Once the package is loaded the user may call SmallGroup(19683,i) and receive either a group if available or a partially constructed group which has the following attributes set
p-class
Rank
Heritage
Order
Additional information can be obtained via SmallGroupsInformation(19683).
gap> G:=SmallGroup(19683,1); <pc group of size 19683 with 9 generators> gap> RankPGroup(G); 1 gap> PClassPGroup(G); 9 gap> GRPS39_Heritage(G); [ 6561, 1, 1 ] gap> H:=SmallGroup(19683,546913); <group> #this is a partially constructed group gap> PClassPGroup(H); 3 gap> RankPGroup(H); 3 gap> GRPS39_Heritage(H); [ 729, 122, 1 ] gap> K:=SmallGroup(19683,546913+1); <group> #this is a partially constructed group gap> PClassPGroup(K); 3 gap> RankPGroup(K); 3 gap> GRPS39_Heritage(K); [ 729, 122, 2 ] #notice that H,K have the same parent group but their age differs
‣ Groups19683Information( arg ) | ( function ) |
prints information on the groups of order 19683 (SmallGroupsInformation(19683) also works).
gap> Groups19683Information(); ###### Groups Information ########################### There are 5,937,876,645 groups of order 19,683 They are sorted by rank, p-class, parent group and then age Group 1 has rank 1 and pclass 9 Groups 2-12 have rank 2 and pclass 3 Groups 13-1431 have rank 2 and pclass 4 Groups 1432-12482 have rank 2 and pclass 5 Groups 12483-16608 have rank 2 and pclass 6 Groups 16609-17592 have rank 2 and pclass 7 Groups 17593-17600 have rank 2 and pclass 8 Group 17601 has rank 3 and pclass 2 Groups 17602-88975117 have rank 3 and pclass 3 Groups with IDs 546913-88541296 are not available and are immediate descendants of 729#122 Groups 88975118-91756404 have rank 3 and pclass 4 Groups 91756405-91883170 have rank 3 and pclass 5 Groups 91883171-91887160 have rank 3 and pclass 6 Groups 91887161-91887185 have rank 3 and pclass 7 Groups 91887186-91953852 have rank 4 and pclass 2 Groups with IDs 91887186-91953852 are not available and are immediate descendants of 81#15 Groups 91953853-340943248 have rank 4 and pclass 3 Groups with IDs 92241644-209726506 are not available and are immediate descendants of 729#425 Groups with IDs 209726507-227782267 are not available and are immediate descendants of 729#440 Groups with IDs 227782268-258162927 are not available and are immediate descendants of 729#453 Groups with IDs 274401608-281443634 are not available and are immediate descendants of 2187#6044 Groups with IDs 304813802-317529080 are not available and are immediate descendants of 2187#6576 Groups 340943249-355792512 have rank 4 and pclass 4 Groups 355792513-355820119 have rank 4 and pclass 5 Groups 355820120-355820169 have rank 4 and pclass 6 Groups 355820170-4032242074 have rank 5 and pclass 2 Groups with IDs 355820170-4032242074 are not available and are immediate descendants of 243#67 Groups 4032242075-4717133004 have rank 5 and pclass 3 Groups with IDs 4032242075-4032242205 are not available and are immediate descendants of 2187#9093 Groups with IDs 4045509432-4062606128 are not available and are immediate descendants of 2187#9118 Groups with IDs 4062606129-4103654980 are not available and are immediate descendants of 2187#9121 Groups with IDs 4110319952-4123052906 are not available and are immediate descendants of 2187#9128 Groups with IDs 4123052907-4451185904 are not available and are immediate descendants of 2187#9131 Groups with IDs 4451185905-4500755636 are not available and are immediate descendants of 2187#9134 Groups with IDs 4500755637-4599685634 are not available and are immediate descendants of 2187#9135 Groups with IDs 4599685635-4612426328 are not available and are immediate descendants of 2187#9138 Groups with IDs 4612426329-4616681990 are not available and are immediate descendants of 2187#9140 Groups 4717133005-4717651022 have rank 5 and pclass 4 Groups 4717651023-4717651101 have rank 5 and pclass 5 Groups 4717651102-5937761741 have rank 6 and pclass 2 Groups with IDs 4717651102-5937761741 are not available and are immediate descendants of 729#504 Groups 5937761742-5937874762 have rank 6 and pclass 3 Groups with IDs 5937761742-5937811556 are not available and are immediate descendants of 2187#9302 Groups 5937874763-5937874872 have rank 6 and pclass 4 Groups 5937874873-5937876585 have rank 7 and pclass 2 Groups with IDs 5937874873-5937876585 are not available and are immediate descendants of 2187#9310 Groups 5937876586-5937876632 have rank 7 and pclass 3 Groups with IDs 5937876593-5937876632 are not available and are immediate descendants of 6561#1396068 Groups 5937876633-5937876644 have rank 8 and pclass 2 Groups with IDs 5937876633-5937876644 are not available and are immediate descendants of 6561#1396077 Group 5937876645 has rank 9 and pclass 1 This library was created by David Burrell (2022).
‣ GRPS39_CheckoutDescendants( arg ) | ( function ) |
‣ GRPS39_Heritage( G ) | ( attribute ) |
Returns: list
Returns as a list the following information for a group of order 19683 loaded from the library [ParentGroupID, ParentGroupOrder, Age]. The Age of a group is the position of the group among its siblings in the ordered list of their standard PC codes (the order is determined by the output of the p-group generation algorithm).
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