Groups of order 720

Factorization and useful forms
The number 720 has prime factors 2, 3, and 5, and prime factorization:

$$\! 720 = 2^4 \cdot 3^2 \cdot 5^1 = 16 \cdot 9 \cdot 5$$.

Other useful expressions for this number are:

$$\! 720 = 6! = 9^3 - 9 = 2^4(2^2 - 1)(2^4 - 1) = (10)(9)(8)$$

GAP implementation
gap> SmallGroupsInformation(720);

There are 840 groups of order 720. They are sorted by their Frattini factors. 1 has Frattini factor [ 30, 1 ]. 2 has Frattini factor [ 30, 2 ]. 3 has Frattini factor [ 30, 3 ]. 4 has Frattini factor [ 30, 4 ]. 5 has Frattini factor [ 60, 6 ]. 6 has Frattini factor [ 60, 7 ]. 7 - 31 have Frattini factor [ 60, 8 ]. 32 has Frattini factor [ 60, 9 ]. 33 - 48 have Frattini factor [ 60, 10 ]. 49 - 64 have Frattini factor [ 60, 11 ]. 65 - 80 have Frattini factor [ 60, 12 ]. 81 - 88 have Frattini factor [ 60, 13 ]. 89 has Frattini factor [ 90, 5 ]. 90 has Frattini factor [ 90, 6 ]. 91 has Frattini factor [ 90, 7 ]. 92 has Frattini factor [ 90, 8 ]. 93 has Frattini factor [ 90, 9 ]. 94 has Frattini factor [ 90, 10 ]. 95 - 101 have Frattini factor [ 120, 36 ]. 102 - 104 have Frattini factor [ 120, 37 ]. 105 - 107 have Frattini factor [ 120, 38 ]. 108 - 110 have Frattini factor [ 120, 39 ]. 111 - 117 have Frattini factor [ 120, 40 ]. 118 - 124 have Frattini factor [ 120, 41 ]. 125 - 151 have Frattini factor [ 120, 42 ]. 152 - 154 have Frattini factor [ 120, 43 ]. 155 - 164 have Frattini factor [ 120, 44 ]. 165 - 174 have Frattini factor [ 120, 45 ]. 175 - 184 have Frattini factor [ 120, 46 ]. 185 - 188 have Frattini factor [ 120, 47 ]. 189 has Frattini factor [ 180, 20 ]. 190 has Frattini factor [ 180, 21 ]. 191 has Frattini factor [ 180, 22 ]. 192 has Frattini factor [ 180, 23 ]. 193 has Frattini factor [ 180, 24 ]. 194 has Frattini factor [ 180, 25 ]. 195 - 219 have Frattini factor [ 180, 26 ]. 220 - 244 have Frattini factor [ 180, 27 ]. 245 - 260 have Frattini factor [ 180, 28 ]. 261 - 285 have Frattini factor [ 180, 29 ]. 286 - 301 have Frattini factor [ 180, 30 ]. 302 has Frattini factor [ 180, 31 ]. 303 - 318 have Frattini factor [ 180, 32 ]. 319 - 334 have Frattini factor [ 180, 33 ]. 335 - 350 have Frattini factor [ 180, 34 ]. 351 - 366 have Frattini factor [ 180, 35 ]. 367 - 382 have Frattini factor [ 180, 36 ]. 383 - 390 have Frattini factor [ 180, 37 ]. 391 has Frattini factor [ 240, 191 ]. 392 has Frattini factor [ 240, 192 ]. 393 has Frattini factor [ 240, 193 ]. 394 has Frattini factor [ 240, 194 ]. 395 has Frattini factor [ 240, 195 ]. 396 has Frattini factor [ 240, 196 ]. 397 has Frattini factor [ 240, 197 ]. 398 has Frattini factor [ 240, 198 ]. 399 has Frattini factor [ 240, 199 ]. 400 has Frattini factor [ 240, 200 ]. 401 has Frattini factor [ 240, 201 ]. 402 has Frattini factor [ 240, 202 ]. 403 has Frattini factor [ 240, 203 ]. 404 has Frattini factor [ 240, 204 ]. 405 has Frattini factor [ 240, 205 ]. 406 has Frattini factor [ 240, 206 ]. 407 has Frattini factor [ 240, 207 ]. 408 has Frattini factor [ 240, 208 ]. 409 has Frattini factor [ 360, 118 ]. 410 - 412 have Frattini factor [ 360, 119 ]. 413 - 415 have Frattini factor [ 360, 120 ]. 416 - 418 have Frattini factor [ 360, 121 ]. 419 - 421 have Frattini factor [ 360, 122 ]. 422 has Frattini factor [ 360, 123 ]. 423 has Frattini factor [ 360, 124 ]. 424 has Frattini factor [ 360, 125 ]. 425 - 431 have Frattini factor [ 360, 126 ]. 432 - 438 have Frattini factor [ 360, 127 ]. 439 - 445 have Frattini factor [ 360, 128 ]. 446 - 452 have Frattini factor [ 360, 129 ]. 453 - 459 have Frattini factor [ 360, 130 ]. 460 - 466 have Frattini factor [ 360, 131 ]. 467 - 471 have Frattini factor [ 360, 132 ]. 472 - 478 have Frattini factor [ 360, 133 ]. 479 - 483 have Frattini factor [ 360, 134 ]. 484 has Frattini factor [ 360, 135 ]. 485 - 486 have Frattini factor [ 360, 136 ]. 487 - 525 have Frattini factor [ 360, 137 ]. 526 - 528 have Frattini factor [ 360, 138 ]. 529 - 531 have Frattini factor [ 360, 139 ]. 532 - 534 have Frattini factor [ 360, 140 ]. 535 - 537 have Frattini factor [ 360, 141 ]. 538 - 540 have Frattini factor [ 360, 142 ]. 541 - 543 have Frattini factor [ 360, 143 ]. 544 - 546 have Frattini factor [ 360, 144 ]. 547 - 553 have Frattini factor [ 360, 145 ]. 554 - 560 have Frattini factor [ 360, 146 ]. 561 - 567 have Frattini factor [ 360, 147 ]. 568 - 574 have Frattini factor [ 360, 148 ]. 575 - 581 have Frattini factor [ 360, 149 ]. 582 - 588 have Frattini factor [ 360, 150 ]. 589 - 615 have Frattini factor [ 360, 151 ]. 616 - 642 have Frattini factor [ 360, 152 ]. 643 - 660 have Frattini factor [ 360, 153 ]. 661 - 687 have Frattini factor [ 360, 154 ]. 688 - 705 have Frattini factor [ 360, 155 ]. 706 - 708 have Frattini factor [ 360, 156 ]. 709 - 718 have Frattini factor [ 360, 157 ]. 719 - 728 have Frattini factor [ 360, 158 ]. 729 - 738 have Frattini factor [ 360, 159 ]. 739 - 748 have Frattini factor [ 360, 160 ]. 749 - 758 have Frattini factor [ 360, 161 ]. 759 - 762 have Frattini factor [ 360, 162 ]. 763 - 840 have trivial Frattini subgroup.

For the selection functions the values of the following attributes are precomputed and stored: IsAbelian, IsNilpotentGroup, IsSupersolvableGroup, IsSolvableGroup, LGLength, FrattinifactorSize and FrattinifactorId.

This size belongs to layer 2 of the SmallGroups library. IdSmallGroup is available for this size.