6. Examples and Notations

In [1,3], two different methods for the Todd-Coxeter enumeration process of cosets, namely Felsch- and HLT-style coset enumeration procedures, were studied and the results for some examples tabulated. We use these examples and compare the results achieved there to the results we obtained by using the frameworks and strategies presented in the previous sections. The examples are based on the groups which are given together with their presentation in Table 1 on page . The examples themselves together with the indeces are given in Table 3 (see page ). Inverse elements are given by capital letters. The results will be summarized in the following sections and are tabulated in the appendix (see the respective sections).

Table 1: Presentations of the Groups used for the Examples

Group	Presentation
$E_1$	$TrtRR = RsrSS = StsTT = 1$
$(2, 5, 7; 2)$	$a^2 = b^5 = (ab)^7 = [a, b]^2 = 1$
$G^{3, 7, 17}$	$a^3 = b^7 = c^{17} = (ab)^2 = (bc)^2 = (ca)^2 = (abc)^2 = 1$
$PSL_2(11)$	$a^{11} = b^2 = (ab)^3 = (a^4bA^5b)^2 = 1$
$(2, 3, 7; 7)$	$a^2 = b^3 = (ab)^7 = [a, b]^7 = 1$
$M_{11}^{(1)}$	$a^{11} = b^5 = c^4 = (a^4c^2)^3 = (bc^2)^2 = (abc)^3 = BabA^4 = CbcB^2 = 1$
$(8, 7 \vert 2, 3)$	$a^8 = b^7 = (ab)^2 = (Ab)^3 = 1$
$Neu$	$a^3 = b^3 = c^3 = (ab)^5 = (Ab)^5 = (ac)^4 = (aC)^4 = aBabCacaC = (bc)^3 = (Bc)^4 = 1$
$Cam(3)$	$r^2srsR^3S = s^2rsrS^3R = 1$
$G^{3,7,16}$	$a^3 = b^7 = c^{16} = (ab)^2 = (bc)^2 = (ca)^2 = (abc)^2 = 1$
$G(2,4)$	$BAbaBabA^2 = ABabAbaB^4 = 1$
$G(2,6)$	$BAbaBabA^2 = ABabAbaB^6 = 1$
$G(3,3)$	$BAbaBabA^3 = ABabAbaB^3 = 1$

We will use ''$n/m$'' to denote the number of cosets defined where $n$ is the maximal number and $m$ the total number of cosets defined.