# Talk:Janko group J3

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Field: Algebra

## Requests for corrections

The correction to the Higman - Mckay reference is omitted.

Further, McKay & Wales gave the groups their names so that their index was the order of their Schur multiplier. —Preceding unsigned comment added by 130.54.16.90 (talk) 2008-06-06T09:32:47.

## Number of elements of each order

Tried installing GAP, and it seems to find conjugacy classes a lot faster, and with a lot less memory than a program I wrote that generates all elements in a group.

Using the generators and this script:

```g := Group(b11, b21);;
Print("Size of group = ", Size(g), "\n\n");;
c := ConjugacyClasses(g);;
#f := [];;
for e in [1..Size(c)] do
r := Representative(c[e]);;
#Add(f, [Order(r), Size(c[e]), 1]);;
Print("Order = ", Order(r), ", Size = ", Size(c[e]), "\n");;
od;
#f := SortedList(f);
Print("\n");;
```

my computer didn't run out of memory. Not sure how to easily check for power equivalence in GAP or how to print factors, and don't even know what a cycle structure is. Maybe it's easy for someone with GAP experience.

```Order = 1, Size = 1
Order = 19, Size = 2643840
Order = 19, Size = 2643840
Order = 15, Size = 3348864
Order = 15, Size = 3348864
Order = 5, Size = 1674432
Order = 5, Size = 1674432
Order = 3, Size = 46512
Order = 10, Size = 5023296
Order = 10, Size = 5023296
Order = 2, Size = 26163
Order = 12, Size = 4186080
Order = 6, Size = 2093040
Order = 4, Size = 523260
Order = 17, Size = 2954880
Order = 17, Size = 2954880
Order = 9, Size = 1860480
Order = 9, Size = 1860480
Order = 9, Size = 1860480
Order = 3, Size = 206720
Order = 8, Size = 6279120
```

Κσυπ Cyp   20:57, 5 July 2008 (UTC)

Robert A, Wilson's ATLAS website lists the conjugacy classes of J3, but it lists orders of centralizers instead of sized of conjugacy classes.

When I said cycle structure I was referring to the structure of an element as a permutation in a well-known permutation representation. For example, in a representation of M24 of degree 24, an element of order 5 is a product of 4 disjoint 5-cycles, written 54.

On the ATLAS website the smallest permutation representation of J3 has degree 6156. I am not inclined to bother with it.

The ATLAS listing shows power equivalence. Scott Tillinghast, Houston TX (talk) 00:06, 9 July 2008 (UTC)

As near as I can tell, wikipedia has no article on "cycle structure". The cycle structure of an element g of a permutation group is the multi-set of sizes of orbits of the cyclic group generated by g. This is related to the cycle decomposition of g when g only has finite orbits, which takes orbit representatives a1, a2, ... and specifies g by (a1, a1^g, a1^(g^2), ... )(a2, a2^g, ...) where each tuple has no duplicate entries. This is also called "cycle notation", and the "cycle structure" is the list of cycle sizes.
For GAP, you can handle much larger groups using its character table library. The following gives all the sizes, orders, and power relations for the O'Nan group. To check that 8a and 8b are not power conjugate, use the last commands.
O'Nan example in GAP
O'Nan example in GAP
```gap> LoadPackage("ctbllib");;
# Get the O'Nan group as a character table
gap> ct := CharacterTable("ON");;
gap> OrdersClassRepresentatives(ct);
gap> SizesConjugacyClasses(ct);
# See which classes are 13th powers of each other
gap> List( Filtered(
> Cycles(PermList(PowerMap(ct,13)), [1..NrConjugacyClasses(ct)]),
> x->Length(x)>1), x->ClassNames(ct){x});
[ [ "15a", "15b" ], [ "16a", "16b" ], [ "16c", "16d" ], [ "19a", "19c", "19b" ],
[ "20a", "20b" ], [ "28a", "28b" ], [ "31a", "31b" ] ]
gap> PrintArray(TransposedMat([ClassNames(ct),
> OrdersClassRepresentatives(ct),
> SizesConjugacyClasses(ct),
> List(SizesConjugacyClasses(ct),StringPP)]));
[[  1a,  1,           1,                       1 ],
[  2a,  2,     2857239,        3^2*7^2*11*19*31 ],
[  3a,  3,   142227008,        2^6*7^3*11*19*31 ],
[  4a,  4,     5714478,      2*3^2*7^2*11*19*31 ],
[  4b,  4,  1800060570,    2*3^4*5*7^3*11*19*31 ],
[  5a,  5,  2560086144,    2^7*3^2*7^3*11*19*31 ],
[  6a,  6,  6400215360,  2^6*3^2*5*7^3*11*19*31 ],
[  7a,  7,   335871360,      2^7*3^4*5*11*19*31 ],
[  7b,  7,  9404398080,    2^9*3^4*5*7*11*19*31 ],
[  8a,  8, 14400484560,  2^4*3^4*5*7^3*11*19*31 ],
[  8b,  8, 14400484560,  2^4*3^4*5*7^3*11*19*31 ],
[ 10a, 10, 23040775296,    2^7*3^4*7^3*11*19*31 ],
[ 11a, 11, 41892318720,     2^9*3^4*5*7^3*19*31 ],
[ 12a, 12, 12800430720,  2^7*3^2*5*7^3*11*19*31 ],
[ 14a, 14, 16457696640,  2^7*3^4*5*7^2*11*19*31 ],
[ 15a, 15, 10240344576,    2^9*3^2*7^3*11*19*31 ],
[ 15b, 15, 10240344576,    2^9*3^2*7^3*11*19*31 ],
[ 16a, 16, 28800969120,  2^5*3^4*5*7^3*11*19*31 ],
[ 16b, 16, 28800969120,  2^5*3^4*5*7^3*11*19*31 ],
[ 16c, 16, 28800969120,  2^5*3^4*5*7^3*11*19*31 ],
[ 16d, 16, 28800969120,  2^5*3^4*5*7^3*11*19*31 ],
[ 19a, 19, 24253447680,     2^9*3^4*5*7^3*11*31 ],
[ 19b, 19, 24253447680,     2^9*3^4*5*7^3*11*31 ],
[ 19c, 19, 24253447680,     2^9*3^4*5*7^3*11*31 ],
[ 20a, 20, 23040775296,    2^7*3^4*7^3*11*19*31 ],
[ 20b, 20, 23040775296,    2^7*3^4*7^3*11*19*31 ],
[ 28a, 28, 16457696640,  2^7*3^4*5*7^2*11*19*31 ],
[ 28b, 28, 16457696640,  2^7*3^4*5*7^2*11*19*31 ],
[ 31a, 31, 14865016320,     2^9*3^4*5*7^3*11*19 ],
[ 31b, 31, 14865016320,     2^9*3^4*5*7^3*11*19 ] ]
gap> PrintArray(TransposedMat([ClassNames(ct),ClassNames(ct){PowerMap(ct,2)}]));
```
If you like to program, GAP can be very helpful. Otherwise the online ATLAS and the paper ATLAS have approximately the same information. JackSchmidt (talk) 15:33, 8 July 2008 (UTC)

The page in the ATLAS website says that elements of orders 5, 10, 15, 17, and 19 all fall into 2 power equivalent conjugacy classes. The 3 conjugacy classes of 9-elements also have what could be called power equivalency. Classes 3A and 3B are not power equivalent. Scott Tillinghast, Houston TX (talk) 14:16, 9 July 2008 (UTC)

I agree, or at least so does GAP and the paper atlas. If we ever want the specific powers: I get that 19A = 19B^-1, 5A=5B^2, 9C=9B^2=9A^4, 15A=15B^2, 10A=10B^3, and 17A=17B^3, but 3A and 3B are different sizes. JackSchmidt (talk) 14:38, 9 July 2008 (UTC)

## Shape of permutations

It is customary to include the fixed points (= orbits of length =1) when using the (permutation- or cycle-) shape of a permutation - thus the M24 example given is written 5^4.1^4 so that the degree ( = 24) may be deduced from the shape.

John McKay 66.130.86.141 (talk) 11:02, 22 September 2009 (UTC)

## Page appearance

Reduced large white space at 1024x768 by moving the infobox below the long [itex] tags. Better to shrink the huge infobox itself, IMHO. Discuss. --Lexein (talk) 10:53, 27 March 2010 (UTC)