Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2243 |
142 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 142 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)(5,9)(8,11)(12,15,16,24,22,27,26,21,19,23,25,14,18,20,17,13);(2,3)(4,6,7,10)(5,8,9,11)(12,14,17,23,18,21,25,27,19,24,26,15,22,13,16,20);(4,5)(6,8)(7,9)(10,11)(12,13)(14,22)(15,17)(16,20)(18,24)(19,21)(23,26)(25,27)Polycyclic presentation:
2^7,3|b^2=e,c^2=d,d^2=fg,e^2=f,f^2=g,b^a=bd,c^a=ce,d^a=df,e^a=efg,f^a=fg,c^b=cg,h^b=h^2,h^c=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10709023, #10709024, #10709030, … |
| maximal normal subgroup of | #10709023, #10709024, #10709030, … |
| maximal quotient of | #10711572, #10713120, #10713127, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3895);