#21778 ⟨a, b | aaa=1, babbbbb=a⟩

Properties

Presentation has sum-of-sides 11
Finite non-Abelian group with 378 elements
Inverses of generators:
- a^-1 = a²
- b^-1 = b¹²⁵

Elements

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

6 element center:
- 1, b²¹, b⁴², b⁶³, b⁸⁴, b¹⁰⁵
Order of generators:
- a: order 3
- b: order 126

Histogram:

order 2	1 element	b⁶³
order 3	44 elements	a, a², b⁶a, b⁶a², b¹²a, ...
order 6	44 elements	b³a, b³a², b⁹a, b⁹a², b¹⁵a, ...
order 7	6 elements	b¹⁸, b³⁶, b⁵⁴, b⁷², b⁹⁰, ...
order 9	90 elements	b²a, b²a², b⁴a, b⁴a², b⁸a, ...
order 14	6 elements	b⁹, b²⁷, b⁴⁵, b⁸¹, b⁹⁹, ...
order 18	90 elements	ba, ba², b⁵a, b⁵a², b⁷, ...
order 21	12 elements	b⁶, b¹², b²⁴, b³⁰, b⁴⁸, ...
order 42	12 elements	b³, b¹⁵, b³³, b³⁹, b⁵¹, ...
order 63	36 elements	b², b⁴, b⁸, b¹⁰, b¹⁶, ...
order 126	36 elements	b, b⁵, b¹¹, b¹³, b¹⁷, ...

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:
- Right-to-left recursive path with deg(b) = 0; deg(a) = 1

b¹²⁶ ⇒ 1
ab ⇒ b²⁵a
a³ ⇒ 1

# ab:aaa=1,babbbbb=a reversed:b/a
bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=1
ab=bbbbbbbbbbbbbbbbbbbbbbbbba
aaa=1

Right Cayley graph

Left Cayley graph

Others with same cardinality

1 unique, 1 total

Σ	#	Presentation	Description	Related
11	14128	⟨a, b \| aaba=b, bbbbbb=1⟩	Finite non-Abelian group with 378 elements

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