301 "set theories"

There are 301 transitive permutation groups of degree 12, any one of
which one could use as a the basis of a 12-et "set theory". Unless
they contain a 12-cycle they will not equate things under
transposition, but even those cases might be interesting, since they
include groups of low order. In fact, there are five different
transitive permutation groups of degree and order 12; one of these,
of course, is the cyclic group of order 12. The others are

E(4) x C(3)

The 2-elementary group of order 4 (Klein 4 group) times the cyclic
group of order 3.

Generators

(0, 4, 8)(1, 5, 9)(2, 6, 10)(3, 7, 11)
(0, 6)(1, 7)(2, 8)(3, 9)(4, 10)(5, 11)
(0, 3)(1, 10)(2, 5)(4, 7)(6, 9)(8, 11)

D6(6) x 2

Dihedral group of order 6, times cyclic group of order 2

Generators

(0, 4, 8)(1, 5, 9)(2, 6, 10)(3, 7, 11)
(0, 1)(2, 3)(4, 5)(6, 7)(8, 9)(10, 11)
(0, 10)(1, 11)(2, 8)(3, 9)(4, 6)(5, 7)

A4(12)

The regular representation of the alternating group of degree four.

Generators

(0, 4, 8)(1, 11, 6)(2, 9, 7)(3, 10, 5)
(0, 11, 10)(1, 9, 5)(2, 4, 3)(6, 8, 7)

(1/2)[3:2]4

Generators: a, h, Z

(0, 4, 8)(1, 5, 9)(2, 6, 10)(3, 7, 11)
(0, 6)(1, 7)(2, 8)(3, 9)(4, 10)(5, 11)
(0, 3, 6, 9)(1, 8, 7, 2)(4, 11, 10, 5)

The names and generators are those found in "On Transitive
Permutation Groups", Conway, Hulpke and McKay.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> There are 301 transitive permutation groups of degree 12, any one of
> which one could use as a the basis of a 12-et "set theory". Unless
> they contain a 12-cycle they will not equate things under
> transposition, but even those cases might be interesting, since they
> include groups of low order. In fact, there are five different
> transitive permutation groups of degree and order 12; one of these,
> of course, is the cyclic group of order 12. The others are
>
> E(4) x C(3)
>
> The 2-elementary group of order 4 (Klein 4 group) times the cyclic
> group of order 3.
>
> Generators
>
> (0, 4, 8)(1, 5, 9)(2, 6, 10)(3, 7, 11)
> (0, 6)(1, 7)(2, 8)(3, 9)(4, 10)(5, 11)
> (0, 3)(1, 10)(2, 5)(4, 7)(6, 9)(8, 11)
>
>
> D6(6) x 2
>
> Dihedral group of order 6, times cyclic group of order 2
>
> Generators
>
> (0, 4, 8)(1, 5, 9)(2, 6, 10)(3, 7, 11)
> (0, 1)(2, 3)(4, 5)(6, 7)(8, 9)(10, 11)
> (0, 10)(1, 11)(2, 8)(3, 9)(4, 6)(5, 7)
>
>
> A4(12)
>
> The regular representation of the alternating group of degree four.
>
> Generators
>
> (0, 4, 8)(1, 11, 6)(2, 9, 7)(3, 10, 5)
> (0, 11, 10)(1, 9, 5)(2, 4, 3)(6, 8, 7)
>
>
> (1/2)[3:2]4
>
> Generators: a, h, Z
>
> (0, 4, 8)(1, 5, 9)(2, 6, 10)(3, 7, 11)
> (0, 6)(1, 7)(2, 8)(3, 9)(4, 10)(5, 11)
> (0, 3, 6, 9)(1, 8, 7, 2)(4, 11, 10, 5)
>
> The names and generators are those found in "On Transitive
> Permutation Groups", Conway, Hulpke and McKay.

I think there is another one: the Mathieu group M12. Or is that the last one on the list?
M12 is quite interesting: not just transitive, but 5-transitive.

Hans Straub

--- In tuning-math@yahoogroups.com, "hstraub64" <hstraub64@t...>
wrote:

> I think there is another one: the Mathieu group M12. Or is that the
last one on the list?
> M12 is quite interesting: not just transitive, but 5-transitive.

I did say there were 301. M12 (order 95040) is number 295, M11(12)
(order 7920) number 272, and M10(12) (order 720) is number 181. Many,
many more, of course, including L(2,11) (order 660) at number 179,
PGL(2,9) (order 720) at number 182, PGL(2,11) (order 1320) at number
218, and even A5(12) (order 60) at number 33.

--- In tuning-math@yahoogroups.com, "hstraub64" <hstraub64@t...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:
> > There are 301 transitive permutation groups of degree 12, any one
of
> > which one could use as a the basis of a 12-et "set theory".
Unless
> > they contain a 12-cycle they will not equate things under
> > transposition, but even those cases might be interesting, since
they
> > include groups of low order. In fact, there are five different
> > transitive permutation groups of degree and order 12; one of
these,
> > of course, is the cyclic group of order 12. The others are
> >
> > E(4) x C(3)
> >
> > The 2-elementary group of order 4 (Klein 4 group) times the
cyclic
> > group of order 3.
> >
> > Generators
> >
> > (0, 4, 8)(1, 5, 9)(2, 6, 10)(3, 7, 11)
> > (0, 6)(1, 7)(2, 8)(3, 9)(4, 10)(5, 11)
> > (0, 3)(1, 10)(2, 5)(4, 7)(6, 9)(8, 11)
> >
> >
> > D6(6) x 2
> >
> > Dihedral group of order 6, times cyclic group of order 2
> >
> > Generators
> >
> > (0, 4, 8)(1, 5, 9)(2, 6, 10)(3, 7, 11)
> > (0, 1)(2, 3)(4, 5)(6, 7)(8, 9)(10, 11)
> > (0, 10)(1, 11)(2, 8)(3, 9)(4, 6)(5, 7)
> >
> >
> > A4(12)
> >
> > The regular representation of the alternating group of degree
four.
> >
> > Generators
> >
> > (0, 4, 8)(1, 11, 6)(2, 9, 7)(3, 10, 5)
> > (0, 11, 10)(1, 9, 5)(2, 4, 3)(6, 8, 7)
> >
> >
> > (1/2)[3:2]4
> >
> > Generators: a, h, Z
> >
> > (0, 4, 8)(1, 5, 9)(2, 6, 10)(3, 7, 11)
> > (0, 6)(1, 7)(2, 8)(3, 9)(4, 10)(5, 11)
> > (0, 3, 6, 9)(1, 8, 7, 2)(4, 11, 10, 5)
> >
> > The names and generators are those found in "On Transitive
> > Permutation Groups", Conway, Hulpke and McKay.
>
> I think there is another one: the Mathieu group M12.

Gene didn't list all 301, so there is much more than just one other
one! He's mentioned the Mathieu group quite a lot before!

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
>
> I did say there were 301. M12 (order 95040) is number 295, M11(12)
> (order 7920) number 272, and M10(12) (order 720) is number 181.
Many,
> many more, of course, including L(2,11) (order 660) at number 179,
> PGL(2,9) (order 720) at number 182, PGL(2,11) (order 1320) at
number
> 218, and even A5(12) (order 60) at number 33.

Ooops, I had overlooked that the five mentioned were those of degree
_and_ order 12...