Endomorphism structure of groups of order 16

From Groupprops
Jump to: navigation, search
This article gives specific information, namely, endomorphism structure, about a family of groups, namely: groups of order 16.
View endomorphism structure of group families | View endomorphism structure of groups of a particular order |View other specific information about groups of order 16

Automorphism groups

Group GAP ID (second part) Order of automorphism group Iso. class of automorphism group Log_2 of largest power of 2 dividing automorphism group Iso. class of 2-Sylow subgroup of automorphism group Log_2 of order of 2-core Iso. class of 2-core Log_2 of order of inner automorphism group Iso. class of inner automorphism group
cyclic group:Z16 1 8 direct product of Z4 and Z2 3 direct product of Z4 and Z2 3 direct product of Z4 and Z2 0 trivial group
direct product of Z4 and Z4 2 96 general linear group:GL(2,Z4) 5 SmallGroup(32,27) 4 elementary abelian group:E16 0 trivial group
SmallGroup(16,3) 3 32 SmallGroup(32,27) 5 SmallGroup(32,27) 5 SmallGroup(32,27) 2 Klein four-group
nontrivial semidirect product of Z4 and Z4 4 32 SmallGroup(32,27) 5 SmallGroup(32,27) 5 SmallGroup(32,27) 2 Klein four-group
direct product of Z8 and Z2 5 16 direct product of D8 and Z2 4 direct product of D8 and Z2 4 direct product of D8 and Z2 0 trivial group
M16 6 16 direct product of D8 and Z2 4 direct product of D8 and Z2 4 direct product of D8 and Z2 2 Klein four-group
dihedral group:D16 7 32 holomorph of Z8 5 holomorph of Z8 5 holomorph of Z8 3 dihedral group:D8
semidihedral group:SD16 8 16 direct product of D8 and Z2 4 direct product of D8 and Z2 4 direct product of D8 and Z2 3 dihedral group:D8
generalized quaternion group:Q16 9 32 holomorph of Z8 5 holomorph of Z8 5 holomorph of Z8 3 dihedral group:D8
direct product of Z4 and V4 10 192 SmallGroup(192,1493) 6 Upper-triangular unipotent matrix group:U(4,2) 5 inner holomorph of D8 (one of the extraspecial groups of order 32) 0 trivial group
direct product of D8 and Z2 11 64 Upper-triangular unipotent matrix group:U(4,2) 6 Upper-triangular unipotent matrix group:U(4,2) 6 Upper-triangular unipotent matrix group:U(4,2) 2 Klein four-group
direct product of Q8 and Z2 12 192 SmallGroup(192,955) 6 Upper-triangular unipotent matrix group:U(4,2) 5 SmallGroup(32,27) 2 Klein four-group
central product of D8 and Z4 13 48 direct product of S4 and Z2 4 direct product of D8 and Z2 3 elementary abelian group:E8 2 Klein four-group
elementary abelian group:E16 14 20160 alternating group:A8 6 Upper-triangular unipotent matrix group:U(4,2) 0 trivial group 0 trivial group