# Regular Maps in the Torus, with Triangular Faces

### Schläfli symbol {3,6}(a,b)

The regular maps with six triangles meeting at each vertex are the duals of those with three hexagons. As for {6,3}(a,b), a and b must be either both odd or both even. The number of faces of these polyhedra is given by (a2+3*b2)/2.

The notation used here {3,6}(a,b) is not consistent with that used in ARM, and is as described above for {6,3}(a,b). Thus our {6,3}(a,b) is the dual of our {3,6}(a,b).

ARM also disallows (in our notation) {3,6}(1,1), in which each of the two triangles shares each of its vertices (but no edge) with itself.

designationno. of
triangles
pictureV
F
E
Eu
dual

Petrie dual

rotational
symmetry
group
{3,6}(1,1}

{3,6}(2,0)

21
2
3
0
{6,3}(1,1)

hemi-3-hosohedron

D6
{3,6}(0,2)

{3,6}(3,1)

63
6
9
0
{6,3}(0,2)

C5{6,6}

D6×C3
{3,6}(2,2)

{3,6}(4,0)

84
8
12
0
{6,3}(2,2)

C4{4,6}

S4
{3,6}(1,3)

{3,6}(4,2)

{3,6}(5,1)

147
14
21
0
{6,3}(1,3}

Frob42

C7⋊C6
K7

{3,6}(3,3)

{3,6}(6,0)

189
18
27
0
{6,3}(3,3)

?K3,3,3

{3,6}(0,4)

{3,6}(6,2)

2412
24
36
0
{6,3}(0,4)

?
{3,6}(2,4)

{3,6}(5,3)

{3,6}(7,1)

2613
26
39
0
{6,3}(2,4)

?the Paley order-13 graph

{3,6}(4,4)

{3,6}(8,0)

3216
32
48
0
{6,3}(4,4)

?the Shrikhande graph

{3,6}(1,5)

{3,6}(7,3)

{3,6}(8,2)

3819
38
57
0
{6,3}(1,5)

C19⋊C3
{3,6}(3,5)

{3,6}(6,4)

{3,6}(9,1)

4221
42
63
0
{6,3}(3,5)

?
{3,6}(5,5)

{3,6}(10,0)

5025
50
75
0
{6,3}(5,5)

?

The following figures have more than 50 faces; they are included because their duals are above.

{3,6}(0,6)

{3,6}(9,3)

5427
54
81
0
{6,3}(0,6)

?
{3,6}(2,6)

{3,6}(8,4)

{3,6}(10,2)

5628
56
84
0
{6,3}(2,6)

?
{3,6}(4,6)

{3,6}(7,5)

{3,6}(11,1)

6231
62
93
0
{6,3}(4,6)

?
{3,6}(6,6)

{3,6}(12,0)

7236
72
108
0
{6,3}(6,6)

?
{3,6}(1,7)

{3,6}(10,4)

{3,6}(11,3)

7437
74
111
0
{6,3}(1,7)

?
{3,6}(3,7)

{3,6}(9,5)

{3,6}(12,2)

7839
78
117
0
{6,3}(3,7)

?
{3,6}(5,7)

{3,6}(8,6)

{3,6}(13,1)

8643
86
126
0
{6,3}(5,7)

?
{3,6}(0,8)

{3,6}(12,4)

9648
96
144
0
{6,3}(0,8)

?
{3,6}(7,7)

{3,6}(14,0)

9849
98
147
0
{6,3}(7,7)

?
{3,6}(2,8)

{3,6}(11,5)

{3,6}(13,3)

9849
98
147
0
{6,3}(2,8)

?

The pink lines, arrows, and shading are explained by the page Representation of 2-manifolds.

Index to other pages on regular maps;
indexes to those on S0 C1 S1 S2 C4 C5 S3 C6 S4.
Some pages on groups