab²	b³	b²ab³	bab⁵	ab³	b⁴	b²ab⁴	ba	ab⁴	b⁵	b²ab⁵	bab	ab⁵	1	a	b	b²
b³	b³	ab³	b⁴	ab⁴	bab³	b⁵	ab⁵	bab⁴	b²ab³	1	a	bab⁵	b²ab⁴	b	ab	ba	b²ab⁵	b²	ab²	bab	b²a	bab²	b²ab	b²ab²
ab³	ab³	b²ab²	ab⁴	b²ab³	bab	ab⁵	b²ab⁴	bab²	b⁴	a	b²ab⁵	bab³	b⁵	ab	b²a	bab⁴	1	ab²	b²ab	bab⁵	b	ba	b²	b³
bab²	bab²	b²	bab³	b³	ab⁵	bab⁴	b⁴	a	b²ab	bab⁵	b⁵	ab	b²ab²	ba	1	ab²	b²ab³	bab	b	ab³	b²ab⁴	ab⁴	b²ab⁵	b²a
b²ab	b²ab	ab	b²ab²	ab²	b³	b²ab³	ab³	b⁴	bab⁵	b²ab⁴	ab⁴	b⁵	ba	b²ab⁵	ab⁵	1	bab	b²a	a	b	bab²	b²	bab³	bab⁴
b⁴	b⁴	bab³	b⁵	bab⁴	b²ab³	1	bab⁵	b²ab⁴	a	b	ba	b²ab⁵	ab	b²	bab	b²a	ab²	b³	bab²	b²ab	ab³	b²ab²	ab⁴	ab⁵
ab⁴	ab⁴	bab	ab⁵	bab²	b⁴	a	bab³	b⁵	b²ab⁵	ab	bab⁴	1	b²a	ab²	bab⁵	b	b²ab	ab³	ba	b²	b²ab²	b³	b²ab³	b²ab⁴
bab³	bab³	ab⁵	bab⁴	a	b²ab	bab⁵	ab	b²ab²	b⁵	ba	ab²	b²ab³	1	bab	ab³	b²ab⁴	b	bab²	ab⁴	b²ab⁵	b²	b²a	b³	b⁴
b²ab²	b²ab²	b³	b²ab³	b⁴	bab⁵	b²ab⁴	b⁵	ba	ab⁴	b²ab⁵	1	bab	ab⁵	b²a	b	bab²	a	b²ab	b²	bab³	ab	bab⁴	ab²	ab³
b⁵	b⁵	b²ab³	1	b²ab⁴	a	b	b²ab⁵	ab	ba	b²	b²a	ab²	bab	b³	b²ab	ab³	bab²	b⁴	b²ab²	ab⁴	bab³	ab⁵	bab⁴	bab⁵
ab⁵	ab⁵	b⁴	a	b⁵	b²ab⁵	ab	1	b²a	bab⁴	ab²	b	b²ab	bab⁵	ab³	b²	b²ab²	ba	ab⁴	b³	b²ab³	bab	b²ab⁴	bab²	bab³
bab⁴	bab⁴	b²ab	bab⁵	b²ab²	b⁵	ba	b²ab³	1	ab²	bab	b²ab⁴	b	ab³	bab²	b²ab⁵	b²	ab⁴	bab³	b²a	b³	ab⁵	b⁴	a	ab
b²ab³	b²ab³	bab⁵	b²ab⁴	ba	ab⁴	b²ab⁵	bab	ab⁵	1	b²a	bab²	a	b	b²ab	bab³	ab	b²	b²ab²	bab⁴	ab²	b³	ab³	b⁴	b⁵
bab⁵	bab⁵	b⁵	ba	1	ab²	bab	b	ab³	b²ab⁴	bab²	b²	ab⁴	b²ab⁵	bab³	b³	ab⁵	b²a	bab⁴	b⁴	a	b²ab	ab	b²ab²	b²ab³
b²ab⁴	b²ab⁴	ab⁴	b²ab⁵	ab⁵	1	b²a	a	b	bab²	b²ab	ab	b²	bab³	b²ab²	ab²	b³	bab⁴	b²ab³	ab³	b⁴	bab⁵	b⁵	ba	bab
b²ab⁵	b²ab⁵	1	b²a	b	bab²	b²ab	b²	bab³	ab	b²ab²	b³	bab⁴	ab²	b²ab³	b⁴	bab⁵	ab³	b²ab⁴	b⁵	ba	ab⁴	bab	ab⁵	a

Right Cayley graph

Left Cayley graph

Others with same cardinality

14 unique, 77 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 \| 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