Linear group L₂(27)

Order = 9 828 = 2².3³.7.13.
Mult = 2.
Out = 6.

Robert Wilson's ATLAS page for L₂(27) is (not yet) available here.

A pair of (2, 3)-generators for L₂(27) satisfies just one of the first four presentations below. In these first four presentations (and others) we have that the field F = GF(27) has a primitive element w which satisfies w³ + 2w + 1 = 0.

L₂(27): Length 67, 2-generator, 4-relator.

< x, y | x² = y³ = (xy)⁷ = (xyxyxy^-1xyxy^-1xyxy^-1xy^-1xyxy^-1xyxy^-1)² = 1 >

Remark: A presentation for L₂(27) on its unique class of (2, 3, 7)-generators (up to automorphisms). Generators are (2, 3, 7; 13). The presentation is available in MAGMA code here. (This applies to the subgroups below too.)

Some subgroups:

M1 = < y, xyxyxy^-1xyxyxy^-1x > = 3³:13 of order 351 and index 28.
M2 = < x, x^yxy > = D₂₈ of order 28 and index 351.
M3 = < x, x^y > = D₂₆ of order 26 and index 378.
M4 = < x, y^{xyxy^-1xyxy^-1xy} > = A₄ of order 12 and index 819.

Realisation: In MAGMA format as 2 × 2 matrices over GF(27) modulo scalars and as 3 × 3 matrices over GF(27) absolutely.

L₂(27): Length 55, 2-generator, 4-relator.

< x, y | x² = y³ = (xy)¹³ = (xyxyxyxy^-1xy^-1xy^-1)² = 1 >

Remark: A presentation for L₂(27) on its (2, 3, 13A/B/C; 14)-generators. The presentation is available in MAGMA code here. (This applies to the subgroups below too.)

Some subgroups:

M1 = < y, xyxy^-1xyxyxyxy^-1x > = 3³:13 of order 351 and index 28.
M2 = < x, x^y > = D₂₈ of order 28 and index 351.
M3 = < x, x^yxy > = D₂₆ of order 26 and index 378.
M4 = < x, y^{xyxy^-1xy^-1} > = A₄ of order 12 and index 819.
H1 = < x, x^{yxy^-1} > = D₁₄ of order 14 and index 702.
H2 = < y, y^yxyx > = A₄ of order 12 and index 819.

Realisation: In MAGMA format as 2 × 2 matrices over GF(27) modulo scalars and as 3 × 3 matrices over GF(27) absolutely.

L₂(27): Length 61, 2-generator, 4-relator.

< x, y | x² = y³ = (xy)¹³ = (xyxyxyxy^-1xy^-1)³ = 1 >

Remark: A presentation for L₂(27) on its (2, 3, 13D/E/F; 13)-generators. The relation (xyxyxyxy^-1xy^-1)³ = 1 is equivalent to the shorter relation (xyxyxy^-1)²(xy^-1xy^-1)²xy^-1 = 1 (new total length 59), but the coset enumeration with the latter relation actually seems to be more difficult for some reason. The presentation (including the extra relation above) is available in MAGMA code here. (This applies to the subgroups below too.)

Some subgroups:

M1 = < y, xyxyxyxyxy^-1xy^-1x > = 3³:13 of order 351 and index 28.
M2 = < x, x^yxy > = D₂₈ of order 28 and index 351.
M3 = < x, x^y > = D₂₆ of order 26 and index 378.
M4 = < x, y^xyxy > = A₄ of order 12 and index 819.
H1 = < y, x^yxyx > = A₄ of order 12 and index 819.

Realisation: In MAGMA format as 2 × 2 matrices over GF(27) modulo scalars and as 3 × 3 matrices over GF(27) absolutely.

L₂(27): Length 91, 2-generator, 5-relator.

< x, y | x² = y³ = (xy)¹⁴ = (xyxyxy^-1xyxy^-1)³ = [y, xyxyxy^-1xy^-1xyxyx] = 1 >

Remark: A presentation for L₂(27) on its (2, 3, 14; 14)-generators. Omitting (xy)¹⁴ = 1 gives L₂(27) × 3 of order 29 484 (and with length 63) and omitting (xyxyxy^-1xyxy^-1)³ = 1 gives the central product 4 o SL₂(27) of order 39 312 = 4 × |L₂(27)| (and with length 61) and omitting both of these gives an infinite group which maps epimorphically onto infinitely many L₂(q) / PGL₂(q). The removal of [y, xyxyxy^-1xy^-1xyxyx] = 1 allows [2⁵⁶].L₂(13) × L₂(27) of order 773 334 781 351 593 284 468 736 > 7.73 × 10²³ as an image. However, we do not know if this group is infinite. The presentation is available in MAGMA code here. (This applies to the subgroups below too.)

Some subgroups:

M1 = < y, xyxyxyxyxyxyxy^-1x > = 3³:13 of order 351 and index 28.
M2 = < x, x^y > = D₂₈ of order 28 and index 351.
M3 = < x, x^yxy > = D₂₆ of order 26 and index 378.
M4 = < x, y^{xyxy^-1} > = A₄ of order 12 and index 819.
H1 = < x, x^{yxy^-1} > = D₁₄ of order 14 and index 702.
H2 = < y, x^{yxy^-1x} > = A₄ of order 12 and index 819.

Realisation: In MAGMA format as 2 × 2 matrices over GF(27) modulo scalars and as 3 × 3 matrices over GF(27) absolutely.

L₂(27): Length 46, 3-generator, 7-relator.

< a, b, c | a³ = b¹³ = a^-1b^-1ab^-1ab^-1ab³ = [a, a^b] = c² = (ac)³ = (bc)² = 1 >

Remark: I'm afraid I don't know the source of this one (or at least of the presentation scheme of groups of type L₂(q) and PGL₂(q) whose presentations look similar to the one above), but I think they are due to J.A.Todd. Traditionally, a, b and c are labelled as $\alpha$, $\beta$ and $\gamma$ in some order. The presentation is available in MAGMA code here. (This applies to the subgroups below too.)

Some subgroups:

M1 = < a, b > = 3³:13 of order 351 and index 28.
M2 = < c, c^bab > = D₂₈ of order 28 and index 351.
M3 = < b, c > = D₂₆ of order 26 and index 378.
M4 = < a, c > = A₄ of order 12 and index 819.

Realisation:
a: \eta -> \eta + 1.
b: \eta -> w².\eta.
c: \eta -> -1 / \eta.

These linear fractional transformations are taken over the field F = GF(27) with primitive element w which satisfies w³ + 2w + 1 = 0. (Strictly, they are taken over the projective line PG₁(27).)

We have converted the above linear fractional transformations into 2 × 2 matrices over GF(27) modulo scalars and also as 3 × 3 matrices over GF(27). (Both of these are in MAGMA format.)

PGL₂(27): Length ??, ?-generator, ??-relator.

L₂(27):3: Length ??, ?-generator, ??-relator.

L₂(27):6: Length 105, 2-generator, 5-relator.

< x, y | x² = y³ = (xy)¹² = (xyxyxy^-1xy^-1xyxy^-1)³ = [x, yxyxyxyxy^-1]² >

Remark: ....

H = < x, y^-1xyxy, yxy^-1xyxyxy^-1 > - index 28.

L₂(27):6: Length ??, 2-generator, ??-relator.

L₂(27):6: Length ??, ?-generator, ??-relator.

Last updated 27^th August, 1997

John N. Bray

Linear group L2(27)

L2(27): Length 67, 2-generator, 4-relator.

L2(27): Length 55, 2-generator, 4-relator.

L2(27): Length 61, 2-generator, 4-relator.

L2(27): Length 91, 2-generator, 5-relator.

L2(27): Length 46, 3-generator, 7-relator.

PGL2(27): Length ??, ?-generator, ??-relator.

L2(27):3: Length ??, ?-generator, ??-relator.

L2(27):6: Length 105, 2-generator, 5-relator.

L2(27):6: Length ??, 2-generator, ??-relator.

L2(27):6: Length ??, ?-generator, ??-relator.

Linear group L₂(27)

L₂(27): Length 67, 2-generator, 4-relator.

L₂(27): Length 55, 2-generator, 4-relator.

L₂(27): Length 61, 2-generator, 4-relator.

L₂(27): Length 91, 2-generator, 5-relator.

L₂(27): Length 46, 3-generator, 7-relator.

PGL₂(27): Length ??, ?-generator, ??-relator.

L₂(27):3: Length ??, ?-generator, ??-relator.

L₂(27):6: Length 105, 2-generator, 5-relator.

L₂(27):6: Length ??, 2-generator, ??-relator.

L₂(27):6: Length ??, ?-generator, ??-relator.