# Groups of order 1440

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## Contents

See pages on algebraic structures of order 1440| See pages on groups of a particular order

## Statistics at a glance

### Factorization and useful forms

The number 1440 has prime factors 2, 3, and 5, and prime factorization:

$\! 1440 = 2^5 \cdot 3^2 \cdot 5^1 = 32 \cdot 9 \cdot 5$

Other useful expressions for this number are:

$1440 = 2(6!) = 2(9^3 - 9)$

### Group counts

Quantity Value Explanation GAP verification
Total number of abelian groups (i.e., finite abelian groups) up to isomorphism 14 (Number of abelian groups of order $2^5$) times (Number of abelian groups of order $3^2$) times (Number of abelian groups of order $5^1$) = (number of unordered integer partitions of 5) times (number of unordered integer partitions of 2) times (number of unordered integer partitions of 1) = $7 \times 2 \times 1 = 14$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups. [SHOW MORE]
Total number of nilpotent groups (i.e., finite nilpotent groups) up to isomorphism 102 (Number of groups of order 32) times (Number of groups of order 9) times (Number of groups of order 5) = $51 \times 2 \times 1 = 102$. See number of nilpotent groups equals product of number of groups of order each maximal prime power divisor, which in turn follows from equivalence of definitions of finite nilpotent group. [SHOW MORE]
Total number of solvable groups (i.e., finite solvable groups) up to isomorphism 5865 [SHOW MORE]
Number of almost simple groups up to isomorphism 1 automorphism group of alternating group:A6 (which can also be described as $P\Gamma L(2,9)$) [SHOW MORE]
Number of almost quasisimple groups up to isomorphism 4 In addition to the almost simple group, there are three others. Two of these are double cover of symmetric group:S6 of minus type and double cover of symmetric group:S6 of plus type. [SHOW MORE]

## GAP implementation

The order 1440 is part of GAP's SmallGroup library. Hence, any group of order 1440 can be constructed using the SmallGroup function by specifying its group ID. Also, IdGroup is available, so the group ID of any group of this order can be queried.

Further, the collection of all groups of order 1440 can be accessed as a list using GAP's AllSmallGroups function. However, the list size may be too large relative to the memory allocation given in typical GAP installations. To overcome this problem, use the IdsOfAllSmallGroups function which stores and manipulates only the group IDs, not the groups themselves.

Here is GAP's summary information about how it stores groups of this order, accessed using GAP's SmallGroupsInformation function:

gap> SmallGroupsInformation(1440);

There are 5958 groups of order 1440.
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 - 73 have Frattini factor [ 60, 8 ].
74 has Frattini factor [ 60, 9 ].
75 - 115 have Frattini factor [ 60, 10 ].
116 - 156 have Frattini factor [ 60, 11 ].
157 - 197 have Frattini factor [ 60, 12 ].
198 - 216 have Frattini factor [ 60, 13 ].
217 has Frattini factor [ 90, 5 ].
218 has Frattini factor [ 90, 6 ].
219 has Frattini factor [ 90, 7 ].
220 has Frattini factor [ 90, 8 ].
221 has Frattini factor [ 90, 9 ].
222 has Frattini factor [ 90, 10 ].
223 - 253 have Frattini factor [ 120, 36 ].
254 - 257 have Frattini factor [ 120, 37 ].
258 - 261 have Frattini factor [ 120, 38 ].
262 - 268 have Frattini factor [ 120, 39 ].
269 - 293 have Frattini factor [ 120, 40 ].
294 - 318 have Frattini factor [ 120, 41 ].
319 - 653 have Frattini factor [ 120, 42 ].
654 - 660 have Frattini factor [ 120, 43 ].
661 - 746 have Frattini factor [ 120, 44 ].
747 - 832 have Frattini factor [ 120, 45 ].
833 - 918 have Frattini factor [ 120, 46 ].
919 - 942 have Frattini factor [ 120, 47 ].
943 has Frattini factor [ 180, 20 ].
944 has Frattini factor [ 180, 21 ].
945 has Frattini factor [ 180, 22 ].
946 has Frattini factor [ 180, 23 ].
947 has Frattini factor [ 180, 24 ].
948 has Frattini factor [ 180, 25 ].
949 - 1015 have Frattini factor [ 180, 26 ].
1016 - 1082 have Frattini factor [ 180, 27 ].
1083 - 1123 have Frattini factor [ 180, 28 ].
1124 - 1190 have Frattini factor [ 180, 29 ].
1191 - 1231 have Frattini factor [ 180, 30 ].
1232 has Frattini factor [ 180, 31 ].
1233 - 1273 have Frattini factor [ 180, 32 ].
1274 - 1314 have Frattini factor [ 180, 33 ].
1315 - 1355 have Frattini factor [ 180, 34 ].
1356 - 1396 have Frattini factor [ 180, 35 ].
1397 - 1437 have Frattini factor [ 180, 36 ].
1438 - 1456 have Frattini factor [ 180, 37 ].
1457 - 1459 have Frattini factor [ 240, 192 ].
1460 - 1462 have Frattini factor [ 240, 193 ].
1463 - 1477 have Frattini factor [ 240, 194 ].
1478 - 1508 have Frattini factor [ 240, 195 ].
1509 - 1519 have Frattini factor [ 240, 196 ].
1520 - 1530 have Frattini factor [ 240, 197 ].
1531 - 1541 have Frattini factor [ 240, 198 ].
1542 has Frattini factor [ 240, 199 ].
1543 - 1555 have Frattini factor [ 240, 200 ].
1556 - 1568 have Frattini factor [ 240, 201 ].
1569 - 1621 have Frattini factor [ 240, 202 ].
1622 - 1628 have Frattini factor [ 240, 203 ].
1629 - 1630 have Frattini factor [ 240, 204 ].
1631 - 1645 have Frattini factor [ 240, 205 ].
1646 - 1660 have Frattini factor [ 240, 206 ].
1661 - 1675 have Frattini factor [ 240, 207 ].
1676 - 1681 have Frattini factor [ 240, 208 ].
1682 - 1684 have Frattini factor [ 360, 119 ].
1685 - 1687 have Frattini factor [ 360, 120 ].
1688 - 1690 have Frattini factor [ 360, 121 ].
1691 - 1693 have Frattini factor [ 360, 122 ].
1694 has Frattini factor [ 360, 123 ].
1695 has Frattini factor [ 360, 124 ].
1696 has Frattini factor [ 360, 125 ].
1697 - 1727 have Frattini factor [ 360, 126 ].
1728 - 1758 have Frattini factor [ 360, 127 ].
1759 - 1789 have Frattini factor [ 360, 128 ].
1790 - 1814 have Frattini factor [ 360, 129 ].
1815 - 1845 have Frattini factor [ 360, 130 ].
1846 - 1870 have Frattini factor [ 360, 131 ].
1871 - 1888 have Frattini factor [ 360, 132 ].
1889 - 1917 have Frattini factor [ 360, 133 ].
1918 - 1935 have Frattini factor [ 360, 134 ].
1936 - 1940 have Frattini factor [ 360, 135 ].
1941 - 1949 have Frattini factor [ 360, 136 ].
1950 - 2520 have Frattini factor [ 360, 137 ].
2521 - 2524 have Frattini factor [ 360, 138 ].
2525 - 2528 have Frattini factor [ 360, 139 ].
2529 - 2532 have Frattini factor [ 360, 140 ].
2533 - 2536 have Frattini factor [ 360, 141 ].
2537 - 2543 have Frattini factor [ 360, 142 ].
2544 - 2550 have Frattini factor [ 360, 143 ].
2551 - 2557 have Frattini factor [ 360, 144 ].
2558 - 2582 have Frattini factor [ 360, 145 ].
2583 - 2607 have Frattini factor [ 360, 146 ].
2608 - 2632 have Frattini factor [ 360, 147 ].
2633 - 2657 have Frattini factor [ 360, 148 ].
2658 - 2682 have Frattini factor [ 360, 149 ].
2683 - 2707 have Frattini factor [ 360, 150 ].
2708 - 3042 have Frattini factor [ 360, 151 ].
3043 - 3377 have Frattini factor [ 360, 152 ].
3378 - 3572 have Frattini factor [ 360, 153 ].
3573 - 3907 have Frattini factor [ 360, 154 ].
3908 - 4102 have Frattini factor [ 360, 155 ].
4103 - 4109 have Frattini factor [ 360, 156 ].
4110 - 4195 have Frattini factor [ 360, 157 ].
4196 - 4281 have Frattini factor [ 360, 158 ].
4282 - 4367 have Frattini factor [ 360, 159 ].
4368 - 4453 have Frattini factor [ 360, 160 ].
4454 - 4539 have Frattini factor [ 360, 161 ].
4540 - 4563 have Frattini factor [ 360, 162 ].
4564 has Frattini factor [ 480, 1188 ].
4565 has Frattini factor [ 480, 1189 ].
4566 has Frattini factor [ 480, 1190 ].
4567 has Frattini factor [ 480, 1191 ].
4568 has Frattini factor [ 480, 1192 ].
4569 has Frattini factor [ 480, 1193 ].
4570 has Frattini factor [ 480, 1194 ].
4571 has Frattini factor [ 480, 1195 ].
4572 has Frattini factor [ 480, 1196 ].
4573 has Frattini factor [ 480, 1197 ].
4574 has Frattini factor [ 480, 1198 ].
4575 has Frattini factor [ 480, 1199 ].
4576 has Frattini factor [ 480, 1200 ].
4577 has Frattini factor [ 480, 1201 ].
4578 has Frattini factor [ 480, 1202 ].
4579 has Frattini factor [ 480, 1203 ].
4580 has Frattini factor [ 480, 1204 ].
4581 has Frattini factor [ 480, 1205 ].
4582 has Frattini factor [ 480, 1206 ].
4583 has Frattini factor [ 480, 1207 ].
4584 has Frattini factor [ 480, 1208 ].
4585 has Frattini factor [ 480, 1209 ].
4586 has Frattini factor [ 480, 1210 ].
4587 has Frattini factor [ 480, 1211 ].
4588 has Frattini factor [ 480, 1212 ].
4589 has Frattini factor [ 480, 1213 ].
4590 - 4591 have Frattini factor [ 720, 763 ].
4592 - 4594 have Frattini factor [ 720, 764 ].
4595 has Frattini factor [ 720, 765 ].
4596 - 4598 have Frattini factor [ 720, 766 ].
4599 - 4613 have Frattini factor [ 720, 767 ].
4614 - 4616 have Frattini factor [ 720, 768 ].
4617 - 4627 have Frattini factor [ 720, 769 ].
4628 - 4638 have Frattini factor [ 720, 770 ].
4639 - 4649 have Frattini factor [ 720, 771 ].
4650 - 4656 have Frattini factor [ 720, 772 ].
4657 - 4663 have Frattini factor [ 720, 773 ].
4664 - 4678 have Frattini factor [ 720, 774 ].
4679 - 4685 have Frattini factor [ 720, 775 ].
4686 - 4692 have Frattini factor [ 720, 776 ].
4693 - 4699 have Frattini factor [ 720, 777 ].
4700 - 4706 have Frattini factor [ 720, 778 ].
4707 - 4709 have Frattini factor [ 720, 779 ].
4710 - 4712 have Frattini factor [ 720, 780 ].
4713 - 4715 have Frattini factor [ 720, 781 ].
4716 - 4718 have Frattini factor [ 720, 782 ].
4719 - 4757 have Frattini factor [ 720, 783 ].
4758 - 4796 have Frattini factor [ 720, 784 ].
4797 - 4805 have Frattini factor [ 720, 785 ].
4806 - 4808 have Frattini factor [ 720, 787 ].
4809 - 4811 have Frattini factor [ 720, 788 ].
4812 - 4814 have Frattini factor [ 720, 789 ].
4815 - 4817 have Frattini factor [ 720, 790 ].
4818 - 4832 have Frattini factor [ 720, 791 ].
4833 - 4847 have Frattini factor [ 720, 792 ].
4848 - 4862 have Frattini factor [ 720, 793 ].
4863 - 4877 have Frattini factor [ 720, 794 ].
4878 - 4892 have Frattini factor [ 720, 795 ].
4893 - 4907 have Frattini factor [ 720, 796 ].
4908 - 4922 have Frattini factor [ 720, 797 ].
4923 - 4924 have Frattini factor [ 720, 798 ].
4925 - 4931 have Frattini factor [ 720, 799 ].
4932 - 4938 have Frattini factor [ 720, 800 ].
4939 - 4945 have Frattini factor [ 720, 801 ].
4946 - 4976 have Frattini factor [ 720, 802 ].
4977 - 5007 have Frattini factor [ 720, 803 ].
5008 - 5038 have Frattini factor [ 720, 804 ].
5039 - 5063 have Frattini factor [ 720, 805 ].
5064 - 5088 have Frattini factor [ 720, 806 ].
5089 - 5119 have Frattini factor [ 720, 807 ].
5120 - 5142 have Frattini factor [ 720, 808 ].
5143 - 5177 have Frattini factor [ 720, 809 ].
5178 - 5200 have Frattini factor [ 720, 810 ].
5201 - 5205 have Frattini factor [ 720, 811 ].
5206 - 5214 have Frattini factor [ 720, 812 ].
5215 - 5361 have Frattini factor [ 720, 813 ].
5362 - 5372 have Frattini factor [ 720, 814 ].
5373 - 5383 have Frattini factor [ 720, 815 ].
5384 - 5394 have Frattini factor [ 720, 816 ].
5395 - 5405 have Frattini factor [ 720, 817 ].
5406 - 5416 have Frattini factor [ 720, 818 ].
5417 - 5427 have Frattini factor [ 720, 819 ].
5428 - 5438 have Frattini factor [ 720, 820 ].
5439 has Frattini factor [ 720, 821 ].
5440 - 5452 have Frattini factor [ 720, 822 ].
5453 - 5465 have Frattini factor [ 720, 823 ].
5466 - 5478 have Frattini factor [ 720, 824 ].
5479 - 5491 have Frattini factor [ 720, 825 ].
5492 - 5504 have Frattini factor [ 720, 826 ].
5505 - 5517 have Frattini factor [ 720, 827 ].
5518 - 5570 have Frattini factor [ 720, 828 ].
5571 - 5623 have Frattini factor [ 720, 829 ].
5624 - 5660 have Frattini factor [ 720, 830 ].
5661 - 5713 have Frattini factor [ 720, 831 ].
5714 - 5750 have Frattini factor [ 720, 832 ].
5751 - 5757 have Frattini factor [ 720, 833 ].
5758 - 5759 have Frattini factor [ 720, 834 ].
5760 - 5774 have Frattini factor [ 720, 835 ].
5775 - 5789 have Frattini factor [ 720, 836 ].
5790 - 5804 have Frattini factor [ 720, 837 ].
5805 - 5819 have Frattini factor [ 720, 838 ].
5820 - 5834 have Frattini factor [ 720, 839 ].
5835 - 5840 have Frattini factor [ 720, 840 ].
5841 - 5958 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 5 of the SmallGroups library.
IdSmallGroup is available for this size.