Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3369 |
338 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 338 conjugacy classes of subgroups — too many to draw. Only its 117 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)(4,6,7,11)(5,9,8,10);(12,15,17,24)(13,19,20,27)(14,21,16,23)(18,26,25,22);(2,3)(4,5,7,8)(6,10,11,9)(12,14,17,16)(13,18,20,25)(15,23,24,21)(19,22,27,26);(5,8)(9,10)(12,13)(14,18)(15,22)(16,25)(17,20)(19,21)(23,27)(24,26)Polycyclic presentation:
2^7,3|b^2=eg,c^2=g,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,c^b=cg,d^b=de,h^b=h^2,f^c=fg57 conjugacy classes — view the full character table.
| maximal subgroup of | #10785149, #10785151, #10785209, … |
| maximal normal subgroup of | #10785149, #10785151, #10785209, … |
| maximal quotient of | #10811843, #10812010, #10814151, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12202);