# Groups of order 1008

## Contents

See pages on algebraic structures of order 1008| See pages on groups of a particular order

## Statistics at a glance

The number 1008 has the prime factors 2, 3, and 7. It has prime factorization:

$\! 1008 = 2^4 \cdot 3^2 \cdot 7^1 = 16 \cdot 9 \cdot 7$

Quantity Value Explanation
Total number of groups up to isomorphism 954
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 $7^1$) = (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 7) = $14 \times 2 \times 1 = 28$. 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 767
Number of solvable groups up to isomorphism 948 The only non-solvable groups are the ones with either projective special linear group:PSL(3,2) (order 168) or projective special linear group:PSL(2,8) as a simple non-abelian composition factor. There are five groups of the former type and one group of the latter type.
Number of simple non-abelian groups up to isomorphism 0

## GAP implementation

The order 1008 is part of GAP's SmallGroup library. Hence, any group of order 1008 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 1008 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(1008);

There are 954 groups of order 1008.
They are sorted by their Frattini factors.
1 has Frattini factor [ 42, 1 ].
2 has Frattini factor [ 42, 2 ].
3 has Frattini factor [ 42, 3 ].
4 has Frattini factor [ 42, 4 ].
5 has Frattini factor [ 42, 5 ].
6 has Frattini factor [ 42, 6 ].
7 - 22 have Frattini factor [ 84, 7 ].
23 - 47 have Frattini factor [ 84, 8 ].
48 - 55 have Frattini factor [ 84, 9 ].
56 has Frattini factor [ 84, 10 ].
57 has Frattini factor [ 84, 11 ].
58 - 73 have Frattini factor [ 84, 12 ].
74 - 89 have Frattini factor [ 84, 13 ].
90 - 105 have Frattini factor [ 84, 14 ].
106 - 113 have Frattini factor [ 84, 15 ].
114 has Frattini factor [ 126, 7 ].
115 has Frattini factor [ 126, 8 ].
116 has Frattini factor [ 126, 9 ].
117 has Frattini factor [ 126, 10 ].
118 has Frattini factor [ 126, 11 ].
119 has Frattini factor [ 126, 12 ].
120 has Frattini factor [ 126, 13 ].
121 has Frattini factor [ 126, 14 ].
122 has Frattini factor [ 126, 15 ].
123 has Frattini factor [ 126, 16 ].
124 - 126 have Frattini factor [ 168, 45 ].
127 - 129 have Frattini factor [ 168, 46 ].
130 - 139 have Frattini factor [ 168, 47 ].
140 - 142 have Frattini factor [ 168, 48 ].
143 - 145 have Frattini factor [ 168, 49 ].
146 - 172 have Frattini factor [ 168, 50 ].
173 - 176 have Frattini factor [ 168, 51 ].
177 - 179 have Frattini factor [ 168, 52 ].
180 - 182 have Frattini factor [ 168, 53 ].
183 - 192 have Frattini factor [ 168, 54 ].
193 - 202 have Frattini factor [ 168, 55 ].
203 - 212 have Frattini factor [ 168, 56 ].
213 - 216 have Frattini factor [ 168, 57 ].
217 - 241 have Frattini factor [ 252, 26 ].
242 has Frattini factor [ 252, 27 ].
243 - 258 have Frattini factor [ 252, 28 ].
259 - 274 have Frattini factor [ 252, 29 ].
275 - 290 have Frattini factor [ 252, 30 ].
291 has Frattini factor [ 252, 31 ].
292 has Frattini factor [ 252, 32 ].
293 - 317 have Frattini factor [ 252, 33 ].
318 - 342 have Frattini factor [ 252, 34 ].
343 - 358 have Frattini factor [ 252, 35 ].
359 - 383 have Frattini factor [ 252, 36 ].
384 - 399 have Frattini factor [ 252, 37 ].
400 - 407 have Frattini factor [ 252, 38 ].
408 has Frattini factor [ 252, 39 ].
409 has Frattini factor [ 252, 40 ].
410 - 425 have Frattini factor [ 252, 41 ].
426 - 441 have Frattini factor [ 252, 42 ].
442 - 457 have Frattini factor [ 252, 43 ].
458 - 473 have Frattini factor [ 252, 44 ].
474 - 489 have Frattini factor [ 252, 45 ].
490 - 497 have Frattini factor [ 252, 46 ].
498 has Frattini factor [ 336, 210 ].
499 has Frattini factor [ 336, 211 ].
500 has Frattini factor [ 336, 212 ].
501 has Frattini factor [ 336, 213 ].
502 has Frattini factor [ 336, 214 ].
503 has Frattini factor [ 336, 215 ].
504 has Frattini factor [ 336, 216 ].
505 has Frattini factor [ 336, 217 ].
506 has Frattini factor [ 336, 218 ].
507 has Frattini factor [ 336, 219 ].
508 has Frattini factor [ 336, 220 ].
509 has Frattini factor [ 336, 221 ].
510 has Frattini factor [ 336, 222 ].
511 has Frattini factor [ 336, 223 ].
512 has Frattini factor [ 336, 224 ].
513 has Frattini factor [ 336, 225 ].
514 has Frattini factor [ 336, 226 ].
515 has Frattini factor [ 336, 227 ].
516 has Frattini factor [ 336, 228 ].
517 has Frattini factor [ 504, 157 ].
518 - 520 have Frattini factor [ 504, 159 ].
521 - 523 have Frattini factor [ 504, 160 ].
524 - 526 have Frattini factor [ 504, 161 ].
527 - 553 have Frattini factor [ 504, 162 ].
554 - 556 have Frattini factor [ 504, 163 ].
557 has Frattini factor [ 504, 164 ].
558 has Frattini factor [ 504, 165 ].
559 - 565 have Frattini factor [ 504, 166 ].
566 - 570 have Frattini factor [ 504, 167 ].
571 - 577 have Frattini factor [ 504, 168 ].
578 - 582 have Frattini factor [ 504, 169 ].
583 has Frattini factor [ 504, 170 ].
584 - 585 have Frattini factor [ 504, 171 ].
586 - 624 have Frattini factor [ 504, 172 ].
625 - 627 have Frattini factor [ 504, 174 ].
628 - 630 have Frattini factor [ 504, 175 ].
631 - 633 have Frattini factor [ 504, 176 ].
634 - 636 have Frattini factor [ 504, 177 ].
637 - 646 have Frattini factor [ 504, 178 ].
647 - 656 have Frattini factor [ 504, 179 ].
657 - 666 have Frattini factor [ 504, 180 ].
667 - 669 have Frattini factor [ 504, 181 ].
670 - 672 have Frattini factor [ 504, 182 ].
673 - 675 have Frattini factor [ 504, 183 ].
676 - 678 have Frattini factor [ 504, 184 ].
679 - 681 have Frattini factor [ 504, 185 ].
682 - 684 have Frattini factor [ 504, 186 ].
685 - 691 have Frattini factor [ 504, 187 ].
692 - 698 have Frattini factor [ 504, 188 ].
699 - 725 have Frattini factor [ 504, 189 ].
726 - 752 have Frattini factor [ 504, 190 ].
753 - 770 have Frattini factor [ 504, 191 ].
771 - 797 have Frattini factor [ 504, 192 ].
798 - 815 have Frattini factor [ 504, 193 ].
816 - 819 have Frattini factor [ 504, 194 ].
820 - 822 have Frattini factor [ 504, 195 ].
823 - 825 have Frattini factor [ 504, 196 ].
826 - 835 have Frattini factor [ 504, 197 ].
836 - 845 have Frattini factor [ 504, 198 ].
846 - 855 have Frattini factor [ 504, 199 ].
856 - 865 have Frattini factor [ 504, 200 ].
866 - 875 have Frattini factor [ 504, 201 ].
876 - 879 have Frattini factor [ 504, 202 ].
880 - 954 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.