Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C2×C3×M32 ⋊ C2 |
| Sylow | C3 ⋊ #2140 |
260 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(8,10,13,12)(9,15,17,21)(11,18,20,19)(14,22,23,16);(2,3)(9,14)(10,12)(11,19)(15,16)(17,23)(18,20)(21,22);(2,3)(4,5,6,7)(8,9)(10,15)(11,14)(12,21)(13,17)(16,19)(18,22)(20,23)Polycyclic presentation:
2^7,3|a^2=e,c^2=g,d^2=fg,f^2=g,b^a=bd,d^a=df,f^a=fg,h^a=h^2,c^b=cg,d^b=df,f^b=fg,h^b=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10710456, #10710459, #10710873, … |
| maximal normal subgroup of | #10710456, #10710459, #10710873, … |
| maximal quotient of | #10704937, #10705181, #10705233, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4016);