Acronym	gahi
Name	grand hecatonicosachoron
Cross sections	©
Circumradius	(1+sqrt(5))/2 = 1.618034
Inradius	(1+sqrt(5))/4 = 0.809017
Coordinates	(τ, 0, 0, 0)             & all permutations, all changes of sign (vertex inscribed f/q-hex) (τ/2, τ/2, τ/2, τ/2)   & all permutations, all changes of sign (vertex inscribed f-tes) (τ2/2, τ/2, 1/2, 0)   & even permutations, all changes of sign (vertex inscribed sadi) where τ = (1+sqrt(5))/2 (a. and b. together define a vertex inscribed f-ico)
General of army	ex
Colonel of regiment	ex
Dual	gishi
Confer	regular   fix+gahi+120id   gahi+gohi   2gahi
External links

As abstract polytope gahi is isomorphic to gishi, thereby replacing doe by gissid, resp. replacing pentagonal faces by pentagrammal ones, resp. replacing pentagrammal edge figures each by pentagonal ones, resp. replacing gike vertex figures by ike ones. – As such gahi is a lieutenant.

Incidence matrix according to Dynkin symbol

x5o3o5/2o

. . .   . | 120 ♦  12 |  30 |  20
----------+-----+-----+-----+----
x . .   . |   2 | 720 |   5 |   5
----------+-----+-----+-----+----
x5o .   . |   5 |   5 | 720 |   2
----------+-----+-----+-----+----
x5o3o   . ♦  20 |  30 |  12 | 120

x5o3o5/3o

. . .   . | 120 ♦  12 |  30 |  20
----------+-----+-----+-----+----
x . .   . |   2 | 720 |   5 |   5
----------+-----+-----+-----+----
x5o .   . |   5 |   5 | 720 |   2
----------+-----+-----+-----+----
x5o3o   . ♦  20 |  30 |  12 | 120

x5o3/2o5/2o

. .   .   . | 120 ♦  12 |  30 |  20
------------+-----+-----+-----+----
x .   .   . |   2 | 720 |   5 |   5
------------+-----+-----+-----+----
x5o   .   . |   5 |   5 | 720 |   2
------------+-----+-----+-----+----
x5o3/2o   . ♦  20 |  30 |  12 | 120

x5o3/2o5/3o

. .   .   . | 120 ♦  12 |  30 |  20
------------+-----+-----+-----+----
x .   .   . |   2 | 720 |   5 |   5
------------+-----+-----+-----+----
x5o   .   . |   5 |   5 | 720 |   2
------------+-----+-----+-----+----
x5o3/2o   . ♦  20 |  30 |  12 | 120

x5/4o3o5/2o

.   . .   . | 120 ♦  12 |  30 |  20
------------+-----+-----+-----+----
x   . .   . |   2 | 720 |   5 |   5
------------+-----+-----+-----+----
x5/4o .   . |   5 |   5 | 720 |   2
------------+-----+-----+-----+----
x5/4o3o   . ♦  20 |  30 |  12 | 120

x5/4o3o5/3o

.   . .   . | 120 ♦  12 |  30 |  20
------------+-----+-----+-----+----
x   . .   . |   2 | 720 |   5 |   5
------------+-----+-----+-----+----
x5/4o .   . |   5 |   5 | 720 |   2
------------+-----+-----+-----+----
x5/4o3o   . ♦  20 |  30 |  12 | 120

x5/4o3/2o5/2o

.   .   .   . | 120 ♦  12 |  30 |  20
--------------+-----+-----+-----+----
x   .   .   . |   2 | 720 |   5 |   5
--------------+-----+-----+-----+----
x5/4o   .   . |   5 |   5 | 720 |   2
--------------+-----+-----+-----+----
x5/4o3/2o   . ♦  20 |  30 |  12 | 120

x5/4o3/2o5/3o

.   .   .   . | 120 ♦  12 |  30 |  20
--------------+-----+-----+-----+----
x   .   .   . |   2 | 720 |   5 |   5
--------------+-----+-----+-----+----
x5/4o   .   . |   5 |   5 | 720 |   2
--------------+-----+-----+-----+----
x5/4o3/2o   . ♦  20 |  30 |  12 | 120