# Groups of order 432

## Contents

See pages on algebraic structures of order 432| See pages on groups of a particular order

## Statistics at a glance

The number 432 has prime factorization:

$\! 432 = 2^4 \cdot 3^3 = 16 \cdot 27$

There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^aq^b$-theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.

Quantity Value Explanation
Total number of groups up to isomorphism 775
Number of abelian groups (i.e., finite abelian groups) up to isomorphism 15 (number of abelian groups of order $2^4$) times (number of abelian groups of order $3^3$) = (number of unordered integer partitions of 4) times (number of unordered integer partitions of 3) = $5 \times 3 = 15$. 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 70 (number of groups of order 16) times (number of groups of order 27) = $14 \times 5 = 70$.
Number of supersolvable groups (i.e., finite supersolvable groups) up to isomorphism 565
Number of solvable groups (i.e., finite solvable groups) up to isomorphism 775 There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^aq^b$-theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.
Number of simple groups up to isomorphism 0

## GAP implementation

The order 432 is part of GAP's SmallGroup library. Hence, any group of order 432 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 432 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(432);

There are 775 groups of order 432.
They are sorted by their Frattini factors.
1 has Frattini factor [ 6, 1 ].
2 has Frattini factor [ 6, 2 ].
3 has Frattini factor [ 12, 3 ].
4 - 19 have Frattini factor [ 12, 4 ].
20 - 27 have Frattini factor [ 12, 5 ].
28 - 31 have Frattini factor [ 18, 3 ].
32 - 33 have Frattini factor [ 18, 4 ].
34 - 36 have Frattini factor [ 18, 5 ].
37 - 39 have Frattini factor [ 24, 12 ].
40 - 42 have Frattini factor [ 24, 13 ].
43 - 52 have Frattini factor [ 24, 14 ].
53 - 56 have Frattini factor [ 24, 15 ].
57 has Frattini factor [ 36, 9 ].
58 - 98 have Frattini factor [ 36, 10 ].
99 - 103 have Frattini factor [ 36, 11 ].
104 - 167 have Frattini factor [ 36, 12 ].
168 - 199 have Frattini factor [ 36, 13 ].
200 - 223 have Frattini factor [ 36, 14 ].
224 has Frattini factor [ 48, 48 ].
225 has Frattini factor [ 48, 49 ].
226 has Frattini factor [ 48, 50 ].
227 has Frattini factor [ 48, 51 ].
228 has Frattini factor [ 48, 52 ].
229 has Frattini factor [ 54, 12 ].
230 has Frattini factor [ 54, 13 ].
231 has Frattini factor [ 54, 14 ].
232 has Frattini factor [ 54, 15 ].
233 has Frattini factor [ 72, 39 ].
234 - 238 have Frattini factor [ 72, 40 ].
239 has Frattini factor [ 72, 41 ].
240 - 251 have Frattini factor [ 72, 42 ].
252 - 260 have Frattini factor [ 72, 43 ].
261 - 272 have Frattini factor [ 72, 44 ].
273 - 279 have Frattini factor [ 72, 45 ].
280 - 324 have Frattini factor [ 72, 46 ].
325 - 339 have Frattini factor [ 72, 47 ].
340 - 379 have Frattini factor [ 72, 48 ].
380 - 399 have Frattini factor [ 72, 49 ].
400 - 411 have Frattini factor [ 72, 50 ].
412 has Frattini factor [ 108, 36 ].
413 has Frattini factor [ 108, 37 ].
414 - 429 have Frattini factor [ 108, 38 ].
430 - 454 have Frattini factor [ 108, 39 ].
455 - 462 have Frattini factor [ 108, 40 ].
463 has Frattini factor [ 108, 41 ].
464 - 479 have Frattini factor [ 108, 42 ].
480 - 495 have Frattini factor [ 108, 43 ].
496 - 511 have Frattini factor [ 108, 44 ].
512 - 519 have Frattini factor [ 108, 45 ].
520 has Frattini factor [ 144, 182 ].
521 - 523 have Frattini factor [ 144, 183 ].
524 - 528 have Frattini factor [ 144, 184 ].
529 has Frattini factor [ 144, 185 ].
530 has Frattini factor [ 144, 186 ].
531 has Frattini factor [ 144, 187 ].
532 - 535 have Frattini factor [ 144, 188 ].
536 - 538 have Frattini factor [ 144, 189 ].
539 - 542 have Frattini factor [ 144, 190 ].
543 has Frattini factor [ 144, 191 ].
544 - 545 have Frattini factor [ 144, 192 ].
546 - 550 have Frattini factor [ 144, 193 ].
551 - 555 have Frattini factor [ 144, 194 ].
556 - 559 have Frattini factor [ 144, 195 ].
560 - 561 have Frattini factor [ 144, 196 ].
562 - 564 have Frattini factor [ 144, 197 ].
565 has Frattini factor [ 216, 154 ].
566 has Frattini factor [ 216, 155 ].
567 - 573 have Frattini factor [ 216, 156 ].
574 - 578 have Frattini factor [ 216, 157 ].
579 - 585 have Frattini factor [ 216, 158 ].
586 - 590 have Frattini factor [ 216, 159 ].
591 has Frattini factor [ 216, 160 ].
592 - 593 have Frattini factor [ 216, 161 ].
594 - 612 have Frattini factor [ 216, 162 ].
613 - 615 have Frattini factor [ 216, 163 ].
616 - 618 have Frattini factor [ 216, 164 ].
619 - 621 have Frattini factor [ 216, 165 ].
622 - 624 have Frattini factor [ 216, 166 ].
625 - 627 have Frattini factor [ 216, 167 ].
628 - 634 have Frattini factor [ 216, 168 ].
635 - 641 have Frattini factor [ 216, 169 ].
642 - 659 have Frattini factor [ 216, 170 ].
660 - 686 have Frattini factor [ 216, 171 ].
687 - 696 have Frattini factor [ 216, 172 ].
697 - 699 have Frattini factor [ 216, 173 ].
700 - 709 have Frattini factor [ 216, 174 ].
710 - 719 have Frattini factor [ 216, 175 ].
720 - 729 have Frattini factor [ 216, 176 ].
730 - 733 have Frattini factor [ 216, 177 ].
734 - 775 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.