Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #2916⋊C3 |
| Fitting, Sylow | #2916 ⋊ C3 |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,3,4,15,8,19,5,16)(2,9,10,14,13,31,20,40)(6,45,22,41,27,48,23,64,7,17,51,42,26,47,24,43,25,18,21,63,28,46,52,44)(11,53,33,38,36,56,49,61,12,29,57,39,35,55,50,60,34,30,32,59,37,54,58,62);(1,36,4,57,8,34,5,58)(2,51,20,27,13,52,10,25)(3,61,15,55,19,59,16,53)(6,12,21,50,26,37,23,33)(7,35,22,32,28,11,24,49)(9,47,40,64,31,45,14,63)(17,39,41,30,46,62,43,56)(18,60,42,54,48,38,44,29)Polycyclic presentation:
2^2,3,2^5|a^2=e,b^2=d,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,d^a=dh,f^c=fg,g^c=f,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #33385⋊C3, #33392⋊C3, #33393⋊C3, … |
| maximal normal subgroup of | #33385⋊C3, #33392⋊C3, #33393⋊C3, … |
| maximal quotient of | #17788⋊C3, #17789⋊C3, #17826⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5824);