Mult = 2.

Out = 2 × 2.

The following classes of O_{8}^{-}(3).2^{2} are described:

**Class 2A**-
An element
*x*of O_{8}^{-}(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 × L_{4}(3)):2_{2}, and class 2A is invariant under all outer automorphisms of O_{8}^{-}(3). **Class 3A**-
An element
*x*of O_{8}^{-}(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 3^{1+8}.(2A_{4}× A_{6}), and class 3A is invariant under all outer automorphisms of O_{8}^{-}(3). A 3A element is conjugate to its inverse. **Class 4F**-
An element
*x*of O_{8}^{-}(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 O_{8}^{-}(3).

**Class 2D/E**-
An element
*x*of O_{8}^{-}(3):2_{1}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 × O_{7}(3). Classes 2D and 2E are fused in the automorphism group of O_{8}^{-}(3):2_{1}.

Classes 2D and 2E are the classes of reflexions and negated reflexions. **Class 8F**-
An element
*x*of O_{8}^{-}(3):2_{1}is in class 8F if it has order 8, its centraliser has order 64, and*x*is not in O_{8}^{-}(3). Class 8F is invariant under all outer automorphisms of O_{8}^{-}(3):2_{1}. An 8F element is conjugate to its third, fifth and seventh powers.

[An 8C element also has centraliser in O_{8}^{-}(3):2_{1}of order 64.]

**Class 2H**-
An element
*x*of O_{8}^{-}(3):2_{2}is in class 2H if it has order 2 and does not reside in O_{8}^{-}(3).The centraliser of a 2H element has shape 2 × L_{2}(81)):2_{1}, order 2 × 531360 = 1062720, and index 19105632. Class 2H is invariant under all outer automorphisms of O_{8}^{-}(3):2_{2}.

**Class 8V/W**-
An element
*x*of O_{8}^{-}(3).2_{3}is in class 8V/W if it has order 8 and its centraliser [in O_{8}^{-}(3).2_{3}] 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 O_{8}^{-}(3).2_{3}. **Class 8X/Y**-
An element
*x*of O_{8}^{-}(3).2_{3}is in class 8X/Y if it has order 8, its centraliser [in O_{8}^{-}(3).2_{3}] has order 384, and the centraliser [in O_{8}^{-}(3).2_{3}] of*x*^{2}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 O_{8}^{-}(3).2_{3}. **Class 8B1**-
An element
*x*of O_{8}^{-}(3).2_{3}is in class 8B1 if it has order 8, its centraliser [in O_{8}^{-}(3).2_{3}] has order 64, and*x*is not in O_{8}^{-}(3). Class 8B1 is invariant under all outer automorphisms of O_{8}^{-}(3).2_{3}. An 8B1 element is conjugate to its third, fifth and seventh powers.

[An element in classes 8C, 8F, 8PQ or 8B1 has centraliser in O_{8}^{-}(3).2^{2}of order 128.]

Return to the main O8-(3) page. Version 2.0 created on 23rd February 2004.

Last updated 23.02.04 by JNB.

R.A.Wilson, R.A.Parker and J.N.Bray.