# Groups of order 1080

## Contents

See pages on algebraic structures of order 1080| See pages on groups of a particular order

## Statistics at a glance

The prime factorization of 1080 is:

$\! 1080 = 2^3 \cdot 3^3 \cdot 5^1 = 8 \cdot 27 \cdot 5$

Quantity Value Explanation
Total number of groups up to isomorphism 583
Number of abelian groups (i.e., finite abelian groups) up to isomorphism 9 (Number of abelian groups of order $2^3$) times (Number of abelian groups of order $3^3$) times (Number of abelian groups of order $5^1$) = (number of unordered integer partitions of 3) times (number of unordered integer partitions of 3) times (number of unordered integer partitions of 1) = $3 \times 3 \times 1 = 9$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups (i.e., finite nilpotent groups) up to isomorphism 25 (Number of groups of order 8) times (Number of groups of order 27) times (Number of groups of order 5) = $5 \times 5 \times 1 = 25$. 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 solvable groups (i.e., finite solvable group)s up to isomorphism 569 (see note on non-solvable groups)
Number of non-solvable groups up to isomorphism 14 There are two possibilities for the composition factors of non-solvable groups:
One is that alternating group:A5 (order 60) is the only simple non-abelian composition factor, and the rest are cyclic group:Z2 (1 time) and cyclic group:Z3 (2 times). There are 12 groups of this type.
The other is that alternating group:A6 (order 360) is the only simple non-abelian composition factor, and the other composition factor is cyclic group:Z3. There are two groups of this type: triple cover of alternating group:A6 and direct product of A6 and Z3.
Number of simple groups up to isomorphism 0
Number of quasisimple groups up to isomorphism 1 triple cover of alternating group:A6 (note that $A_6$ is unique among alternating groups in having a triple cover)
Number of almost simple groups up to isomorphism 0

## GAP implementation

The order 1080 is part of GAP's SmallGroup library. Hence, any group of order 1080 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 1080 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(1080);

There are 583 groups of order 1080.
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 - 13 have Frattini factor [ 60, 8 ].
14 has Frattini factor [ 60, 9 ].
15 - 19 have Frattini factor [ 60, 10 ].
20 - 24 have Frattini factor [ 60, 11 ].
25 - 29 have Frattini factor [ 60, 12 ].
30 - 32 have Frattini factor [ 60, 13 ].
33 - 35 have Frattini factor [ 90, 5 ].
36 - 39 have Frattini factor [ 90, 6 ].
40 - 43 have Frattini factor [ 90, 7 ].
44 - 45 have Frattini factor [ 90, 8 ].
46 - 47 have Frattini factor [ 90, 9 ].
48 - 50 have Frattini factor [ 90, 10 ].
51 has Frattini factor [ 120, 36 ].
52 has Frattini factor [ 120, 37 ].
53 has Frattini factor [ 120, 38 ].
54 has Frattini factor [ 120, 39 ].
55 has Frattini factor [ 120, 40 ].
56 has Frattini factor [ 120, 41 ].
57 has Frattini factor [ 120, 42 ].
58 has Frattini factor [ 120, 43 ].
59 has Frattini factor [ 120, 44 ].
60 has Frattini factor [ 120, 45 ].
61 has Frattini factor [ 120, 46 ].
62 has Frattini factor [ 120, 47 ].
63 has Frattini factor [ 180, 19 ].
64 - 66 have Frattini factor [ 180, 20 ].
67 - 70 have Frattini factor [ 180, 21 ].
71 - 72 have Frattini factor [ 180, 22 ].
73 has Frattini factor [ 180, 23 ].
74 has Frattini factor [ 180, 24 ].
75 has Frattini factor [ 180, 25 ].
76 - 103 have Frattini factor [ 180, 26 ].
104 - 117 have Frattini factor [ 180, 27 ].
118 - 129 have Frattini factor [ 180, 28 ].
130 - 150 have Frattini factor [ 180, 29 ].
151 - 162 have Frattini factor [ 180, 30 ].
163 - 167 have Frattini factor [ 180, 31 ].
168 - 182 have Frattini factor [ 180, 32 ].
183 - 202 have Frattini factor [ 180, 33 ].
203 - 222 have Frattini factor [ 180, 34 ].
223 - 232 have Frattini factor [ 180, 35 ].
233 - 242 have Frattini factor [ 180, 36 ].
243 - 251 have Frattini factor [ 180, 37 ].
252 has Frattini factor [ 270, 23 ].
253 has Frattini factor [ 270, 24 ].
254 has Frattini factor [ 270, 25 ].
255 has Frattini factor [ 270, 26 ].
256 has Frattini factor [ 270, 27 ].
257 has Frattini factor [ 270, 28 ].
258 has Frattini factor [ 270, 29 ].
259 has Frattini factor [ 270, 30 ].
260 has Frattini factor [ 360, 118 ].
261 has Frattini factor [ 360, 119 ].
262 has Frattini factor [ 360, 120 ].
263 has Frattini factor [ 360, 121 ].
264 has Frattini factor [ 360, 122 ].
265 has Frattini factor [ 360, 123 ].
266 has Frattini factor [ 360, 124 ].
267 has Frattini factor [ 360, 125 ].
268 - 271 have Frattini factor [ 360, 126 ].
272 - 273 have Frattini factor [ 360, 127 ].
274 - 276 have Frattini factor [ 360, 128 ].
277 - 278 have Frattini factor [ 360, 129 ].
279 has Frattini factor [ 360, 130 ].
280 has Frattini factor [ 360, 131 ].
281 has Frattini factor [ 360, 132 ].
282 has Frattini factor [ 360, 133 ].
283 has Frattini factor [ 360, 134 ].
284 has Frattini factor [ 360, 135 ].
285 has Frattini factor [ 360, 136 ].
286 - 287 have Frattini factor [ 360, 137 ].
288 - 291 have Frattini factor [ 360, 138 ].
292 - 295 have Frattini factor [ 360, 139 ].
296 - 298 have Frattini factor [ 360, 140 ].
299 - 301 have Frattini factor [ 360, 141 ].
302 - 306 have Frattini factor [ 360, 142 ].
307 - 310 have Frattini factor [ 360, 143 ].
311 - 314 have Frattini factor [ 360, 144 ].
315 - 317 have Frattini factor [ 360, 145 ].
318 - 321 have Frattini factor [ 360, 146 ].
322 - 323 have Frattini factor [ 360, 147 ].
324 has Frattini factor [ 360, 148 ].
325 has Frattini factor [ 360, 149 ].
326 has Frattini factor [ 360, 150 ].
327 - 330 have Frattini factor [ 360, 151 ].
331 - 332 have Frattini factor [ 360, 152 ].
333 - 334 have Frattini factor [ 360, 153 ].
335 - 337 have Frattini factor [ 360, 154 ].
338 - 339 have Frattini factor [ 360, 155 ].
340 - 344 have Frattini factor [ 360, 156 ].
345 - 347 have Frattini factor [ 360, 157 ].
348 - 351 have Frattini factor [ 360, 158 ].
352 - 355 have Frattini factor [ 360, 159 ].
356 - 357 have Frattini factor [ 360, 160 ].
358 - 359 have Frattini factor [ 360, 161 ].
360 - 362 have Frattini factor [ 360, 162 ].
363 has Frattini factor [ 540, 88 ].
364 has Frattini factor [ 540, 89 ].
365 has Frattini factor [ 540, 90 ].
366 has Frattini factor [ 540, 91 ].
367 has Frattini factor [ 540, 92 ].
368 has Frattini factor [ 540, 93 ].
369 has Frattini factor [ 540, 94 ].
370 has Frattini factor [ 540, 95 ].
371 has Frattini factor [ 540, 96 ].
372 has Frattini factor [ 540, 97 ].
373 has Frattini factor [ 540, 98 ].
374 - 380 have Frattini factor [ 540, 99 ].
381 - 387 have Frattini factor [ 540, 100 ].
388 - 392 have Frattini factor [ 540, 101 ].
393 - 399 have Frattini factor [ 540, 102 ].
400 - 404 have Frattini factor [ 540, 103 ].
405 - 411 have Frattini factor [ 540, 104 ].
412 - 418 have Frattini factor [ 540, 105 ].
419 - 425 have Frattini factor [ 540, 106 ].
426 - 432 have Frattini factor [ 540, 107 ].
433 - 439 have Frattini factor [ 540, 108 ].
440 - 442 have Frattini factor [ 540, 109 ].
443 - 447 have Frattini factor [ 540, 110 ].
448 has Frattini factor [ 540, 111 ].
449 - 453 have Frattini factor [ 540, 112 ].
454 - 458 have Frattini factor [ 540, 113 ].
459 - 463 have Frattini factor [ 540, 114 ].
464 - 468 have Frattini factor [ 540, 115 ].
469 - 473 have Frattini factor [ 540, 116 ].
474 - 478 have Frattini factor [ 540, 117 ].
479 - 483 have Frattini factor [ 540, 118 ].
484 - 486 have Frattini factor [ 540, 119 ].
487 - 583 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.