ATLAS: Unitary group U_{6}(2), Fischer group Fi_{21}
The information on this page was prepared with help from Ibrahim Suleiman.
Order = 9196830720 = 2^{15}.3^{6}.5.7.11.
Mult = 2^{2} × 3.
Out = S_{3}.
Standard generators
U6(2) and covers
 Standard generators of U6(2) are
a
and b where
a is in class 2A,
b has order 7,
ab has order 11
and abb has order 18.

Standard generators of the double cover 2.U6(2) are preimages
A
and B where
B has order 7, AB has order 11 and ABBB has order 11.

Standard generators of the triple cover 3.U6(2) are preimages
A
and B where
A has order 2
and B has order 7.

Standard generators of the sextuple cover 6.U6(2) are preimages
A
and B where
A has order 2,
B has order 7,
AB has order 33 and ABBB has order 11.

Standard generators of 2^{2}.U6(2) are preimages
A
and B where
B has order 7 and AB has order 11.

Standard generators of (2^{2} × 3).U6(2) are preimages
A
and B where
A has order 2,
B has order 7 and
AB has order 33.
U6(2):2 and covers

Standard generators of U6(2):2 are
c
and d where
c is in class 2D,
d is in class 6J,
and cd has order 11.

Standard generators of the double cover 2.U6(2):2 are preimages
C
and D where
CD has order 11.

Standard generators of the triple cover 3.U6(2):2 are preimages
C
and D where
CD has order 11.

Standard generators of the sixfold cover 6.U6(2):2 are preimages
C
and D where
CD has order 11.

Standard generators of 2^{2}.U6(2):2 are preimages
C
and D where
C has order 2,
D has order 6,
and CDCDCDCDCDDCDCDDCDDCDD has order 7.
U6(2):3 and covers

Standard generators of U6(2):3 are
e
and f where
e is in class 3D,
f has order 11,
ef has order 21,
and eff has order 18.
 Standard generators of 3.U6(2):3 are preimages
E
and F where
F has order 11.
 Standard generators of 2^{2}.U6(2):3 are preimages
E
and F where
F has order 11.
U6(2):S3 and covers
 Standard generators of U6(2):S3 are
g
and h where
g is in class 2D,
h is in class 6J',
and gh has order 21.
 Standard generators of 3.U6(2):S3 are preimages
G
and H. No extra conditions are required, as all such pairs are
automorphic.
 Standard generators of 2^{2}.U6(2):S3 are preimages
G
and H where
Automorphisms
An automorphism of order 3 can be obtained by mapping
(a,b) to
((abb)^4a(abb)^4,
(abababbab)^1babababbab).
An automorphism of order 2 can be obtained by mapping
(a,b) to
(a,
b^1).
Representations
U6(2) and covers

The representations of U6(2) available are
 a and
b as
permutations on 672 points.
 a and
b as
permutations on 693 points.
 a and
b as
permutations on 891 points.
 a and
b as
permutations on 1408 points.
 a and
b as
permutations on 1408 points.
 a and
b as
permutations on 1408 points.
 a and
b as
permutations on 6237 points.
 a and
b as
permutations on 20736 points.
 a and
b as
permutations on 59136 points.
 a and
b as
20 × 20 matrices over GF(2).
 a and
b as
21 × 21 matrices over GF(3).
 a and
b as
22 × 22 matrices over GF(5).
 The representations of 2.U6(2) available are

A and
B as
permutations on 1344 points.
 A and
B as
56 × 56 matrices over GF(3).
 A and
B as
56 × 56 matrices over GF(7).
 The representations of 3.U6(2) available are
 A and
B as
6 × 6 matrices over GF(4)  the natural representation.
 A and
B as
21 × 21 matrices over GF(7).

A and
B as
permutations on 2016 points.
 The representations of 6.U6(2) available are
 A and
B as
120 × 120 matrices over GF(7).
U6(2):2 and covers
 The representations of U6(2):2 available are
 c and
d as
20 × 20 matrices over GF(2).
 c and
d as
permutations on 1408 points.
 The representation of 2.U6(2):2 available is
 C and
D as
56 × 56 matrices over GF(3).
 The representation of 3.U6(2):2 available is
 C and
D as
12 × 12 matrices over GF(2).
 The representation of 6.U6(2):2 available is
 C and
D as
240 × 240 matrices over GF(7).
 The representations of 2^{2}.U6(2):2 available are
 C and
D as
112 × 112 matrices over GF(3).
 C and
D as
240 × 240 matrices over GF(3).
U6(2):3 and covers
 The representation of U6(2):3 available is
 e and
f as
20 × 20 matrices over GF(2).
 The representation of 3.U6(2):3 available is
 E and
F as
6 × 6 matrices over GF(4).
 The representation of 2^{2}.U6(2):3 available is
 E and
F as
168 × 168 matrices over GF(3).
 The representation of (2^{2} × 3).U6(2):3 available is
 E and
F as
360 × 360 matrices over GF(7).
U6(2):S3 and covers
 The representation of U6(2):S3 available is
 g and
h as
20 × 20 matrices over GF(2).
 The representation of 3.U6(2).S3 available is
 G and
H as
12 × 12 matrices over GF(2).
 The representation of 2^{2}.U6(2):S3 available is
 G and
H as
168 × 168 matrices over GF(3).
 The representation of (2^{2} × 3).U6(2):S3 available is
 G and
H as
720 × 720 matrices over GF(7).
Maximal subgroups
The maximal subgroups of U6(2) include:
Return to main ATLAS page.
Last updated 26.05.00
R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk