#5728 ⟨a, b | aaaa=1, babbb=a⟩

Properties

Presentation has sum-of-sides 10
Finite non-Abelian group with 320 elements
Inverses of generators:
- a^-1 = a³
- b^-1 = b⁷⁹

Elements

Elements in the center commute with all other elements.
The order of x is the least n such that xⁿ = 1.

4 element center:
- 1, b²⁰, b⁴⁰, b⁶⁰
Order of generators:
- a: order 4
- b: order 80

Histogram:

order 2	11 elements	a², a²b⁸, a²b¹⁶, a²b²⁴, a²b³², ...
order 4	52 elements	a, a³, ab⁴, a²b⁴, a³b⁴, ...
order 5	4 elements	b¹⁶, b³², b⁴⁸, b⁶⁴
order 8	64 elements	ab², a²b², a³b², ab⁶, a²b⁶, ...
order 10	4 elements	b⁸, b²⁴, b⁵⁶, b⁷²
order 16	128 elements	ab, a²b, a³b, ab³, a²b³, ...
order 20	8 elements	b⁴, b¹², b²⁸, b³⁶, b⁴⁴, ...
order 40	16 elements	b², b⁶, b¹⁴, b¹⁸, b²², ...
order 80	32 elements	b, b³, b⁷, b⁹, b¹¹, ...

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

b⁸⁰ ⇒ 1
ba ⇒ ab⁷⁷
a⁴ ⇒ 1

# ab:aaaa=1,babbb=a b/a
bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=1
ba=abbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb
aaaa=1

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#5728 ⟨a, b | aaaa=1, babbb=a⟩

Properties

Elements

Complete rewriting system

Right Cayley graph

Left Cayley graph