# Characters of almost quasisimple groups

A group $$Q$$ is called quasisimple if it is a perfect central extension of a simple group.

A group $$A$$ is called almost simple if there exists a series of containments $$S \subset A \subset Aut(S)$$ where $$S$$ is a simple group and $$Aut(S)$$ is the automorphism group of $$S$$. In other words, a group $$A$$ is almost simple if it is the extension of a simple group by some subgroup of the outer automorphism group.

For a quasisimple group $$Q$$, all characters of the corresponding simple group $$Q/Z(Q)$$ lift to characters of $$Q$$. Since the characters of finite simple groups are well known, these are easy to obtain. There are some other characters of $$Q$$ which cannot be obtained from the characters of the simple group $$Q/Z(Q)$$. Even these are not too hard to find since quasisimple groups are perfect and there are good libraries of characters for (small) perfect groups. For example GAP has the characters tables of all Perfect Groups of size up to two million. https://www.gap-system.org/Datalib/perfect.html

The characters of finite almost simple groups are also not too hard to obtain. They can usually be found in the same references as the characters tables of finite simple groups. In GAP one can just use the command AutomorphismGroup(S) where $$S$$ is the simple group in question. For a given almost simple group $$A$$ most (but not all) of its characters can be found by just restricting the characters of $$Aut(S)$$ to the subgroup $$A$$.

For an almost quasisimple group $$G$$ I am unsure how to find characters. Some characters can be lifted from the character table of $$Aut(G)$$. However I think there are many that cannot. For example there is a group of size $$1440$$ which is an extension of $$SL(2,9)$$ by an outer automoprhism of order $$2$$ and it appears to have a degree $$4$$ character. However I do not know how to compute this character.

Even easier than computing the full character table, I would just be curious how to find all the irrep dimensions for an almost quasisimple group.

TLDR: I'm curious how to find the character tables of almost quasisimple groups. Even figuring out all the irrep dimensions for (small) almost quasisimple groups would be interesting to me. I would be happy with a reference for the character tables, or a way to find them in GAP, or some direct method to find irrep dimensions for almost quasisimple groups, as long is works for a few interesting small groups, say of size less than 10,000.

Edit 1: very interesting point raised in the comments by Derek Holt. In response here is a (perhaps slightly vague but hopefully no longer incorrect) definition: A group $$G$$ is called almost quasisimple if it is the extension of a quasisimple group by some subgroup of its outer automorphism group, with the outer automorphisms acting in the natural way.

• The group that you describe as $Q.{\rm Out}(Q)$ does not always exist. For example when $Q = {\rm SL}(2,9) = 2.A_6$, we have ${\rm Out}(Q) = C_2 \times C_2$. But there is no group $Q.{\rm Out}(Q)$. More precisely, there are groups $2.S_6$ and $2.{\rm PGL}(2,9)$, but no group $2.M_{11}$. You can find information on the characters of a lot of almost quasisimple groups in the ATLAS, but it requires some expertise to extract it. Alternatively, they are all in the GAP character table library. Commented May 12, 2023 at 17:04
• @DerekHolt Thanks so much! This comment helped me answer math.stackexchange.com/a/4698409/758507 Just a heads up I think you made a small typo and meant to say $2.M_{10}$ Commented May 13, 2023 at 15:19
• Yes I meant to write $2.M_{10}$! Commented May 13, 2023 at 15:52

Since you have the [gap] tag: In GAP you could construct the group, calculate the automorphism group, and then its character table. Say, for $$SL(2,9)$$:

gap> g:=PerfectGroup(720,1);
A6 2^1
gap> IsomorphismGroups(g,SL(2,9))<>fail; # verify its the right one
true
gap> a:=AutomorphismGroup(g);
<group of size 1440 with 4 generators>
gap> p:=Image(IsomorphismPermGroup(a),a); # convert to permutation (since faster)
<permutation group of size 1440 with 4 generators>
gap> c:=CharacterTable(p);
CharacterTable( <permutation group of size 1440 with 4 generators> )
gap> Irr(c);; # completes character calculation
gap> Display(c); # show table
CT1

2  5  5  4  3  3  1  4  3  4  1  1  3   1
3  2  .  .  .  .  .  .  .  1  2  1  .   .
5  1  .  .  .  1  1  .  .  .  .  .  .   1

1a 2a 4a 4b 2b 5a 4c 8a 2c 3a 6a 8b 10a
2P 1a 1a 2a 2a 1a 5a 2a 4c 1a 3a 3a 4c  5a
3P 1a 2a 4a 4b 2b 5a 4c 8a 2c 1a 2c 8b 10a
5P 1a 2a 4a 4b 2b 1a 4c 8a 2c 3a 6a 8b  2b
7P 1a 2a 4a 4b 2b 5a 4c 8a 2c 3a 6a 8b 10a

X.1      1  1  1  1  1  1  1  1  1  1  1  1   1
X.2      1  1 -1 -1  1  1  1  1 -1  1 -1 -1   1
X.3      1  1 -1  1 -1  1  1 -1 -1  1 -1  1  -1
X.4      1  1  1 -1 -1  1  1 -1  1  1  1 -1  -1
X.5      9  1 -1 -1  1 -1  1 -1  3  .  .  1   1
X.6      9  1 -1  1 -1 -1  1  1  3  .  . -1  -1
X.7      9  1  1 -1 -1 -1  1  1 -3  .  .  1  -1
X.8      9  1  1  1  1 -1  1 -1 -3  .  . -1   1
X.9     10  2  2  .  .  . -2  .  2  1 -1  .   .
X.10    10  2 -2  .  .  . -2  . -2  1  1  .   .
X.11    16  .  .  . -4  1  .  .  . -2  .  .   1
X.12    16  .  .  .  4  1  .  .  . -2  .  .  -1
X.13    20 -4  .  .  .  .  .  .  .  2  .  .   .

• Ah I see so I can find character tables for $2.S_6$ using c:=CharacterTable("2.A6.2_1"); and for $2.PGL(2,9)$ using c:=CharacterTable("2.A6.2_2"); etc... Thanks! Commented Jun 10, 2023 at 13:16