We give three little examples for the conventions stated in ATLAS Tables, listing both the ATLAS format and the table displayed by GAP.
First, let G be the trivial group. We consider the cyclic group C6 of order 6. It can be viewed in several ways, namely
Situation 1. is shown here.
------- ------- ; @ ; ; @ 2 1 1 1 1 1 1
|| || || || 1 1 3 1 1 1 1 1 1
|| G || || G.2 || p power A
|| || || || p' part A 1a 3a 3b 2a 6a 6b
------- ------- ind 1A fus ind 2A 2P 1a 3b 3a 1a 3b 3a
------- ------- 3P 1a 1a 1a 2a 2a 2a
|| || || || chi1 + 1 : ++ 1
|| 3.G || || 3.G.2 || X.1 1 1 1 1 1 1
|| || || || ind 1 fus ind 2 X.2 1 1 1 -1 -1 -1
------- ------- 3 6 X.3 1 A /A 1 A /A
3 6 X.4 1 A /A -1 -A -/A
X.5 1 /A A 1 /A A
chi2 o2 1 : oo2 1 X.6 1 /A A -1 -/A -A
A = E(3)
= (-1+ER(-3))/2 = b3
X.1, X.2 extend c1.
X.3, X.4 extend the proxy character c2.
X.5, X.6 extend the not printed character with proxy c2.
The classes 1a, 3a, 3b are preimages of 1A,
and 2a, 6a, 6b are preimages of 2A.
Situation 2. is shown here.
------- ------- ; @ ; ; @ 2 1 1 1 1 1 1
|| || || || 1 1 3 1 1 1 1 1 1
|| G || || G.3 || p power A
|| || || || p' part A 1a 2a 3a 6a 3b 6b
------- ------- ind 1A fus ind 3A 2P 1a 1a 3b 3b 3a 3a
------- ------- 3P 1a 2a 1a 2a 1a 2a
|| || || || chi1 + 1 : +oo 1
|| 2.G || || 2.G.3 || X.1 1 1 1 1 1 1
|| || || || ind 1 fus ind 3 X.2 1 1 A A /A /A
------- ------- 2 6 X.3 1 1 /A /A A A
X.4 1 -1 1 -1 1 -1
chi2 + 1 : +oo 1 X.5 1 -1 A -A /A -/A
X.6 1 -1 /A -/A A -A
A = E(3)
= (-1+ER(-3))/2 = b3
X.1--X.3 extend c1, X.4--X.6 extend c2.
The classes 1a and 2a are preimages of 1A,
3a and 6a are preimages of the proxy class 3A,
and 3b and 6b are preimages of the not printed class with proxy 3A.
Situation 3. is shown here.
------- ; @ 2 1 1 1 1 1 1
|| || 1 3 1 1 1 1 1 1
|| G || p power
|| || p' part 1a 6a 3a 2a 3b 6b
------- ind 1A 2P 1a 3a 3b 1a 3a 3b
------- 3P 1a 2a 1a 2a 1a 2a
|| || chi1 + 1
|| 2.G || X.1 1 1 1 1 1 1
|| || ind 1 X.2 1 -1 1 -1 1 -1
------- 2 X.3 1 A /A 1 A /A
------- X.4 1 /A A 1 /A A
|| || chi2 + 1 X.5 1 -A /A -1 A -/A
|| 3.G || X.6 1 -/A A -1 /A -A
|| || ind 1
------- 3 A = E(3)
------- 3 = (-1+ER(-3))/2 = b3
|| ||
|| 6.G || chi3 o2 1
|| ||
------- ind 1
6
3
2
3
6
chi4 o2 1
X.1, X.2 correspond to c1, c2, respectively;
X.3, X.5 correspond to the proxies c3, c4, and
X.4, X.6 to the not printed characters with these proxies.
followers.
The factor fusion onto 3\.G is given by [ 1, 2, 3, 1, 2, 3 ],
that onto G\.2 by [ 1, 2, 1, 2, 1, 2 ].
Finally, situation 4. is shown here.
------- ------- ------- -------
|| || || || || || || ||
|| G || || G.2 || || G.3 || || G.6 ||
|| || || || || || || ||
------- ------- ------- -------
; @ ; ; @ ; ; @ ; ; @
1 1 1 1
p power A A AA
p' part A A AA
ind 1A fus ind 2A fus ind 3A fus ind 6A
chi1 + 1 : ++ 1 : +oo 1 :+oo+oo 1
2 1 1 1 1 1 1
3 1 1 1 1 1 1
1a 2a 3a 3b 6a 6b
2P 1a 1a 3b 3a 3b 3a
3P 1a 2a 1a 1a 2a 2a
X.1 1 1 1 1 1 1
X.2 1 -1 A /A -A -/A
X.3 1 1 /A A /A A
X.4 1 -1 1 1 -1 -1
X.5 1 1 A /A A /A
X.6 1 -1 /A A -/A -A
A = E(3)
= (-1+ER(-3))/2 = b3
The classes 1a, 2a correspond to 1A, 2A, respectively.
3a, 6a correspond to the proxies 3A, 6A,
and 3b, 6b to the not printed classes with these proxies.
The second example explains the fusion case; again, G is the trivial group.
------- ------- ; @ ; ; @ 3.G.2
|| || || || 1 1
|| G || || G.2 || p power A 2 1 . 1
|| || || || p' part A 3 1 1 .
------- ------- ind 1A fus ind 2A
------- ------- 1a 3a 2a
|| || || || X1 + 1 : ++ 1 2P 1a 3a 1a
|| 2.G || || 2.G.2 || 3P 1a 1a 2a
|| || || || ind 1 fus ind 2
------- ------- 2 2 X.1 1 1 1
------- ------- X.2 1 1 -1
|| || || X2 + 1 : ++ 1 X.3 2 -1 .
|| 3.G || || 3.G.2
|| || || ind 1 fus ind 2
------- 3 6.G.2
------- ------- 3
|| || || 2 2 1 1 2 2 2
|| 6.G || || 6.G.2 X3 o2 1 * + 3 1 1 1 1 . .
|| || ||
------- ind 1 fus ind 2 1a 6a 3a 2a 2b 2c
6 2 2P 1a 3a 3a 1a 1a 1a
3 3P 1a 2a 1a 2a 2b 2c
2
3 Y.1 1 1 1 1 1 1
6 Y.2 1 1 1 1 -1 -1
Y.3 1 -1 1 -1 1 -1
X4 o2 1 * + Y.4 1 -1 1 -1 -1 1
Y.5 2 -1 -1 2 . .
Y.6 2 1 -1 -2 . .
The tables of G, 2\.G, 3\.G, 6\.G and G\.2 are known from the first
example, that of 2\.G\.2 will be given in the next one.
So here we print only the GAP tables of 3\.G\.2 @ D6 and
6\.G\.2 @ D12.
In 3\.G\.2, the characters X.1, X.2 extend c1;
c3 and its non-printed partner fuse to give X.3,
and the two preimages of 1A of order 3 collapse.
In 6\.G\.2, Y.1--Y.4 are extensions of c1, c2,
so these characters are the inflated characters from 2\.G\.2
(with respect to the factor fusion [ 1, 2, 1, 2, 3, 4 ]).
Y.5 is inflated from 3\.G\.2
(with respect to the factor fusion [ 1, 2, 2, 1, 3, 3 ]),
and Y.6 is the result of the fusion of c4 and its non-printed
partner.
For the last example, let G be the elementary abelian group 22 of order 4. Consider the following tables.
------- ------- ; @ @ @ @ ; ; @
|| || || || 4 4 4 4 1
|| G || || G.3 || p power A A A A
|| || || || p' part A A A A
------- ------- ind 1A 2A 2B 2C fus ind 3A
------- -------
|| || || || chi1 + 1 1 1 1 : +oo 1
|| 2.G || || 2.G.3 || chi2 + 1 1 -1 -1 . + 0
|| || || || chi3 + 1 -1 1 -1 ||
------- ------- chi4 + 1 -1 -1 1 ||
ind 1 4 4 4 fus ind 3
2 6
chi5 - 2 0 0 0 : -oo 1
G.3
2 2 2 . .
3 1 . 1 1
1a 2a 3a 3b
2P 1a 1a 3b 3a
3P 1a 2a 1a 1a
X.1 1 1 1 1
X.2 1 1 A /A
X.3 1 1 /A A
X.4 3 -1 . .
A = E(3)
= (-1+ER(-3))/2 = b3
2.G 2.G.3
2 3 3 2 2 2 2 3 3 2 1 1 1 1
3 1 1 . 1 1 1 1
1a 2a 4a 4b 4c
2P 1a 1a 2a 1a 1a 1a 2a 4a 3a 6a 3b 6b
3P 1a 2a 4a 4b 4c 2P 1a 1a 2a 3b 3b 3a 3a
3P 1a 2a 4a 1a 2a 1a 2a
X.1 1 1 1 1 1
X.2 1 1 1 -1 -1 X.1 1 1 1 1 1 1 1
X.3 1 1 -1 1 -1 X.2 1 1 1 A A /A /A
X.4 1 1 -1 -1 1 X.3 1 1 1 /A /A A A
X.5 2 -2 . . . X.4 3 3 -1 . . . .
X.5 2 -2 . 1 1 1 1
X.6 2 -2 . A -A /A -/A
X.7 2 -2 . /A -/A A -A
A = E(3)
= (-1+ER(-3))/2 = b3
In the table of G\.3 @ A4, the characters c2, c3, and
c4 fuse, and the classes 2A, 2B and 2C collapse.
To get the table of 2\.G @ Q8, one just has to split the class 2A
and adjust the representative orders.
Finally, the table of 2\.G\.3 @ SL2(3) is given;
the class fusion corresponding to the injection 2\.G \hookrightarrow 2\.G\.3
is [ 1, 2, 3, 3, 3 ], and the factor fusion corresponding to the
epimorphism 2\.G\.3 ® G\.3 is [ 1, 1, 2, 3, 3, 4, 4 ].
(The beautiful LaTeX pictures that were part of the GAP 3 manual will be reintroduced as soon as the decision to use TeX for the manual will be revised.)
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