ATLAS: Held group He

The following information is available for He:

• Standard generators.
• Automorphisms.
• Black box algorithms.
• Presentations.
• Representations.
• Maximal subgroups.
• Conjugacy classes.

Standard generators

A pair of elements conjugate to (a, b) may be obtained as a' = (cdcdcddcdcddcd)^{14}, b' = (cdd)^{-2}(cdcdcdd)^3(cdd)^2.

Automorphisms

Black box algorithms

Finding generators

To find standard generators for He:

• Find any element of order 10 or 28. This then powers to a 2A-element x.
• Find any element of order 8. Its fourth power is a 2B-element z, say.
• Find two conjugates of z, whose product has order 7 or 21. This powers to a 7C-element y, say.
• Find a conjugate of x and a conjugate of y, whose product has order 17. These are standard generators of He.

To find standard generators for He.2:

• Find any element of order 16. Its eighth power is a 2B-element x, say.
• Find any element of order 30. Its fifth power is a 6C-element y, say
• Find a conjugate of x and a conjugate of y, whose product has order 30. These are standard generators of He.2.

Checking generators

To check that elements x and y of He are standard generators:

• Check o(x) = 2
• Check o(y) = 7
• Check o(xy) = 17
• Check o(xyxyxyy) = 23
• Let z = xyyxyxyyxyy
• Check o(z) = 10
• Check o(xz⁵) = 3

To check that elements x and y of He.2 are standard generators:

• Check o(x) = 2
• Check o(y) = 6
• Check o(xy) = 30
• Let z = xyyxyyxy
• Check o(z) = 24
• Check o(xz¹²) = 17
• Let u = y³
• Let t = (u.u^x)⁴((u.u^xyyxy)²)
• Check o(t) = 15
• Check o([t,y]) = 1

Presentations

< a, b | a² = b⁷ = (ab)¹⁷ = [a, b]⁶ = [a, b³]⁵ = [a, babab^-1abab] = (ab)⁴ab²ab^-3ababab^-1ab³ab^-2ab² = 1 >.

< c, d | c² = d⁶ = ... = 1 > - we haven't found this one yet.

These presentations are available in Magma format as follows: He on a and b and .

Representations

• Some primitive permutation representations.
  • Permutations on 2058 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Permutations on 8330 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Permutations on 29155 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Permutations on 244800 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some irreducible representations in characteristic 2.
  • Dimension 51 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 101 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 246 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 680 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some irreducible representations in characteristic 3.
  • Dimension 51 over GF(9): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 153 over GF(9): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 153 over GF(9): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 679 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some irreducible representations in characteristic 5.
  • Dimension 51 over GF(25): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 104 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 153 over GF(25): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 153 over GF(25): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 680 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 925 over GF(25): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 925 over GF(25): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some irreducible representations in characteristic 7.
  • Dimension 50 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 153 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 426 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 798 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some irreducible representations in characteristic 17.
  • Dimension 102 over GF(17): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 306 over GF(17): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 680 over GF(17): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).

• Permutations on 2058 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 8330 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 102 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 202 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 492 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 680 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 102 over GF(3): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 306 over GF(3): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 679 over GF(3): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 102 over GF(5): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 104 over GF(5): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP). - Corrected (and renamed) on 6.10.04; previous representation generated 3.S7.
• Dimension 306 over GF(5): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 680 over GF(5): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 50 over GF(7): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 153 over GF(7): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 426 over GF(7): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 798 over GF(7): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 102 over GF(17): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 306 over GF(17): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 680 over GF(17): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).

Maximal subgroups

• S4(4):2, with standard generators a, (abb)^-7(bababababbababb)^7(abb)^7.
• 2^2.L3(4).S3, with generators here.
• 2^6:3.S6, with generators here.
• 2^6:3.S6, with generators here.
• 2^1+6L3(2), with generators here.
• 7^2:2.L2(7), with generators here.
• 3.S7, with generators here.
• 7^1+2:(3 × S3), with generators here.
• S4 × L3(2), with generators here.
• 7:3 × L3(2), with generators here.
• 5^2:4A4, with generators here.

• He, with standard generators (cdcdcddcdcddcd)^{14}, (cdd)^{-2}(cdcdcdd)^3(cdd)^2.
• S4(4):4. with generators here.
• 2^2.L3(4).D12. with generators here.
• 2^1+6.L3(2).2. with generators here.
• 7^2:2.L2(7).2. with generators here.
• 3.S7 × 2. with generators here.
• (S5 × S5).2. with generators here.
• 2^4+4.(S3 × S3).2. with generators here.
• 7^1+2:(6 × S3). with generators here.
• S4 × L3(2):2. with generators here.
• 7:6 × L3(2). with generators here.
• 5^2:4S4. with generators here.

Conjugacy classes

• The element 21AB is in class 21A, as it has trace 2 in the 104-dimensional representation over GF(5).
• The element 17AB is wlog in class 17A, independently of all other choices here.
• The elements 14CD, 21CD, and 28AB are linked already by power maps, and are wlog respectively 14C, 21D, and 28A.

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