The existence of symmetric designs with parameters (189, 48, 12)
type: Item
instance of: scholarly article
title: The existence of symmetric designs with parameters (189, 48, 12)
zbMATH Open document ID: 0883.05015
author: Zvonimir Janko
published in: Journal of Combinatorial Theory. Series A
publication date: 1998-01-04T00:00:00Z
review text: Since 1994, I have searched for the possible existence of symmetric designs with parameters (189, 48, 12) (of order 36). Finally, the following result was proved: Let \(G\) be the wreathed product of a Frobenius group \(F_{21}\) of order 21 with a cyclic group \(\mathbb{Z}_2\) of order 2. Suppose that \(G\) acts as an automorphism group on a symmetric design with parameters (198, 48, 12). Then there exists (up to isomorphism) exactly one such design \(D\) which is self-dual. The full automorphism group of \(D\) is our group \(G\) of order 882 and acts in three orbits of lengths 21, 21, and 147 for points and blocks. The group \(G\) has three conjugacy classes of subgroups of order 7. The elements of order 7 in one class act fixed-point-free and in the other two classes have exactly 7 fixed points each. The group \(G\) has three conjugacy classes of subgroups of order 3. The elements of order 3 in one class act fixed-point-free and in the other two classes have exactly 3 and 6 fixed points, respectively. An involution in \(G\) has exactly 29 fixed points. The 2-rank of \(D\) is 63.
Mathematics Subject Classification ID: 05B05
zbMATH DE Number: 1093107
zbMATH Keywords: existence of symmetric designs
zbMATH Keywords: automorphism group
zbMATH Keywords: conjugacy classes of subgroups
zbMATH Keywords: fixed points
MaRDI profile type: Publication
full work available at URL: jcta.1997.2806
OpenAlex ID: W2080893328
cites work: Q4028438
DOI: 10.1006/JCTA.1997.2806
description: scientific article; zbMATH DE number 1093107