ATLAS: Alternating group A15
Order = 653837184000
Mult = 2.
Out = 2.
Standard generators of A15 are a in class 3A,
b of order 13 such that ab has order 15 and abb
has order 15. In the natural
representation we may take
a = (1, 2, 3) and
b = (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15).
Standard generators of S15 are c in class 2D,
d in class 14E such that cd has order 15. In the natural
representation we may take
c = (1, 2) and
d = (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15).
To find standard generators for A15:
- Find an element of order 33 or 105 (probability 92/1155, or approximately 1/13) and power it up to give an element a in class 3A.
- Find an element t of order 13 (probability 1/13).
- Find a conjugate b of t such that ab has order 15 and abb has order 15 (probability 1/35).
- The elements a and b are standard generators of A15.
To find standard generators for S15:
- Find an element of order 26 or 70 (probability 24/455, or approximately 1/18) and power it up to give an element c in class 2D. There are other
possibilities if you are able to distinguish outer and inner elements (such
as powering up an outer 22 or 42 element).
- Find an element t of order 14 (probability approximately 1/14).
This is probably in class 14E.
- Find a conjugate c of t such that cd has order 15
(probability 2/15, or about 1/7). If you cannot find such a d
after several attempts, then t was probably in the wrong class,
so go back to the second step.
- The elements c and d are standard generators of S15.
The representations of A15 available are:
- Some primitive permutation representations
-
Permutations on 15 points - the natural representation above:
a and
b (GAP).
-
Permutations on 105 points
a and
b (GAP).
-
Permutations on 455 points
a and
b (GAP).
-
Permutations on 1365 points
a and
b (GAP).
- Some integer matrix representations
-
Dimension 14 (partition [2,113]):
a and b (GAP).
-
Dimension 90 (partition [22,111]):
a and b (GAP).
-
Dimension 91 (partition [3, 112]):
a and b (GAP).
-
Dimension 350 (partition [23, 19]):
a and b (GAP).
-
Dimension 364 (partition [4, 111]):
a and b (GAP).
The representations of S15 available are:
- Some primitive permutation representations
-
Permutations on 15 points - the natural representation above:
c and
d (GAP).
-
Permutations on 105 points
c and
d (GAP).
-
Permutations on 455 points
c and
d (GAP).
-
Permutations on 1365 points
c and
d (GAP).
- Some integer matrix representations
-
Dimension 14 (partition [2,113]):
c and d (GAP).
-
Dimension 90 (partition [22,111]):
c and d (GAP).
-
Dimension 91 (partition [3, 112]):
c and d (GAP).
-
Dimension 350 (partition [23, 19]):
c and d (GAP).
-
Dimension 364 (partition [4, 111]):
c and d (GAP).
Go to main ATLAS (version 2.0) page.
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Anonymous ftp access is also available on
sylow.mat.bham.ac.uk.
Version 2.0 created on 17th February 2004.
Last updated 18.02.04 by SN.