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In this answer to a question from a while back, it says that the Valentiner group is isomorphic to $PGL_3(\mathbb{F}_4)$. However, when I implement in sage:

G=PGL(3, 4)
G.order()

I get that the order is 60,480 , while the Wikipedia page gives the order as 1080. One paper I could find on the topic doesn't mention the isomorphism at all, instead saying it's a subgroup of SL(3, F) where F is $\mathbb{Q}(\exp(2\pi i/15))$ and under a projection to PGL(3, F) the Valentiner group maps to $A_6$. So are these groups really isomorphic, or is there an error in that old question, or an error in my implementation?

As a corollary, I'm looking for a way to implement the Valentiner group in sage, if anyone has any thoughts on how to do that?

Thank you, have a nice day.

FamisherCaterpillar
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    The Wikipedia page you link to does not say anywhere that the Valentiner group is isomorphic to ${\rm PGL}_3({\mathbb F}_4)$. It is clear from the description there that it is the quasisimple group $3.A_6$. – Derek Holt Nov 20 '24 at 17:15
  • For the orders see also here. Of course, $|PGL(3,4)|=\frac{(4^3-1)(4^3-4)(4^3-4^2)}{3}=60480$. – Dietrich Burde Nov 20 '24 at 17:17
  • @DerekHolt but it does say that at the answer in the first link OP posted, which is what they were saying. I'll point this out to the answerer, see if they want to edit their answer. – verret Nov 20 '24 at 19:55
  • @DerekHolt It is not clear to me that it is the quasisimplegroup $3.A_6$. In fact, I am not even sure what that period notation means. Can you please expand upon this comment? – FamisherCaterpillar Nov 21 '24 at 14:45
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    It is stated on the Wikipedia page that you linked to that the Valentiner group is the perfect triple cover of $A_6$. That is what the notation $3.A_6$ means. The same thing is stated at the beginning of this paper – Derek Holt Nov 21 '24 at 16:06
  • @DerekHolt I see, thank you. – FamisherCaterpillar Nov 23 '24 at 00:29
  • It's been long enough that I have no idea what my source was for that. Thanks for spotting the error. – Alexander Gruber Nov 27 '24 at 23:38

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