# Groups of order 720

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## Contents

See pages on algebraic structures of order 720| See pages on groups of a particular order

## Statistics at a glance

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

### Group counts

Quantity Value Explanation
Total number of groups up to isomorphism 840
Number of abelian groups (i.e., finite 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 $5^1$) = (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 (i.e., finite 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 5) = $14 \times 2 \times 1 = 28$. 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 solvable groups (i.e., finite solvable groups) up to isomorphism 817 See note on non-solvable groups
Number of non-solvable groups up to isomorphism 23 There are 23 non-solvable groups, of which 5 have alternating group:A6 as one of the composition factors (including direct product of A6 and Z2, the one quasisimple group $SL(2,9)$ and the three almost simple groups). The remaining 17 have alternating group:A5 as the unique non-abelian composition factor in them, with the other factors being cyclic of prime order (two cyclic group:Z2s, one cyclic group:Z3).
Number of simple groups up to isomorphism 0
Number of quasisimple groups up to isomorphism 1 special linear group:SL(2,9) (ID: (720,409))
Number of almost simple groups up to isomorphism 3 symmetric group:S6 (ID: (720,763)), projective general linear group:PGL(2,9) (ID: (720,764)), Mathieu group:M10 (ID: (720,765)). These are precisely the three subgroups of order 720 between alternating group:A6 and its automorphism group, which has order 1440 and where the quotient group (the outer automorphism group) is a Klein four-group.
Number of almost quasisimple groups up to isomorphism 4 The almost simple groups and the quasisimple group
Number of perfect groups up to isomorphism 1 special linear group:SL(2,9) (ID: (720,409))

## GAP implementation

The order 720 is part of GAP's SmallGroup library. Hence, any group of order 720 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 720 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(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.