Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #2889⋊C3 |
| Fitting, Sylow | #2889 ⋊ C3 |
184 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(2,7)(3,11)(4,14,5)(9,24,10,19,23,20)(12,28,13,25,27,26)(15,17,16)(21,31,22)(29,32,30);(1,23,11,32,6,24,3,31)(2,27,18,14,7,28,8,15)(4,10,25,30,16,20,12,22)(5,19,26,21,17,9,13,29)Polycyclic presentation:
2^2,3,2^5|a^2=e,e^2=h,f^2=h,g^2=h,b^a=bd,d^a=dh,e^b=eh,f^c=fgh,g^c=f,g^f=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #33349⋊C3, #33351⋊C3, C2×#2889⋊C3, … |
| maximal normal subgroup of | #33349⋊C3, #33351⋊C3, C2×#2889⋊C3, … |
| maximal quotient of | #17352⋊C3, #17414⋊C3, #17654⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5810);