# Difference between revisions of "Groups of order 480"

## Contents

See pages on algebraic structures of order 480| See pages on groups of a particular order

## Statistics at a glance

The number 480 has the prime factorization:

$\! 480 = 2^5 \cdot 3^1 \cdot 5^1 =64\cdot 3 \cdot 5$

Quantity Value Explanation
Total number of groups up to isomorphism 1213
Number of abelian groups (i.e., finite abelian groups) up to isomorphism 7 (Number of abelian groups of order $2^5$) times (Number of abelian groups of order $3^1$) times (Number of abelian groups of order $5^1$) = (number of unordered integer partitions of 5) times (number of unordered integer partitions of 1) times (number of unordered integer partitions of 1) = $7 \times 1 \times 1 = 7$. 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 51 (Number of groups of order 32) times (Number of groups of order 3) times (Number of groups of order 5) = $51 \times 1 \times 1 = 51$. 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 (i.e., finite supersolvable groups) up to isomorphism 1082
Number of solvable groups (i.e., finite solvable groups) up to isomorphism 1187 See note on non-solvable groups
Number of non-solvable groups up to isomorphism 26 All the non-solvable groups have alternating group:A5 as the unique simple non-abelian composition factor and three cyclic group:Z2s as the other composition factors
Number of simple groups up to isomorphism 0
Number of quasisimple groups up to isomorphism 0
Number of almost simple groups up to isomorphism 0
Number of perfect groups up to isomorphism 0

## GAP implementation

The order 840 is part of GAP's SmallGroup library. Hence, any group of order 840 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 840 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(480);

There are 1213 groups of order 480.
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 - 73 have Frattini factor [ 60, 8 ].
74 has Frattini factor [ 60, 9 ].
75 - 115 have Frattini factor [ 60, 10 ].
116 - 156 have Frattini factor [ 60, 11 ].
157 - 197 have Frattini factor [ 60, 12 ].
198 - 216 have Frattini factor [ 60, 13 ].
217 - 219 have Frattini factor [ 120, 34 ].
220 - 222 have Frattini factor [ 120, 35 ].
223 - 253 have Frattini factor [ 120, 36 ].
254 - 257 have Frattini factor [ 120, 37 ].
258 - 261 have Frattini factor [ 120, 38 ].
262 - 268 have Frattini factor [ 120, 39 ].
269 - 293 have Frattini factor [ 120, 40 ].
294 - 318 have Frattini factor [ 120, 41 ].
319 - 653 have Frattini factor [ 120, 42 ].
654 - 660 have Frattini factor [ 120, 43 ].
661 - 746 have Frattini factor [ 120, 44 ].
747 - 832 have Frattini factor [ 120, 45 ].
833 - 918 have Frattini factor [ 120, 46 ].
919 - 942 have Frattini factor [ 120, 47 ].
943 - 953 have Frattini factor [ 240, 189 ].
954 - 960 have Frattini factor [ 240, 190 ].
961 - 963 have Frattini factor [ 240, 192 ].
964 - 966 have Frattini factor [ 240, 193 ].
967 - 981 have Frattini factor [ 240, 194 ].
982 - 1012 have Frattini factor [ 240, 195 ].
1013 - 1023 have Frattini factor [ 240, 196 ].
1024 - 1034 have Frattini factor [ 240, 197 ].
1035 - 1045 have Frattini factor [ 240, 198 ].
1046 has Frattini factor [ 240, 199 ].
1047 - 1059 have Frattini factor [ 240, 200 ].
1060 - 1072 have Frattini factor [ 240, 201 ].
1073 - 1125 have Frattini factor [ 240, 202 ].
1126 - 1132 have Frattini factor [ 240, 203 ].
1133 - 1134 have Frattini factor [ 240, 204 ].
1135 - 1149 have Frattini factor [ 240, 205 ].
1150 - 1164 have Frattini factor [ 240, 206 ].
1165 - 1179 have Frattini factor [ 240, 207 ].
1180 - 1185 have Frattini factor [ 240, 208 ].
1186 - 1213 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.