ATLAS: Harada-Norton group HN
Order = 273030912000000 = 2^{14}.3^{6}.5^{6}.7.11.19.
Mult = 1.
Out = 2.
Standard generators
Standard generators of the Harada-Norton group HN are a
and b where
a is in class 2A,
b is in class 3B,
ab has order 22,
and ababb has order 5.
Standard generators of its automorphism group HN:2 are
c
and d where
c is in class 2C,
d is in class 5A,
and cd has order 42.
A pair of elements conjugate to
(a,b)
may be obtained as
a' = (cd)^{-3}(cdcdcdcddcdcddcdd)^{10}(cd)^3,
b' = (cdd)^{8}(cdcdd)^5(cdd)^{10}.
Black box algorithms
To find standard generators for HN:
- Find any element of order 14 or 22. It powers up to a 2A-element.
- Find any element of order 9. This powers up to a 3B-element, y, say.
- Find a conjugate a of x and a conjugate b of y, whose product has order 22,
and whose commutator has order 5.
To find standard generators for HN.2:
- Find any element of order 14, 18 or 22. It powers up to a 2C-element.
- Find any element of order 35 or 60. This powers up to a 5A-element, y, say.
- Find a conjugate a of x and a conjugate b of y, whose product has order 42.
Representations
The representations of HN available are
- a and
b as
permutations on 1140000 points.
- Some representations in characteristic 2
- a and
b as
132 x 132 matrices over GF(4).
- a and
b as
132 x 132 matrices over GF(4).
- a and
b as
133 x 133 matrices over GF(4) - indecomposable with constituents 132.1.
- a and
b as
760 x 760 matrices over GF(2).
- a and
b as
2650 x 2650 matrices over GF(4).
- a and
b as
133 x 133 matrices over GF(9) - kindly provided by J.N.Bray.
- a and
b as
133 x 133 matrices over GF(9).
- a and
b as
760 x 760 matrices over GF(3) - kindly provided by Jürgen Müller.
- a and
b as
133 x 133 matrices over GF(5).
- a and
b as
626 x 626 matrices over GF(5).
- a and
b as
627 x 627 matrices over GF(5) - uniserial 626.1.
- a and
b as
133 x 133 matrices over GF(49).
- a and
b as
133 x 133 matrices over GF(49).
- a and
b as
133 x 133 matrices over GF(11).
- a and
b as
133 x 133 matrices over GF(11).
- a and
b as
133 x 133 matrices over GF(19).
- a and
b as
133 x 133 matrices over GF(19).
The representations of HN:2 available are
- c and
d as
264 x 264 matrices over GF(2).
- c and
d as
133 x 133 matrices over GF(5).
The maximal subgroups of HN are
The maximal subgroups of HN:2 are
Conjugacy classes
A set of generators for the maximal cyclic subgroups can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
Problems of algebraic conjugacy are not yet dealt with.
Return to main ATLAS page.
Last updated 1.12.99
R.A.Wilson, R.A.Parker and J.N.Bray.