# ATLAS: Orthogonal group O7(3)

Order = 4585351680 = 29.39.5.7.13.
Mult = 6.
Out = 2.

The following information is available for O7(3):

### Standard generators

Standard generators of O7(3) are a and b where a is in class 2A, b has order 7, ab has order 13 and abb has order 20.
Standard generators of the double cover 2.O7(3) are preimages A and B where B has order 7 and AB has order 13.
Standard generators of the triple cover 3.O7(3) are preimages A and B where A has order 2 and B has order 7.
Standard generators of the sextuple cover 6.O7(3) are preimages A and B where A has order 4, B has order 7, and AB has order 39.

Standard generators of O7(3):2 are c and d where c is in class 2D, d has order 7, and cd has order 26, and cdcdd has order 14.
Standard generators of the double cover 2.O7(3):2 are preimages C and D where D has order 7.
Standard generators of the triple cover 3O7(3):2 are preimages C and D where D has order 7.
Standard generators of the sixfold cover 6.O7(3):2 are preimages C and D where D has order 7.

### Automorphisms

The outer automorphism of O7(3) may be realised by mapping (a, b) to (a, b-1). This automorphism resides in Class 2F.

### Black box algorithms

To find standard generators for O7(3):
• Find an element of order 14. This powers up to x in class 2A and y of order 7.
• Find a conjugate a of x and a conjugate b of y such that ab has order 13 and abb has order 20.
To find standard generators for O7(3).2:
• Find an element of order 26. This powers up to x in class 2D
• Find an element of order 7, 14 or 28. This powers up to y of order 7.
• Find a conjugate c of x and a conjugate d of y such that cd has order 26 and cdcdd has order 14.

### Representations

The representations of O7(3) available are
• Some primitive permutation representations.
• Some faithful irreducibles in characteristic 2.
• Some faithful irreducibles in characteristic 3.
• Dimension 78 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 78 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 78 over GF(13): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
The representations of 2.O7(3) available are
The representation of 3.O7(3) available is
The representations of O7(3):2 available are
The representation of 2.O7(3).2 available is
The representation of 3.O7(3):2 available is

### Maximal subgroups

The maximal subgroups of O7(3) are as follows.
• 2U4(3).2
• 3^5:U4(2):2
• L4(3):2
• G2(3)
• G2(3)
• 3^3+3:L3(3)
• S6(2)
• S6(2)
• 3^1+6:(2A4 x A4).2
• S9
• S9
• (2^2 x U4(2)):2
• 2^6:A7
• S4 x S6
• S4 x 2(A4 x A4).2
The maximal subgroups of O7(3):2 are as follows.
• O7(3)
• 2U4(3).2.2
• 3^5:(U4(2):2 x 2)
• L4(3):2 x 2
• 3^3+3:(L3(3) x 2)
• 3^1+6:(2S4 x S4)
• D8 x U4(2):2
• 2^6:S7
• S4 x S6 x 2
• S4 x 2(A4 x A4).4

### Checks applied

CheckDateBy whomRemarks
Links to (meataxe) representations work and have right degree and field
All info from v1 is included
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Word program syntax
Word programs applied
All necessary standard generators are defined24.01.01RAW
All representations are in standard generators

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Version 2.0 created on 24th January 2001.
Last updated 25.01.01 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.