Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Sylow | C3 ⋊ #3504 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×C4×#41 | C2 |
| prime radical | C2×C3×#41 | C22 |
| hypercenter | C4×#41 | D3 |
| hypocenter | C3 | #3504 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4×#41 | |
988 conjugacy classes of subgroups; only the 549 normal subgroups are shown.
This group has 988 conjugacy classes of subgroups — too many to draw. Only its 549 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 — 5 generators on 27 points:
(1,2,3)(12,15,16,24)(13,19,18,23)(14,22,21,27)(17,25,20,26);(14,21)(17,20)(22,27)(25,26);(4,6,7,11)(5,8,9,10)(13,18)(17,20)(19,23)(25,26);(12,14)(13,17)(15,22)(16,21)(18,20)(19,25)(23,26)(24,27);(2,3)(4,5,7,9)(6,10,11,8)(12,13,16,18)(14,20,21,17)(15,23,24,19)(22,25,27,26)Polycyclic presentation:
2^7,3|a^2=g,c^2=fg,e^2=f,b^a=bf,c^a=cg,e^a=ef,h^a=h^2,d^b=df84 conjugacy classes — view the full character table.
| maximal subgroup of | #10687452, #10763434, #10763461, … |
| maximal normal subgroup of | #10687452, #10763434, #10763461, … |
| maximal quotient of | #10955298, #10955647, #10955667, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18405);