Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C3×#476 ⋊ C2 |
| Sylow | C3 ⋊ #1634 |
218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,5,7,9)(6,11,10,8)(12,15,17,16)(13,19,20,24)(14,22,23,25)(18,26,27,21);(2,3)(6,10)(8,11)(12,14,17,23)(13,18,20,27)(15,22,16,25)(19,26,24,21);(5,8)(6,10)(9,11)(12,13,17,20)(14,21,23,26)(15,19,16,24)(18,25,27,22)Polycyclic presentation:
2^7,3|a^2=f,b^2=f,c^2=fg,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,d^b=df,e^b=eg,h^b=h^2,d^c=dg,e^c=eg45 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1634, C2×#71145, #10771080, … |
| maximal normal subgroup of | D3×#1634, C2×#71145, #10771080, … |
| maximal quotient of | #10684019, #10684029, #10684202, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3721);