[7] It is a highly totient number, as there are 17 solutions to the equation φ(x) = 72, more than any integer under 72.
[8] It is equal to the sum of its preceding smaller highly totient numbers 24 and 48, and contains the first six highly totient numbers 1, 2, 4, 8, 12 and 24 as a subset of its proper divisors.
[9] It also is a perfect indexed Harshad number in decimal (twenty-eighth), as it is divisible by the sum of its digits (9).
[10] 72 plays a role in the Rule of 72 in economics when approximating annual compounding of interest rates of a round 6% to 10%, due in part to its high number of divisors.
Lie algebras: There are 72 compact and paracompact Coxeter groups of ranks four through ten: 14 of these are compact finite representations in only three-dimensional and four-dimensional spaces, with the remaining 58 paracompact or noncompact infinite representations in dimensions three through nine.
These terminate with three paracompact groups in the ninth dimension, of which the most important is
: it contains the final semiregular hyperbolic honeycomb 621 made of only regular facets and the 521 Euclidean honeycomb as its vertex figure, which is the geometric representation of the
shares the same fundamental symmetries with the Coxeter-Dynkin over-extended form
++ equivalent to the tenth-dimensional symmetries of Lie algebra
72 lies between the 8th pair of twin primes (71, 73), where 71 is the largest supersingular prime that is a factor of the largest sporadic group (the friendly giant
), and 73 the largest indexed member of a definite quadratic integer matrix representative of all prime numbers[23][a] that is also the number of distinct orders (without multiplicity) inside all 194 conjugacy classes of