Database of Soluble Groups

The database solgps contains pc-presentations of soluble groups constructed by S.P. Glasby and translated from Cayley into Magma by S. Collins. Specifically, it contains 13 functions G1, G2, ..., G11, B26, F2332 which construct groups which are briefly described below. (January 1994) The group G designated NAME is created by typing

         G := NAME();

G1

The group BO.7^(2+1).13^(14+1) of order 2^4 * 3 * 7^3 * 13^15 where BO is the binary octahedral group of order 48 and 7^(2+1) and 13^(14+1) denote extraspecial groups of prime exponent.

G2

S4.2^5 of order 768

G3

A group of order 2^11*3^13 and derived length 8. See S.P. Glasby and R.B. Howlett, Extraspecial towers and Weil representations, J. Algebra (1992), 236-260 for more details.

G4

S3 wr S3 where |G| = 2^4*3^4 and degree 9.

G5

GL(2,3).3^2.5^9 where |G| = 2^4*3^3*5^9.

G6

S4 wr S4 wr S4 where |G| = 2^63*3^21 and degree 4^3.

G7

Aut(F).F*.F+ where F=GF(2^11). |G| = 11*23*89*2^11.

G8

Aut(F).F*.F+ where F=GF(3^10). |G| = 2^4*5*11^2*61*3^10

G9

The Borel subgroup of GL(5, 3) which has order 2^5*3^10.

G10

The Borel subgroup of GL(4, 8) which has order 2^18*7^4.

G11

The group Aut(F).F*.F+ where F=GF(2^13). |G| = 13*8191*2^13.

B26

The two generator Burnside group B(2,6) of exponent 6 and order 2^28*3^25.

F2332

If F denotes the free gp of rank 2, then below is a pc-presentation for the gp G = F/(F^2)^3 x F/(F^3)^2 of exponent 6 and order 2^30*3^28. The subgp <G.1*G.4*G.28*G.31,G.2*G.6*G.29*G.32*G.58> is a 2-generator subgp of exponent 6 and order 2^28*3^25, and hence it is B(2,6). A pc-presentation for B(2,6) may be found above in function B26. Note that <G.1,...,G.27> and <G.28,...,G.58> are the two direct factors.

Example

> G := G2();

Constructing a group G = S4.2^5 of order 768.

A group with factorized order [ <2, 8>, <3, 1> ]
has been constructed