Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #2225 |
130 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 130 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(12,13,16,19)(14,21,23,27)(15,24,22,18)(17,25,26,20);(2,3)(4,6,7,11)(5,8,9,10);(4,5,7,9)(6,10,11,8)(13,17)(14,20)(15,22)(18,27)(19,26)(21,24)(23,25)Polycyclic presentation:
2^7,3|a^2=d,b^2=d,c^2=g,e^2=fg,f^2=g,b^a=bd,c^a=ce,e^a=ef,f^a=fg,h^b=h^2,e^c=ef,f^c=fg51 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2225, C2×#69210, #10949829, … |
| maximal normal subgroup of | D3×#2225, C2×#69210, #10949829, … |
| maximal quotient of | #10711231, #10711798, #10711802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1786);