ATLAS: Alternating group A21
Order = 25545471085854720000
Mult = 2.
Out = 2.
The following information is available for A21:
Standard generators of A21 are a of order 3
(in the smallest conjugacy class of elements of elements of order 3),
b of order 19 such that ab has order 21.
In the natural representation we may take
a = (1,2,3) and
b = (3,4,5,6,7,8, ... ,21).
Standard generators of S21 are c of order 2
(belonging to the smallest conjugacy class of outer involutions),
d of order 20 such that cd has order 21.
In the natural
representation we may take
c = (1, 2) and
d = (2, 3, 4, 5, 6, 7, 8, ..., 21).
To find standard generators of A21:
- Find an element of order 48, 51, 165, 195, 231 or 420
(probability about 1/11) and
power up to give an element a in class 3A.
- Find an element t of order 19 (probability about 1/19).
- Find a conjugate b of t such that ab has
order 21 (probability about 1/140).
- The elements a and b are standard generators
for A17.
To find standard generators of S21
- Find an element of order 38, 130, 154 or 330 (probability about
1/23) and power up to give an element c of order 2
in the conjugacy class of transpositions. Alternatively,
if you look among outer elements only, then elements of order
34 and 126 also work, and the probability rises to about 1/7.
- Find an element t of order 19 (probability 1/38).
- Find a conjugate u of t such that cu has order
20 (probability about 1/6).
- Find a conjugate d of cu such that
cd has order 21 (probability about 1/11).
- The elements c and d are standard generators
for S17.
The representations of A21 available are:
- Some primitive permutation representations
-
Permutations on 21 points - the natural representation above:
a and
b (GAP).
-
Permutations on 210 points:
a and
b (GAP).
- Some integer matrix representations
-
Dimension 20 (partition [2, 119]):
a and b (GAP).
-
Dimension 189 (partition [22, 117]):
a and b (GAP).
-
Dimension 190 (partition [3, 118]):
a and b (GAP).
The representations of S21 available are:
- Some primitive permutation representations
-
Permutations on 21 points - the natural representation above:
c and
d (GAP).
-
Permutations on 210 points:
c and
d (GAP).
- Some integer matrix representations
-
Dimension 20 (partition [2, 119]):
c and d (GAP).
-
Dimension 189 (partition [22, 117]
c and d (GAP).
-
Dimension 190 (partition [3, 118]):
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 24th February 2004.
Last updated 01.03.04 by SN.