#5009 ⟨a, b | aba=bb, aabbb=1⟩

Properties

Presentation has sum-of-sides 10
Finite non-Abelian group with 24 elements
Inverses of generators:
- a^-1 = ab³
- b^-1 = b¹¹

Element profile

4 element center:
- 1, b³, b⁶, b⁹

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(b) = 0; deg(a) = 1

b¹² ⇒ 1
ba ⇒ ab⁵
a² ⇒ b⁹

# ab:aba=bb,aabbb=1 b/a
bbbbbbbbbbbb=1
ba=abbbbb
aa=bbbbbbbbb

Cayley table

	1	a	b	ab	b²	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	ab⁷	b⁸	ab⁸	b⁹	ab⁹	b¹⁰	ab¹⁰	b¹¹	ab¹¹
1	1	a	b	ab	b²	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	ab⁷	b⁸	ab⁸	b⁹	ab⁹	b¹⁰	ab¹⁰	b¹¹	ab¹¹
a	a	b⁹	ab	b¹⁰	ab²	b¹¹	ab³	1	ab⁴	b	ab⁵	b²	ab⁶	b³	ab⁷	b⁴	ab⁸	b⁵	ab⁹	b⁶	ab¹⁰	b⁷	ab¹¹	b⁸
b	b	ab⁵	b²	ab⁶	b³	ab⁷	b⁴	ab⁸	b⁵	ab⁹	b⁶	ab¹⁰	b⁷	ab¹¹	b⁸	a	b⁹	ab	b¹⁰	ab²	b¹¹	ab³	1	ab⁴
ab	ab	b²	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	ab⁷	b⁸	ab⁸	b⁹	ab⁹	b¹⁰	ab¹⁰	b¹¹	ab¹¹	1	a	b
b²	b²	ab¹⁰	b³	ab¹¹	b⁴	a	b⁵	ab	b⁶	ab²	b⁷	ab³	b⁸	ab⁴	b⁹	ab⁵	b¹⁰	ab⁶	b¹¹	ab⁷	1	ab⁸	b	ab⁹
ab²	ab²	b⁷	ab³	b⁸	ab⁴	b⁹	ab⁵	b¹⁰	ab⁶	b¹¹	ab⁷	1	ab⁸	b	ab⁹	b²	ab¹⁰	b³	ab¹¹	b⁴	a	b⁵	ab	b⁶
b³	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	ab⁷	b⁸	ab⁸	b⁹	ab⁹	b¹⁰	ab¹⁰	b¹¹	ab¹¹	1	a	b	ab	b²	ab²
ab³	ab³	1	ab⁴	b	ab⁵	b²	ab⁶	b³	ab⁷	b⁴	ab⁸	b⁵	ab⁹	b⁶	ab¹⁰	b⁷	ab¹¹	b⁸	a	b⁹	ab	b¹⁰	ab²	b¹¹
b⁴	b⁴	ab⁸	b⁵	ab⁹	b⁶	ab¹⁰	b⁷	ab¹¹	b⁸	a	b⁹	ab	b¹⁰	ab²	b¹¹	ab³	1	ab⁴	b	ab⁵	b²	ab⁶	b³	ab⁷
ab⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	ab⁷	b⁸	ab⁸	b⁹	ab⁹	b¹⁰	ab¹⁰	b¹¹	ab¹¹	1	a	b	ab	b²	ab²	b³	ab³	b⁴
b⁵	b⁵	ab	b⁶	ab²	b⁷	ab³	b⁸	ab⁴	b⁹	ab⁵	b¹⁰	ab⁶	b¹¹	ab⁷	1	ab⁸	b	ab⁹	b²	ab¹⁰	b³	ab¹¹	b⁴	a
ab⁵	ab⁵	b¹⁰	ab⁶	b¹¹	ab⁷	1	ab⁸	b	ab⁹	b²	ab¹⁰	b³	ab¹¹	b⁴	a	b⁵	ab	b⁶	ab²	b⁷	ab³	b⁸	ab⁴	b⁹
b⁶	b⁶	ab⁶	b⁷	ab⁷	b⁸	ab⁸	b⁹	ab⁹	b¹⁰	ab¹⁰	b¹¹	ab¹¹	1	a	b	ab	b²	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵
ab⁶	ab⁶	b³	ab⁷	b⁴	ab⁸	b⁵	ab⁹	b⁶	ab¹⁰	b⁷	ab¹¹	b⁸	a	b⁹	ab	b¹⁰	ab²	b¹¹	ab³	1	ab⁴	b	ab⁵	b²
b⁷	b⁷	ab¹¹	b⁸	a	b⁹	ab	b¹⁰	ab²	b¹¹	ab³	1	ab⁴	b	ab⁵	b²	ab⁶	b³	ab⁷	b⁴	ab⁸	b⁵	ab⁹	b⁶	ab¹⁰
ab⁷	ab⁷	b⁸	ab⁸	b⁹	ab⁹	b¹⁰	ab¹⁰	b¹¹	ab¹¹	1	a	b	ab	b²	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷
b⁸	b⁸	ab⁴	b⁹	ab⁵	b¹⁰	ab⁶	b¹¹	ab⁷	1	ab⁸	b	ab⁹	b²	ab¹⁰	b³	ab¹¹	b⁴	a	b⁵	ab	b⁶	ab²	b⁷	ab³
ab⁸	ab⁸	b	ab⁹	b²	ab¹⁰	b³	ab¹¹	b⁴	a	b⁵	ab	b⁶	ab²	b⁷	ab³	b⁸	ab⁴	b⁹	ab⁵	b¹⁰	ab⁶	b¹¹	ab⁷	1
b⁹	b⁹	ab⁹	b¹⁰	ab¹⁰	b¹¹	ab¹¹	1	a	b	ab	b²	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	ab⁷	b⁸	ab⁸
ab⁹	ab⁹	b⁶	ab¹⁰	b⁷	ab¹¹	b⁸	a	b⁹	ab	b¹⁰	ab²	b¹¹	ab³	1	ab⁴	b	ab⁵	b²	ab⁶	b³	ab⁷	b⁴	ab⁸	b⁵
b¹⁰	b¹⁰	ab²	b¹¹	ab³	1	ab⁴	b	ab⁵	b²	ab⁶	b³	ab⁷	b⁴	ab⁸	b⁵	ab⁹	b⁶	ab¹⁰	b⁷	ab¹¹	b⁸	a	b⁹	ab
ab¹⁰	ab¹⁰	b¹¹	ab¹¹	1	a	b	ab	b²	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	ab⁷	b⁸	ab⁸	b⁹	ab⁹	b¹⁰
b¹¹	b¹¹	ab⁷	1	ab⁸	b	ab⁹	b²	ab¹⁰	b³	ab¹¹	b⁴	a	b⁵	ab	b⁶	ab²	b⁷	ab³	b⁸	ab⁴	b⁹	ab⁵	b¹⁰	ab⁶
ab¹¹	ab¹¹	b⁴	a	b⁵	ab	b⁶	ab²	b⁷	ab³	b⁸	ab⁴	b⁹	ab⁵	b¹⁰	ab⁶	b¹¹	ab⁷	1	ab⁸	b	ab⁹	b²	ab¹⁰	b³

Right Cayley graph

Left Cayley graph

Others with same cardinality

6 unique, 47 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	6935	⟨a, b \| bb=aa, aaabab=1⟩	Finite non-Abelian group with 24 elements	2 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.

30 total

Σ	#	Presentation	Mapping
10	5014	⟨a, b \| aba=bb, abbba=1⟩	φ(a) = a, φ(b) = b
10	5017	⟨a, b \| aba=bb, baabb=1⟩	φ(a) = a, φ(b) = b
11	11327	⟨a, b \| aabaa=b, aabba=1⟩	φ(a) = b, φ(b) = a
11	11337	⟨a, b \| aabaa=b, baaab=1⟩	φ(a) = b, φ(b) = a
11	11341	⟨a, b \| aabaa=b, bbaaa=1⟩	φ(a) = b, φ(b) = a
11	13466	⟨a, b \| aaabb=1, abbbab=1⟩	φ(a) = b, φ(b) = a
11	13480	⟨a, b \| aaabb=1, bababb=1⟩	φ(a) = b, φ(b) = a
11	13483	⟨a, b \| aaabb=1, babbba=1⟩	φ(a) = b, φ(b) = a
11	13490	⟨a, b \| aaabb=1, bbabab=1⟩	φ(a) = b, φ(b) = a
11	13495	⟨a, b \| aaabb=1, bbbaba=1⟩	φ(a) = b, φ(b) = a
11	13624	⟨a, b \| aabba=1, ababbb=1⟩	φ(a) = b, φ(b) = a
11	13630	⟨a, b \| aabba=1, abbbab=1⟩	φ(a) = b, φ(b) = a
11	13644	⟨a, b \| aabba=1, bababb=1⟩	φ(a) = b, φ(b) = a
11	13647	⟨a, b \| aabba=1, babbba=1⟩	φ(a) = b, φ(b) = a
11	13654	⟨a, b \| aabba=1, bbabab=1⟩	φ(a) = b, φ(b) = a
11	13659	⟨a, b \| aabba=1, bbbaba=1⟩	φ(a) = b, φ(b) = a
11	13770	⟨a, b \| abbba=1, aaabab=1⟩	φ(a) = a, φ(b) = b
11	13774	⟨a, b \| abbba=1, aababa=1⟩	φ(a) = a, φ(b) = b
11	13780	⟨a, b \| abbba=1, abaaab=1⟩	φ(a) = a, φ(b) = b
11	15370	⟨a, b \| aba=bb, aaabab=1⟩	φ(a) = a, φ(b) = b
11	15374	⟨a, b \| aba=bb, aababa=1⟩	φ(a) = a, φ(b) = b
11	15380	⟨a, b \| aba=bb, abaaab=1⟩	φ(a) = a, φ(b) = b
11	18631	⟨a, b \| aba=b, aaabbbb=1⟩	φ(a) = bb, φ(b) = a
11	18641	⟨a, b \| aba=b, aabbabb=1⟩	φ(a) = bb, φ(b) = a
11	18644	⟨a, b \| aba=b, aabbbba=1⟩	φ(a) = bb, φ(b) = a
11	18659	⟨a, b \| aba=b, abbaabb=1⟩	φ(a) = bb, φ(b) = a
11	18661	⟨a, b \| aba=b, abbabba=1⟩	φ(a) = bb, φ(b) = a
11	18671	⟨a, b \| aba=b, baaabbb=1⟩	φ(a) = bb, φ(b) = a
11	18674	⟨a, b \| aba=b, baabbab=1⟩	φ(a) = bb, φ(b) = a
11	18682	⟨a, b \| aba=b, bbaaabb=1⟩	φ(a) = bb, φ(b) = a

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