# Groups of order 1200

See pages on algebraic structures of order 1200| See pages on groups of a particular order

## Statistics at a glance

The number 1200 has prime factors 2, 3, and 5. The prime factorization is:

$\! 1200 = 2^4 \cdot 3^1 \cdot 5^2 = 16 \cdot 3 \cdot 25$

There are both solvable and non-solvable groups of this order.

Quantity Value Explanation
Total number of groups up to isomorphism 1040
Number of abelian groups up to isomorphism 10 (number of abelian groups of order $2^4$) $\times$ (number of abelian groups of order $3^1$) $\times$ (number of abelian groups of order $5^2$) = (number of unordered integer partitions of 4) $\times$ (number of unordered integer partitions of 1) $\times$ (number of unordered integer partitions of 2) = $5 \times 1 \times 2 = 10$.
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 3) $\times$ (number of groups of order 25) = $14 \times 1 \times 2 = 28$.
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 765
Number of solvable groups up to isomorphism 1021

## GAP implementation

The order 1200 is part of GAP's SmallGroup library. Hence, any group of order 1200 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 1200 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(1200);

There are 1040 groups of order 1200.
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 - 95 have Frattini factor [ 120, 36 ].
96 - 98 have Frattini factor [ 120, 37 ].
99 - 101 have Frattini factor [ 120, 38 ].
102 - 104 have Frattini factor [ 120, 39 ].
105 - 111 have Frattini factor [ 120, 40 ].
112 - 118 have Frattini factor [ 120, 41 ].
119 - 145 have Frattini factor [ 120, 42 ].
146 - 148 have Frattini factor [ 120, 43 ].
149 - 158 have Frattini factor [ 120, 44 ].
159 - 168 have Frattini factor [ 120, 45 ].
169 - 178 have Frattini factor [ 120, 46 ].
179 - 182 have Frattini factor [ 120, 47 ].
183 has Frattini factor [ 150, 5 ].
184 has Frattini factor [ 150, 6 ].
185 has Frattini factor [ 150, 7 ].
186 has Frattini factor [ 150, 8 ].
187 has Frattini factor [ 150, 9 ].
188 has Frattini factor [ 150, 10 ].
189 has Frattini factor [ 150, 11 ].
190 has Frattini factor [ 150, 12 ].
191 has Frattini factor [ 150, 13 ].
192 has Frattini factor [ 240, 191 ].
193 has Frattini factor [ 240, 192 ].
194 has Frattini factor [ 240, 193 ].
195 has Frattini factor [ 240, 194 ].
196 has Frattini factor [ 240, 195 ].
197 has Frattini factor [ 240, 196 ].
198 has Frattini factor [ 240, 197 ].
199 has Frattini factor [ 240, 198 ].
200 has Frattini factor [ 240, 199 ].
201 has Frattini factor [ 240, 200 ].
202 has Frattini factor [ 240, 201 ].
203 has Frattini factor [ 240, 202 ].
204 has Frattini factor [ 240, 203 ].
205 has Frattini factor [ 240, 204 ].
206 has Frattini factor [ 240, 205 ].
207 has Frattini factor [ 240, 206 ].
208 has Frattini factor [ 240, 207 ].
209 has Frattini factor [ 240, 208 ].
210 has Frattini factor [ 300, 23 ].
211 has Frattini factor [ 300, 24 ].
212 - 227 have Frattini factor [ 300, 25 ].
228 - 243 have Frattini factor [ 300, 26 ].
244 - 259 have Frattini factor [ 300, 27 ].
260 has Frattini factor [ 300, 28 ].
261 has Frattini factor [ 300, 29 ].
262 has Frattini factor [ 300, 30 ].
263 has Frattini factor [ 300, 31 ].
264 has Frattini factor [ 300, 32 ].
265 has Frattini factor [ 300, 33 ].
266 has Frattini factor [ 300, 34 ].
267 has Frattini factor [ 300, 35 ].
268 - 283 have Frattini factor [ 300, 36 ].
284 - 308 have Frattini factor [ 300, 37 ].
309 - 333 have Frattini factor [ 300, 38 ].
334 - 358 have Frattini factor [ 300, 39 ].
359 - 374 have Frattini factor [ 300, 40 ].
375 - 382 have Frattini factor [ 300, 41 ].
383 has Frattini factor [ 300, 42 ].
384 has Frattini factor [ 300, 43 ].
385 - 400 have Frattini factor [ 300, 44 ].
401 - 416 have Frattini factor [ 300, 45 ].
417 - 432 have Frattini factor [ 300, 46 ].
433 - 448 have Frattini factor [ 300, 47 ].
449 - 464 have Frattini factor [ 300, 48 ].
465 - 472 have Frattini factor [ 300, 49 ].
473 - 475 have Frattini factor [ 600, 144 ].
476 - 478 have Frattini factor [ 600, 145 ].
479 - 481 have Frattini factor [ 600, 146 ].
482 - 484 have Frattini factor [ 600, 147 ].
485 has Frattini factor [ 600, 148 ].
486 has Frattini factor [ 600, 149 ].
487 - 491 have Frattini factor [ 600, 151 ].
492 - 498 have Frattini factor [ 600, 152 ].
499 - 505 have Frattini factor [ 600, 153 ].
506 - 523 have Frattini factor [ 600, 154 ].
524 has Frattini factor [ 600, 155 ].
525 has Frattini factor [ 600, 156 ].
526 - 530 have Frattini factor [ 600, 157 ].
531 - 537 have Frattini factor [ 600, 158 ].
538 - 544 have Frattini factor [ 600, 159 ].
545 - 551 have Frattini factor [ 600, 160 ].
552 - 558 have Frattini factor [ 600, 161 ].
559 - 563 have Frattini factor [ 600, 162 ].
564 - 568 have Frattini factor [ 600, 163 ].
569 - 573 have Frattini factor [ 600, 164 ].
574 - 580 have Frattini factor [ 600, 165 ].
581 - 587 have Frattini factor [ 600, 166 ].
588 - 594 have Frattini factor [ 600, 167 ].
595 - 601 have Frattini factor [ 600, 168 ].
602 - 606 have Frattini factor [ 600, 169 ].
607 - 611 have Frattini factor [ 600, 170 ].
612 - 618 have Frattini factor [ 600, 171 ].
619 has Frattini factor [ 600, 172 ].
620 - 621 have Frattini factor [ 600, 173 ].
622 - 660 have Frattini factor [ 600, 174 ].
661 - 670 have Frattini factor [ 600, 175 ].
671 - 673 have Frattini factor [ 600, 176 ].
674 - 676 have Frattini factor [ 600, 177 ].
677 - 679 have Frattini factor [ 600, 178 ].
680 - 682 have Frattini factor [ 600, 179 ].
683 - 692 have Frattini factor [ 600, 180 ].
693 - 695 have Frattini factor [ 600, 181 ].
696 - 698 have Frattini factor [ 600, 182 ].
699 - 701 have Frattini factor [ 600, 183 ].
702 - 708 have Frattini factor [ 600, 184 ].
709 - 715 have Frattini factor [ 600, 185 ].
716 - 722 have Frattini factor [ 600, 186 ].
723 - 729 have Frattini factor [ 600, 187 ].
730 - 736 have Frattini factor [ 600, 188 ].
737 - 743 have Frattini factor [ 600, 189 ].
744 - 750 have Frattini factor [ 600, 190 ].
751 - 757 have Frattini factor [ 600, 191 ].
758 - 775 have Frattini factor [ 600, 192 ].
776 - 802 have Frattini factor [ 600, 193 ].
803 - 829 have Frattini factor [ 600, 194 ].
830 - 856 have Frattini factor [ 600, 195 ].
857 - 874 have Frattini factor [ 600, 196 ].
875 - 878 have Frattini factor [ 600, 197 ].
879 - 881 have Frattini factor [ 600, 198 ].
882 - 884 have Frattini factor [ 600, 199 ].
885 - 894 have Frattini factor [ 600, 200 ].
895 - 904 have Frattini factor [ 600, 201 ].
905 - 914 have Frattini factor [ 600, 202 ].
915 - 924 have Frattini factor [ 600, 203 ].
925 - 934 have Frattini factor [ 600, 204 ].
935 - 938 have Frattini factor [ 600, 205 ].
939 - 1040 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.