# Groups of order 1584

## Contents

See pages on algebraic structures of order 1584| See pages on groups of a particular order

## Statistics at a glance

The number 1584 has prime factors 2, 3, and 11, with the following prime factorization: $\! 1584 = 2^4 \cdot 3^2 \cdot 11^1 = 16 \cdot 9 \cdot 11$

Quantity Value Explanation
Total number of groups up to isomorphism 697
Number of abelian groups up to isomorphism 10 (number of abelian groups of order $2^4$) $\times$ (number of abelian groups of order $3^2$) $\times$ (number of abelian groups of order $11^1$) = (number of unordered integer partitions of 4) $\times$ (number of unordered integer partitions of 2) $\times$ (number of unordered integer partitions of 1) = $5 \times 2 \times 1 = 10$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups up to isomorphism 28 (number of groups of order 16) $\times$ (number of groups of order 9) $\times$ (number of groups of order 11) = $14 \times 2 \times 1 = 28$. 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.
Number of supersolvable groups up to isomorphism 574
Number of solvable groups up to isomorphism 697 all groups of this order are solvable.
Number of simple non-abelian groups up to isomorphism 0 Follows from all groups of the order being solvable

## GAP implementation

The order 1584 is part of GAP's SmallGroup library. Hence, any group of order 1584 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 1584 can be accessed as a list using GAP's AllSmallGroups function.

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

gap> SmallGroupsInformation(1584);

There are 697 groups of order 1584.
They are sorted by their Frattini factors.
1 has Frattini factor [ 66, 1 ].
2 has Frattini factor [ 66, 2 ].
3 has Frattini factor [ 66, 3 ].
4 has Frattini factor [ 66, 4 ].
5 - 29 have Frattini factor [ 132, 5 ].
30 has Frattini factor [ 132, 6 ].
31 - 46 have Frattini factor [ 132, 7 ].
47 - 62 have Frattini factor [ 132, 8 ].
63 - 78 have Frattini factor [ 132, 9 ].
79 - 86 have Frattini factor [ 132, 10 ].
87 has Frattini factor [ 198, 5 ].
88 has Frattini factor [ 198, 6 ].
89 has Frattini factor [ 198, 7 ].
90 has Frattini factor [ 198, 8 ].
91 has Frattini factor [ 198, 9 ].
92 has Frattini factor [ 198, 10 ].
93 - 95 have Frattini factor [ 264, 31 ].
96 - 98 have Frattini factor [ 264, 32 ].
99 - 101 have Frattini factor [ 264, 33 ].
102 - 128 have Frattini factor [ 264, 34 ].
129 - 131 have Frattini factor [ 264, 35 ].
132 - 141 have Frattini factor [ 264, 36 ].
142 - 151 have Frattini factor [ 264, 37 ].
152 - 161 have Frattini factor [ 264, 38 ].
162 - 165 have Frattini factor [ 264, 39 ].
166 has Frattini factor [ 396, 17 ].
167 has Frattini factor [ 396, 18 ].
168 - 192 have Frattini factor [ 396, 19 ].
193 - 217 have Frattini factor [ 396, 20 ].
218 - 233 have Frattini factor [ 396, 21 ].
234 - 258 have Frattini factor [ 396, 22 ].
259 - 274 have Frattini factor [ 396, 23 ].
275 has Frattini factor [ 396, 24 ].
276 - 291 have Frattini factor [ 396, 25 ].
292 - 307 have Frattini factor [ 396, 26 ].
308 - 323 have Frattini factor [ 396, 27 ].
324 - 339 have Frattini factor [ 396, 28 ].
340 - 355 have Frattini factor [ 396, 29 ].
356 - 363 have Frattini factor [ 396, 30 ].
364 has Frattini factor [ 528, 160 ].
365 has Frattini factor [ 528, 161 ].
366 has Frattini factor [ 528, 162 ].
367 has Frattini factor [ 528, 163 ].
368 has Frattini factor [ 528, 164 ].
369 has Frattini factor [ 528, 165 ].
370 has Frattini factor [ 528, 166 ].
371 has Frattini factor [ 528, 167 ].
372 has Frattini factor [ 528, 168 ].
373 has Frattini factor [ 528, 169 ].
374 has Frattini factor [ 528, 170 ].
375 has Frattini factor [ 792, 108 ].
376 has Frattini factor [ 792, 109 ].
377 - 383 have Frattini factor [ 792, 110 ].
384 - 388 have Frattini factor [ 792, 111 ].
389 - 395 have Frattini factor [ 792, 112 ].
396 - 400 have Frattini factor [ 792, 113 ].
401 has Frattini factor [ 792, 114 ].
402 - 403 have Frattini factor [ 792, 115 ].
404 - 442 have Frattini factor [ 792, 116 ].
443 - 445 have Frattini factor [ 792, 117 ].
446 - 448 have Frattini factor [ 792, 118 ].
449 - 451 have Frattini factor [ 792, 119 ].
452 - 454 have Frattini factor [ 792, 120 ].
455 - 457 have Frattini factor [ 792, 121 ].
458 - 460 have Frattini factor [ 792, 122 ].
461 - 463 have Frattini factor [ 792, 123 ].
464 - 470 have Frattini factor [ 792, 124 ].
471 - 477 have Frattini factor [ 792, 125 ].
478 - 504 have Frattini factor [ 792, 126 ].
505 - 531 have Frattini factor [ 792, 127 ].
532 - 549 have Frattini factor [ 792, 128 ].
550 - 576 have Frattini factor [ 792, 129 ].
577 - 594 have Frattini factor [ 792, 130 ].
595 - 597 have Frattini factor [ 792, 131 ].
598 - 607 have Frattini factor [ 792, 132 ].
608 - 617 have Frattini factor [ 792, 133 ].
618 - 627 have Frattini factor [ 792, 134 ].
628 - 637 have Frattini factor [ 792, 135 ].
638 - 647 have Frattini factor [ 792, 136 ].
648 - 651 have Frattini factor [ 792, 137 ].
652 - 697 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.