Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #1700 |
146 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 49 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)(5,9)(6,10)(12,14,15,21)(13,18,17,26)(16,25,23,22)(19,27,24,20);(2,3)(4,6,7,5,8,10,11,9)(13,17)(18,26)(19,24)(20,27);(2,3)(4,5,8,9)(6,11,10,7)(12,13)(14,20)(15,17)(16,24)(18,22)(19,23)(21,27)(25,26)Polycyclic presentation:
2^7,3|a^2=f,b^2=dg,c^2=g,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^2,c^b=cf,h^b=h^2,e^c=eg,h^c=h^242 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1700, C2×#68477, #10777866, … |
| maximal normal subgroup of | D3×#1700, C2×#68477, #10777866, … |
| maximal quotient of | #10688295, #10691400, #10691418, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1053);