Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#446 ⋊ C2 |
| Sylow | C3 ⋊ #3389 |
348 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 129 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 — 4 generators on 27 points:
(1,2,3)(4,7,8,5)(6,9,11,10)(14,22)(17,21)(23,27)(25,26);(2,3)(13,18)(14,22)(15,24)(17,25)(19,20)(21,26);(4,6,8,11)(5,9,7,10)(12,14,16,22)(13,17,20,21)(15,23,24,27)(18,26,19,25);(2,3)(4,5,8,7)(6,10,11,9)(12,13)(14,21)(15,18)(16,20)(17,22)(19,24)(23,25)(26,27)Polycyclic presentation:
2^7,3|a^2=e,b^2=eg,d^2=e,f^2=g,b^a=beg,c^a=cf,f^a=fg,h^a=h^2,c^b=cg,d^b=deg,f^c=fg,h^c=h^257 conjugacy classes — view the full character table.
| maximal subgroup of | #10708917, #10708921, #10708931, … |
| maximal normal subgroup of | #10708917, #10708921, #10708931, … |
| maximal quotient of | #10811860, #10813487, #10813527, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12555);