The monster group[1] is the largest of the 26 sporadic groups, which are one of the classes in the classification of the finite simple groups. Its order is approximately \(8 \cdot 10^{53}\), with a full decimal expansion of:
808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
.
Conway and Guy suggested that the order of the monster group is the "largest undeflatable number" (in comparison to Graham's number and Skewes' number which are upper bounds). Arguably, numbers such as TREE(3) discredit this claim, but the order of the monster group is still interesting in that it is a naturally occurring large number, and there does not appear to be any obvious way to generalize the value.
Monster Group (John Conway) - Numberphile
Approximations in other notations[]
| Notation | Lower bound | Upper bound |
|---|---|---|
| Scientific notation | \(8.08\times10^{53}\) | \(8.081\times10^{53}\) |
| Arrow notation | \(43\uparrow33\) | \(3\uparrow113\) |
| Steinhaus-Moser Notation | 34[3] | 35[3] |
| Copy notation | 7[54] | 8[54] |
| Taro's multivariable Ackermann function | A(3,176) | A(3,177) |
| Pound-Star Notation | #*(7,3,5)*15 | #*(4,7,6,4,0,2)*6 |
| BEAF | {43,33} | {3,113} |
| Hyper-E notation | 8E53 | E[3]113 |
| Bashicu matrix system | (0)(0)(0)(0)[2339] | (0)(0)(0)(0)[2340] |
| Hyperfactorial array notation | 43! | 44! |
| Fast-growing hierarchy | \(f_2(171)\) | \(f_2(172)\) |
| Hardy hierarchy | \(H_{\omega^2}(171)\) | \(H_{\omega^2}(172)\) |
| Slow-growing hierarchy | \(g_{\omega^{32}5}(46)\) | \(g_{\omega^{\omega2+9}10}(17)\) |