Perfect Groups with faithful complex irreps

finite-groupsgroup-theoryrepresentation-theory

Is it true that a perfect finite group $ G $ has a faithful complex irrep if and only if the center of $ G $ is cyclic?

The corresponding fact is true for quasisimple finite $ G $

Does every quasisimple finite group have a faithful complex irrep?

And certainly it is necessary to have cyclic center for a finite group to have a faithful complex irrep.

So the question is really: Does every perfect group with cyclic center have a faithful complex irrep?

An example of a non perfect group with cyclic center which has no faithful complex irrep is given here

https://mathoverflow.net/q/57129/387190

Best Answer

No. Take PerfectGroup(245760,4) (this requires GAP 4.12 :-) ), generated by

(1,11,10,8,14)(2,4,12,15,16)(3,13,5,6,7)(17,25,28,21,23)(18,19,24,30,29)(20,27,22,26,31)(33,45,39,41,35)(34,47,42,36,46)(38,40,48,44,43),
(1,13,12,16,9)(2,7,5,6,14)(3,10,15,8,4)(17,21,23,27,30)(18,24,32,22,28)(19,20,29,25,31)(33,40,45,36,37)(34,38,48,39,41)(35,46,42,44,47),
(1,14,2)(3,12,16)(4,9,7)(5,15,8)(6,13,10)(17,31,18)(19,28,29)(20,25,23)(21,32,24)(22,30,26)(33,39,38)(34,43,47)(35,36,45)(37,42,48)(41,46,44)

It has a structure $(2^4\times 2^4\times 2^4):A_5$, where $2^4$ is the irreducible $A_5$-module of dimension 4 that does not come from the permutation representation. It has 21 minimal normal subgroups (all of type $2^4$). The solvable radical $(2^4)^3$ is composed from 21 classes of order 15 (which each generate one of the minimal normal subgroups) and 63 classes of order 60 (that each generate a normal subgroup of order 256), and of course the identity.

This group has trivial center (thus cyclic) but has no faithful irreducible representation.

This is the smallest possible example, and the only one of order 245760. (There are further ones in order 491520, not neccessarily just extensions with this one.)

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