Biggs, Rory2021-07-092021-07-0920212021*S2021http://hdl.handle.net/2263/80760Dissertation (MSc)--University of Pretoria, 2021.In this dissertation we study a five-dimensional two-step nilpotent matrix Lie group. Some basic group properties are investigated. The structure of the Lie algebra’s subspaces is investigated; a complete set of scalar invariants is given for the Lie algebra’s subspace structure. Following this, we classify the left-invariant sub-Riemannian structures on this Lie group up to isometry. The normal geodesics of the rank three left-invariant sub-Riemannian structure are determined as an illustrative case.en© 2019 University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria.MathematicsUCTDDissertation