This regular, non-compact hyperbolic tesselation uses the euclidean tiling that in the sense of an infinite horohedron as vertex figure and likewise trat as one of its cells.

Incidence matrix according to Dynkin symbol

x3o3o *b6o   (N,M,K → ∞)

. . .    . | 2NK ♦   3M |   6M |  2M   M
-----------+-----+------+------+--------
x . .    . |   2 | 3NMK |    4 |   2   2
-----------+-----+------+------+--------
x3o .    . |   3 |    3 | 4NMK |   1   1
-----------+-----+------+------+--------
x3o3o    . ♦   4 |    6 |    4 | NMK   *
x3o . *b6o ♦   K |   3K |   2K |   * 2NM

o3x3o3o3*a3*c   (N,M,K → ∞)

. . . .       | 4NK ♦   3M |   3M   3M |   M   M   M
--------------+-----+------+-----------+------------
. x . .       |   2 | 6NMK |    2    2 |   2   1   1
--------------+-----+------+-----------+------------
o3x . .       |   3 |    3 | 4NMK    * |   1   1   0
. x3o .       |   3 |    3 |    * 4NMK |   1   0   1
--------------+-----+------+-----------+------------
o3x3o . *a3*c ♦   K |   3K |    K    K | 4NM   *   *
o3x . o3*a    ♦   4 |    6 |    4    0 |   * NMK   *
. x3o3o       ♦   4 |    6 |    0    4 |   *   * NMK

s4o3o6o   (N,M,K → ∞)

hemi( . . . . ) | 2NK ♦   3M |   6M |  2M   M
----------------+-----+------+------+--------
      s4o . .   |   2 | 3NMK |    4 |   2   2
----------------+-----+------+------+--------
sefa( s4o3o . ) |   3 |    3 | 4NMK |   1   1
----------------+-----+------+------+--------
      s4o3o .   ♦   4 |    6 |    4 | NMK   *
sefa( s4o3o6o ) ♦   K |   3K |   2K |   * 2NM

starting figure: x4o3o6o

s4o3o3o3*b   (N,M,K → ∞)

hemi( . . . .    ) | 4NK ♦   3M |   3M   3M |   M   M   M
-------------------+-----+------+-----------+------------
      s4o . .      |   2 | 6NMK |    2    2 |   1   1   2
-------------------+-----+------+-----------+------------
sefa( s4o3o .    ) |   3 |    3 | 4NMK    * |   1   0   1
sefa( s4o . o3*b ) |   3 |    3 |    * 4NMK |   0   1   1
-------------------+-----+------+-----------+------------
      s4o3o .      ♦   4 |    6 |    4    0 | NMK   *   *
      s4o . o3*b   ♦   4 |    6 |    0    4 |   * NMK   *
sefa( s4o3o3o3*b ) ♦   K |   3K |    K    K |   *   * 4NM

starting figure: x4o3o3o3b