Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C3×#329 ⋊ C2 |
| Sylow | C3 ⋊ #1659 |
228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,5)(6,11)(7,9)(8,10)(15,25)(18,23)(20,27)(21,24);(2,3)(6,10)(8,11)(12,14)(13,17)(15,24)(16,22)(18,27)(19,26)(20,23)(21,25);(2,3)(5,8)(6,10)(9,11)(12,13,16,19)(14,20,22,27)(15,23,25,18)(17,24,26,21)Polycyclic presentation:
2^7,3|a^2=f,d^2=f,e^2=fg,b^a=bd,c^a=ce,d^a=df,e^a=efg,h^a=h^2,d^b=df,e^b=eg,h^b=h^2,d^c=dg,e^c=efg39 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1659, C2×#68352, #10776340, … |
| maximal normal subgroup of | D3×#1659, C2×#68352, #10776340, … |
| maximal quotient of | #10683976, #10684005, #10684022, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 928);