The Fischer-Clifford matrices and character table of the maximal subgroup $2^9{:}(L_3(4){:}S_3)$ of $U_6(2){:}S_3$

Document Type : Research Paper

Author

Department of‎ ‎Mathematics‎, ‎Faculty of Military Science, Stellenbosch‎ University‎‎, ‎Private Bag X2, Saldanha‎, ‎7395‎, ‎South Africa.

Abstract

The full automorphism group of $U_6(2)$ is a group of the form $U_6(2){:}S_3$. The group $U_6(2){:}S_3$ has a maximal subgroup $2^9{:}(L_3(4){:}S_3)$ of order 61931520. In the present paper, we determine the Fischer-Clifford matrices (which are not known yet) and hence compute the character table of the split extension $2^9{:}(L_3(4){:}S_3)$.

Keywords

Main Subjects


F‎. ‎Ali‎, ‎Fischer-Clifford Theory for Split and Non-Split Group Extensions‎, ‎PhD Thesis‎, ‎University of Natal‎, ‎2001‎.
F‎. ‎Ali and J‎. ‎Moori‎, ‎The Fischer-Clifford matrices of a maximal subgroup of $Fi'_24$‎, Represent‎. ‎Theory 7 (2003) 300--321‎.
F‎. ‎Ali and J‎. ‎Moori‎, ‎The Fischer-Clifford matrices and character table of a maximal subgroup of $Fi_24$‎, Algebra Colloq. 17 (2010)‎, ‎no‎. ‎3‎, ‎389--414‎.
F‎. ‎Ali and J‎. ‎Moori‎, ‎Fischer-Clifford matrices and character table of‎ ‎the split extension $2^6:S_8$‎, Hacet‎. ‎J‎. ‎Math‎. ‎Stat. 43 (2014)‎, ‎no‎. ‎2‎, ‎153--171‎
W‎. ‎Bosma and J‎. ‎J‎. ‎Canon‎, ‎Handbook of Magma Functions‎, ‎Department of Mathematics‎, ‎University of Sydney‎, ‎1994‎.
J‎. ‎J‎. ‎Cannon‎, ‎An introduction to the group theory language‎, Cayley.Computational group theory (Durham‎, ‎1982)‎, ‎145--183‎, ‎Academic Press‎, ‎London‎, ‎1984‎.
J‎. ‎H‎. ‎Conway‎, ‎R‎. ‎T‎. ‎Curtis‎, ‎S‎. ‎P‎. ‎Norton‎, ‎R‎. ‎A‎. ‎Parker‎, ‎and R‎. ‎A‎. ‎Wilson‎, ‎Atlas of Finite Groups‎,
‎Oxford University Press‎, ‎Eynsham‎, ‎1985‎.
B‎. ‎Fischer‎, ‎Clifford-matrices‎, ‎Representation theory of finite groups and finite-dimensional algebras (Bielefeld‎, ‎1991)‎, ‎1--16‎, ‎Progr‎. ‎Math.‎, ‎95‎, ‎Birkh\"auser‎, ‎Basel‎, ‎1991‎.
R‎. ‎L‎. ‎Fray and A‎. ‎L‎. ‎Prins‎, ‎The Fischer-Clifford matrices of the inertia group $2^7:O^-(6,2)$ of a maximal subgroup $2^7:SP(6,2)$ in $Sp_8(2)$‎, Int‎. ‎J‎. ‎Group Theory 2 (2013)‎, ‎no‎. ‎3‎, ‎19--38‎.
R‎. ‎L‎. ‎Fray and A‎. ‎L‎. ‎Prins‎, ‎The Fischer-Clifford matrices of an extension group of the form $2^7:(2^5:S_6)$‎, Int‎. ‎J‎. ‎Group Theory 3 (2014)‎, ‎no‎. ‎2‎, ‎21--39‎.
R‎. ‎L‎. ‎Fray and A‎. ‎L‎. ‎Prins‎, ‎On the inertia groups of the maximal subgroup $2^7:SP(6,2)$ in $Aut(Fi_22)$‎, Quaest‎. ‎Math. 38 (2015)‎, ‎no‎. ‎1‎, ‎83--102‎.
D‎. ‎Gorenstein‎, ‎Finite Groups‎, ‎Harper and Row Publishers‎, ‎New York‎, ‎1968‎.
G‎. ‎Karpilovsky‎, ‎Group Representations‎: ‎Introduction to Group Representations and Characters‎, ‎North-Holland Publishing Co.‎, ‎Amsterdam‎, ‎1992‎.
J‎. ‎Moori‎, ‎On the Groups $G^+$ and $\overlineG$ of the Forms $2^10:M_22$ and $2^10:\overlineM_22$‎, ‎PhD Thesis‎, ‎University of Birmingham‎, ‎1975‎.
J‎. ‎Moori‎, ‎On certain groups associated with the smallest Fischer group‎, J‎. ‎Lond‎. ‎Math‎. ‎Soc‎. ‎(2) 23 (1981)‎, ‎no‎. ‎1‎, ‎61--67‎.
J‎. ‎Moori and Z‎. ‎E‎. ‎Mpono‎, ‎The Fischer-Clifford matrices of the group $2^6:SP_6(2)$‎, Quaest‎. ‎Math. 22 (1999)‎, ‎no‎. ‎2‎, ‎257--298‎.
J‎. ‎Moori and Z‎. ‎E‎. ‎Mpono‎, ‎Fischer-Clifford matrices and the character table of a maximal subgroup of $\overlineF_22$‎, Int‎. ‎J‎. ‎Math‎. ‎Game Theory and Algebra 10 (2000)‎, ‎no‎. ‎1‎, ‎1--12‎.
Z‎. ‎Mpono‎, ‎Fischer-Clifford Theory and Character Tables of Group‎ ‎Extensions‎, ‎PhD Thesis‎, ‎University of Natal‎, ‎1998‎.
H‎. ‎Pahlings‎, ‎The character table of $2^1+22^\cdotCo_2$‎, J‎. ‎Algebra 315 (2007)‎, ‎no‎. ‎1‎, ‎301--325‎.
‎A‎. ‎L‎. ‎Prins‎, ‎Fischer-Clifford Matrices and Character Tables of Inertia Groups of Maximal Subgroups of Finite Simple Groups of Extension Type‎, ‎PhD Thesis‎, ‎University of the Western Cape‎, ‎2011‎.
‎The GAP Group‎, ‎\em GAP‎ ‎--Groups,‎ ‎Algorithms‎, ‎and Programming‎, ‎Version 4.6.3; 2013‎. ‎(http://www.gap-system.org)‎.
T‎. ‎T‎. ‎Seretlo‎, ‎Fischer Clifford Matrices and Character Tables of Certain Groups Associated with Simple Groups $O^+_10(2)$‎, ‎$HS$ and $Ly$‎, ‎PhD Thesis‎, ‎University of KwaZulu Natal‎, ‎2011‎.
R‎. ‎A‎. ‎Wilson‎, ‎P‎. ‎Walsh‎, ‎J‎. ‎Tripp‎, ‎I‎. ‎Suleiman‎, ‎S‎. ‎Rogers‎, ‎R‎. ‎Parker‎, ‎S‎. ‎Norton‎, ‎S‎. ‎Nickerson‎, ‎S‎. ‎Linton‎, ‎J‎. ‎Bray and R‎. ‎Abbot‎, ‎\em ATLAS of Finite Group Representations‎, \http://brauer.maths.qmul.ac.uk/Atlas/v3/‎.