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Janko group J4 AI simulator
(@Janko group J4_simulator)
Hub AI
Janko group J4 AI simulator
(@Janko group J4_simulator)
Janko group J4
In the area of modern algebra known as group theory, the Janko group J4 is a sporadic simple group of order
J4 is one of the 26 Sporadic groups. Zvonimir Janko found J4 in 1975 by studying groups with an involution centralizer of the form 21 + 12.3.(M22:2). Its existence and uniqueness was shown using computer calculations by Simon P. Norton and others in 1980. It has a modular representation of dimension 112 over the finite field with 2 elements and is the stabilizer of a certain 4995 dimensional subspace of the exterior square, a fact which Norton used to construct it, and which is the easiest way to deal with it computationally. Aschbacher & Segev (1991) and Ivanov (1992) gave computer-free proofs of uniqueness. Ivanov & Meierfrankenfeld (1999) and Ivanov (2004) gave a computer-free proof of existence by constructing it as an amalgams of groups 211:M24, 210:L5(2), and 23+12.(L3(2)xS5) over their intersections.
The Schur multiplier and the outer automorphism group are both trivial.
Since 37 and 43 are not supersingular primes, J4 cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.
The smallest faithful complex representation has dimension 1333; there are two complex conjugate representations of this dimension. The smallest faithful representation over any field is a 112 dimensional representation over the field of 2 elements.
The smallest permutation representation is on 173067389=112 · 29 · 31 · 37 · 43 points, with point stabilizer of the form 211:M24. This permutation representation has rank 7; the suborbit lengths are 1, 15180=22 · 3 · 5 · 11 · 23, 28336=24 · 7 · 11 · 23, 3400320=27 · 3 · 5 · 7 · 11 · 23, 32643072=211 · 32 · 7 · 11 · 23, 54405120=211 · 3 · 5 · 7 · 11 · 23, and 82575360=218 · 32 · 7. The points can be identified with certain "special vectors" in the 112 dimensional representation.
The degrees of irreducible representations of the Janko group J4 are 1, 1333, 1333, 299367, 299367, ... (sequence A003907 in the OEIS).
It has a presentation in terms of three generators a, b, and c as
Janko group J4
In the area of modern algebra known as group theory, the Janko group J4 is a sporadic simple group of order
J4 is one of the 26 Sporadic groups. Zvonimir Janko found J4 in 1975 by studying groups with an involution centralizer of the form 21 + 12.3.(M22:2). Its existence and uniqueness was shown using computer calculations by Simon P. Norton and others in 1980. It has a modular representation of dimension 112 over the finite field with 2 elements and is the stabilizer of a certain 4995 dimensional subspace of the exterior square, a fact which Norton used to construct it, and which is the easiest way to deal with it computationally. Aschbacher & Segev (1991) and Ivanov (1992) gave computer-free proofs of uniqueness. Ivanov & Meierfrankenfeld (1999) and Ivanov (2004) gave a computer-free proof of existence by constructing it as an amalgams of groups 211:M24, 210:L5(2), and 23+12.(L3(2)xS5) over their intersections.
The Schur multiplier and the outer automorphism group are both trivial.
Since 37 and 43 are not supersingular primes, J4 cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.
The smallest faithful complex representation has dimension 1333; there are two complex conjugate representations of this dimension. The smallest faithful representation over any field is a 112 dimensional representation over the field of 2 elements.
The smallest permutation representation is on 173067389=112 · 29 · 31 · 37 · 43 points, with point stabilizer of the form 211:M24. This permutation representation has rank 7; the suborbit lengths are 1, 15180=22 · 3 · 5 · 11 · 23, 28336=24 · 7 · 11 · 23, 3400320=27 · 3 · 5 · 7 · 11 · 23, 32643072=211 · 32 · 7 · 11 · 23, 54405120=211 · 3 · 5 · 7 · 11 · 23, and 82575360=218 · 32 · 7. The points can be identified with certain "special vectors" in the 112 dimensional representation.
The degrees of irreducible representations of the Janko group J4 are 1, 1333, 1333, 299367, 299367, ... (sequence A003907 in the OEIS).
It has a presentation in terms of three generators a, b, and c as
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