Talk:Dicyclic group

Dicylic groups are in general not nilpotent. Consider the case where A is cyclic of order 6. In this case the corresponding dicyclic group of order 12 is not nilpotent. — Preceding unsigned comment added by 141.51.166.3 (talk • contribs) 16:00, 7 July 2003‎

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Is the restriction that n > 1 actually necessary? It seems to me like the dicyclic group of order 4 should simply be the cyclic group. —Caesura^(t) 00:09, 24 June 2006 (UTC)

Different definitions

It appears different authors have different definitions. Some call all of these groups generalized quaternion groups. See my edits there for more info. RobHar (talk) 09:53, 7 July 2009 (UTC)

Connection between generalized quaternion group and dicyclic group

I think "More generally, when n is a power of 2, the dicyclic group is isomorphic to the generalized quaternion group." is wrong. Because the conditions a^(2n) = 1 (if I interpret a^(2n) = 1 so, that a has order 2n) and b^2 = a^n for dicyclic groups implies order(b) = 4 and so it has the same presentation as sie quaternion group. Where is my mistake? — Preceding unsigned comment added by 141.51.131.52 (talk) 16:01, 9 January 2013 (UTC)

Hmmmm.. this article isn't well referenced! I have a reference that defines the first part, "When n = 2, the dicyclic group is isomorphic to the quaternion group Q." but not on the second above, with the generalized quaternion group. Okay, Coxeter talks about it at [1] in Regular Complex Polytopes p. 82. But perhaps there are different definitions so maybe someone else can help clarify? Tom Ruen (talk) 23:52, 9 January 2013 (UTC)

Visualising this group

I've been trying to figure out how to visualise this group for a while. Clearly Dic_n is a variation of Dih_2n in which x is of order 4 instead of 2. But I think I've finally come up with something. Consider a regular n-gonal prism, and then powers of a are represented by rotations about the prism's central axis, and x by an unusual kind of flip in which the top face and bottom face are rotated about perpendicular axes.

What do you think? Can anyone here do better? — Smjg (talk) 20:06, 2 January 2014 (UTC)

I'm outside my domain here, but I see the Quaternion group, Q8, is Dic₂. And there are Cayley graphs at Quaternion_group#Cayley_graph comparing to Dih₂. So that's the best visualization I know. Tom Ruen (talk) 20:35, 2 January 2014 (UTC)

This page has Cayley graphs for Dic8, Dic12, Dic16, Dic20 [2] It seems like this is a better start. Tom Ruen (talk) 21:14, 2 January 2014 (UTC)

Oh yes, I'd almost forgotten about Cayley graphs. When they're readable they're a good way to visualise groups. Thank you! One of these days I'll try and get a few of them into this article.

BTW I've noticed that the source uses different notation from what we have here - there Dic12 means the dicyclic group of order 12, which we write as Dic₃.

This might also be the source we need to extend List of small groups a bit more! — Smjg (talk) 01:05, 3 January 2014 (UTC)