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EXCEPTIONAL GROUPS OF ORDER 
$p^6$ FOR PRIMES 
$p\geq 5$
Published online by Cambridge University Press: 24 February 2025
Abstract
The minimal faithful permutation degree 
$\mu (G)$ of a finite group G is the least integer n such that G is isomorphic to a subgroup of the symmetric group 
$S_n$. If G has a normal subgroup N such that 
$\mu (G/N)> \mu (G)$, then G is exceptional. We prove that the proportion of exceptional groups of order 
$p^6$ for primes 
$p \geq 5$ is asymptotically zero. We identify 
$(11p+107)/2$ such groups and conjecture that there are no others.
MSC classification
Primary:
20D15: Nilpotent groups, $p$-groups
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
O’Brien was supported by the Marsden Fund of New Zealand Grant 23-UOA-080 and by a Research Award of the Alexander von Humboldt Foundation. Prajapati acknowledges the Science and Engineering Research Board, Government of India, for financial support through grant MTR/2019/000118. Udeep thanks the Indian Institute of Science Education and Research Mohali for his Postdoctoral Fellowship.
References
Behravesh, H. and Ghaffarzadeh, G., ‘Minimal degree of faithful quasi-permutation representations of
$p$
-groups’, Algebra Colloq. 18 (2011), 843–846.CrossRefGoogle Scholar
$p$
-groups’, Algebra Colloq. 18 (2011), 843–846.CrossRefGoogle ScholarBesche, H. U., Eick, B. and O’Brien, E. A., ‘A millennium project: constructing small groups’, Internat. J. Algebra Comput. 12(05) (2002), 623–644.CrossRefGoogle Scholar
Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system I: the user language’, J. Symbolic Comput. 24(3–4) (1997), 235–265.CrossRefGoogle Scholar
Britnell, J. R., Saunders, N. and Skyner, T., ‘On exceptional groups of order
${p}^5$
’, J. Pure Appl. Algebra 221(11) (2017), 2647–2665.CrossRefGoogle Scholar
${p}^5$
’, J. Pure Appl. Algebra 221(11) (2017), 2647–2665.CrossRefGoogle ScholarCannon, J. J., Holt, D. F. and Unger, W. R., ‘The use of permutation representations in structural computations in large finite matrix groups’, J. Symbolic Comput. 95 (2019), 26–38.CrossRefGoogle Scholar
Chamberlain, R., ‘Minimal exceptional
$p$
-groups’, Bull. Aust. Math. Soc. 98(3) (2018), 434–438.CrossRefGoogle Scholar
$p$
-groups’, Bull. Aust. Math. Soc. 98(3) (2018), 434–438.CrossRefGoogle ScholarEasdown, D. and Hendriksen, M., ‘Minimal permutation representations of semidirect products of groups’, J. Group Theory 19(6) (2016), 1017–1048.CrossRefGoogle Scholar
Easdown, D. and Praeger, C. E., ‘On minimal faithful permutation representations of finite groups’, Bull. Aust. Math. Soc. 38(2) (1988), 207–220.CrossRefGoogle Scholar
Elias, B., Silberman, L. and Takloo-Bighash, R., ‘Minimal permutation representations of nilpotent groups’, Exp. Math. 19(1) (2010), 121–128.CrossRefGoogle Scholar
Franchi, C., ‘On minimal degrees of permutation representations of abelian quotients of finite groups’, Bull. Aust. Math. Soc. 84(3) (2011), 408–413.CrossRefGoogle Scholar
Girnat, B., ‘Die Klassifikation der Gruppen bis zur Ordnung
${p}^5$
’, Preprint, 2018, arXiv:1806.07462.Google Scholar
${p}^5$
’, Preprint, 2018, arXiv:1806.07462.Google ScholarHolt, D. F. and Walton, J., ‘Representing the quotient groups of a finite permutation group’, J. Algebra 248(1) (2002), 307–333.CrossRefGoogle Scholar
James, R., ‘The groups of order
${p}^6$
(
$p$
an odd prime)’, Math. Comp. 34(150) (1980), 613–637.Google Scholar
${p}^6$
(
$p$
an odd prime)’, Math. Comp. 34(150) (1980), 613–637.Google ScholarJohnson, D. L., ‘Minimal permutation representations of finite groups’, Amer. J. Math. 93(4) (1971), 857–866.CrossRefGoogle Scholar
Karpilovsky, G. I., ‘The least degree of a faithful representation of abelian groups’, Vestnik Khar’kov Gos. Univ. 53 (1970), 107–115.Google Scholar
Kovács, L. G. and Praeger, C. E., ‘On minimal faithful permutation representations of finite groups’, Bull. Aust. Math. Soc. 62(2) (2000), 311–317.CrossRefGoogle Scholar
Lemieux, S., ‘Finite exceptional
$p$
-groups of small order’, Comm. Algebra 35(6) (2007), 1890–1894.CrossRefGoogle Scholar
$p$
-groups of small order’, Comm. Algebra 35(6) (2007), 1890–1894.CrossRefGoogle ScholarLewis, M. L., ‘Character tables of groups where all nonlinear irreducible characters vanish off the center’, in: Ischia Group Theory 2008 (eds. Bianchi, M., Longobardi, P., Maj, M. and Scoppola, C. M.) (World Scientific, Singapore, 2009), 174–182.CrossRefGoogle Scholar
Lewis, M. L., ‘Generalizing Camina groups and their character tables’, J. Group Theory 12(2) (2009), 209–218.CrossRefGoogle Scholar
Newman, M. F., O’Brien, E. A. and Vaughan-Lee, M. R., ‘Groups and nilpotent Lie rings whose order is the sixth power of a prime’, J. Algebra 278(1) (2004), 383–401.CrossRefGoogle Scholar
Newman, M. F., O’Brien, E. A. and Vaughan-Lee, M. R., ‘Presentations for the groups of order
${p}^6$
for prime
$p\ge 7$
’, Preprint, 2023, arXiv:2302.02677v2.Google Scholar
${p}^6$
for prime
$p\ge 7$
’, Preprint, 2023, arXiv:2302.02677v2.Google ScholarO’Brien, E. A., Prajapati, S. K. and Udeep, A., ‘Minimal degrees for faithful permutation representations of groups of order
${p}^6$
where
$p$
is an odd prime’, J. Algebraic Combin. 60(2) (2024), 319–388.CrossRefGoogle Scholar
${p}^6$
where
$p$
is an odd prime’, J. Algebraic Combin. 60(2) (2024), 319–388.CrossRefGoogle ScholarO’Brien, E. A., Prajapati, S. K. and Udeep, A., ‘Minimal degree permutation representations for groups of order
${p}^6$
’. https://github.com/eamonnaobrien/Minimal-Degree.Google Scholar
${p}^6$
’. https://github.com/eamonnaobrien/Minimal-Degree.Google ScholarPrajapati, S. K. and Udeep, A., ‘Minimal faithful quasi-permutation representation degree of
$p$
-groups with cyclic center’, Proc. Indian Acad. Sci. Math. Sci. 133 (2023), Article no. 38.CrossRefGoogle Scholar
$p$
-groups with cyclic center’, Proc. Indian Acad. Sci. Math. Sci. 133 (2023), Article no. 38.CrossRefGoogle ScholarPrajapati, S. K. and Udeep, A., ‘On faithful quasi-permutation representations of VZ-groups and Camina
$p$
-groups’, Comm. Algebra 51(4) (2023), 1431–1446.CrossRefGoogle Scholar
$p$
-groups’, Comm. Algebra 51(4) (2023), 1431–1446.CrossRefGoogle ScholarPrajapati, S. K. and Udeep, A., ‘Minimal degrees for faithful permutation representations of groups of order
${p}^5$
where
$p$
is an odd prime’, Preprint, 2024, https://github.com/ayushudeep/MinimalDegreeGroupsofOrderp5.Google Scholar
${p}^5$
where
$p$
is an odd prime’, Preprint, 2024, https://github.com/ayushudeep/MinimalDegreeGroupsofOrderp5.Google ScholarWong, W. J., ‘Linear groups analogous to permutation groups’, J. Aust. Math. Soc. 3(2) (1963),180–184.CrossRefGoogle Scholar