AboutHap

About HAP: Third homology of some superperfect groups

A group G is said to be superperfect if it is perfect and its second integral homology is trivial. Using the library of perfect groups compiled by Holt and Pleskin, we (very partially) list below the superperfect groups of order at most 68880 together with their third integral homology. Further information on these superperfect groups can be obtained using the GAP command

DisplayInformationPerfectGroup(order,number);

where "order" is the order of the group and "number" is the number of the group given in the table.

(Work on the table is to be continued ... )

	List of superperfect groups
Order	Perfect Group Number	Third integral homology

120	1	[ 8, 3, 5 ]
336	1	[16,3]
504	1	[ 2, 9, 7 ]
1320	1	[ 8, 3, 5 ]
2160	1	[ 16, 3, 3, 5 ]
2184	1	[ 8, 3, 7 ]
2688	3	[ 8, 8, 3 ]
3840	6	???
3840	7	???
4080	1	[ 3, 5, 17 ]
4896	1	[ 32, 9 ]
5376	1	???
5616	1	[ 8, 3 ]
6048	1	[ 8, 9, 5 ]
7920	1	[ 8 ]
9720	2	[ 8, 9, 5 ]
12144	1	[ 16, 3, 11 ]
14400	1	[ 8, 8, 3, 3, 5, 5 ]
15000	2	[ 8, 3, 5, 25 ]
15000	3	???
15120	1	[ 16, 3, 3 ]
15360	4,6-7: Not yet checked if these are superperfect
15600	1	[ 16, 3, 13 ]
19656	1	[ 8, 3, 7, 13 ]
21504	5-22: Not yet checked if these are superferfect
24360	1	[ 8, 3, 5, 7 ]
29160	1	???
29160	3: Not yet checked if this is superperfect
29760	1	[ 64, 3, 5 ]
32256	1-2: Not yet checked if these are superperfect
32736	1	[ 3, 11, 31 ]
34560	4	???
37500	1: Not yet checked if this is superperfect
40320	1	[ 8, 16, 3, 3, 5 ]
40320	3	???
43008	1-25: Not yet checked if these are superperfect
46080	1: Not yet checked if this is superperfect
50616	1	[ 8, 9, 19 ]
51840	1	???
57624	1-2: Not yet checked if these are superperfect
58320	2: Not yet checked if this is superperfect
62400	1	???
64512	1-4: Not yet checked if these are superperfect
68880	1	[ 16, 3, 5, 7 ]

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