ATLAS: O₈^-(3): Class definitions

The following classes of O₈^-(3).2² are described:

• Classes 2A, 3A and 4F.
• Classes 2D/E and 8F.
• Class 2H.
• Classes 8V/W, 8X/Y and 8B1.

Classes in O₈^-(3)

Class 2A

An element x of O₈^-(3) is in class 2A if it has order 2 and its centraliser has order 48522240 (index 209223). The centraliser of a 2A element has shape (4 × L₄(3)):2₂, and class 2A is invariant under all outer automorphisms of O₈^-(3).

Class 3A

An element x of O₈^-(3) is in class 3A if it has order 3 and its centraliser has order 170061120 (index 59696). The centraliser of a 3A element has shape 3¹⁺⁸.(2A₄ × A₆), and class 3A is invariant under all outer automorphisms of O₈^-(3). A 3A element is conjugate to its inverse.

Class 4F

An element x of O₈^-(3) is in class 4F if it has order 4 and its centraliser has order 768. A 4F element is conjugate to its inverse and class 4F is invariant under all outer automorphisms of O₈^-(3).

Classes in O₈^-(3):2₁ but not in O₈^-(3)

Class 2D/E

An element x of O₈^-(3):2₁ is in class 2D/E if it has order 2 and its centraliser has order 18341406720 = 2 × 9170703360 (index 1107). The centraliser of a 2D/E element has shape 2 × O₇(3). Classes 2D and 2E are fused in the automorphism group of O₈^-(3):2₁.
Classes 2D and 2E are the classes of reflexions and negated reflexions.

Class 8F

An element x of O₈^-(3):2₁ is in class 8F if it has order 8, its centraliser has order 64, and x is not in O₈^-(3). Class 8F is invariant under all outer automorphisms of O₈^-(3):2₁. An 8F element is conjugate to its third, fifth and seventh powers.
[An 8C element also has centraliser in O₈^-(3):2₁ of order 64.]

Classes in O₈^-(3):2₂ but not in O₈^-(3)

Class 2H

An element x of O₈^-(3):2₂ is in class 2H if it has order 2 and does not reside in O₈^-(3).The centraliser of a 2H element has shape 2 × L₂(81)):2₁, order 2 × 531360 = 1062720, and index 19105632. Class 2H is invariant under all outer automorphisms of O₈^-(3):2₂.

Classes in O₈^-(3).2₃ but not in O₈^-(3)

Class 8V/W

An element x of O₈^-(3).2₃ is in class 8V/W if it has order 8 and its centraliser [in O₈^-(3).2₃] has order 768. An 8V element is conjugate to its third power, but its fifth and seventh powers lie class 8W. Classes 8V and 8W are fused in the automorphism group of O₈^-(3).2₃.

Class 8X/Y

An element x of O₈^-(3).2₃ is in class 8X/Y if it has order 8, its centraliser [in O₈^-(3).2₃] has order 384, and the centraliser [in O₈^-(3).2₃] of x² has order 9216. An 8X element is conjugate to its third power, but its fifth and seventh powers lie class 8Y. Classes 8X and 8Y are fused in the automorphism group of O₈^-(3).2₃.

Class 8B1

An element x of O₈^-(3).2₃ is in class 8B1 if it has order 8, its centraliser [in O₈^-(3).2₃] has order 64, and x is not in O₈^-(3). Class 8B1 is invariant under all outer automorphisms of O₈^-(3).2₃. An 8B1 element is conjugate to its third, fifth and seventh powers.
[An element in classes 8C, 8F, 8PQ or 8B1 has centraliser in O₈^-(3).2² of order 128.]

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