Some people may find this helpful.
It is a list of a the hexagons and heptagons in the 2nd Hamiltonian.

HEXAGONS:
( 1, 7, 11, 12, 19, 8)

( 2, 3, 4, 23, 24, 25)

( 4, 5, 6, 21, 22, 23)

(12, 15, 20, 26, 18, 19)


HEPTAGONS:
( 1, 2, 3, 4, 5, 6, 7)

( 1, 2, 3, 9, 13, 16, 8)

( 1, 2, 25, 24, 17, 11, 7)

( 1, 2, 25, 26, 18, 19, 8)

( 1, 7, 6, 21, 22, 16, 8)

( 2, 3, 9, 15, 20, 26, 25)

( 3, 4, 5, 10, 14, 13, 9)

( 3, 4, 23, 22, 16, 13, 9)

( 4, 5, 10, 14, 17, 24, 23)

( 5, 6, 7, 11, 17, 14, 10)

( 5, 6, 21, 20, 26, 18, 10)

( 6, 7, 11, 12, 15, 20, 21)

( 8, 16, 13, 9, 15, 12, 19)

( 8, 16, 13, 14, 10, 18, 19)

( 9, 13, 14, 17, 11, 12, 15)

( 9, 13, 16, 22, 21, 20, 15)

(10, 14, 17, 11, 12, 19, 18)

(10, 14, 17, 24, 25, 26, 18)

(13, 14, 17, 24, 23, 22, 16)

(20, 21, 22, 23, 24, 25, 26)