Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C3×#519 ⋊ C2 |
| Sylow | C3 ⋊ #3428 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator | C2×C3 | C24×C4 |
| Fitting | C3×#519 | C2 |
| socle, cosocle | C22×C3 | C25 |
| prime radical | C22×#239 | C2 |
| hypercenter | #519 | D3 |
| hypocenter | C3 | #3428 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #519 | |
1540 conjugacy classes of subgroups; only the 805 normal subgroups are shown.
This group has 1540 conjugacy classes of subgroups — too many to draw. Only its 805 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2)(8,11,12,19)(9,15,14,21)(10,18,17,23)(13,20,16,22);(9,14)(13,16)(15,21)(20,22);(2,3)(10,17)(13,16)(18,23)(20,22);(2,3)(8,10)(9,13)(11,18)(12,17)(14,16)(15,20)(19,23)(21,22);(4,5,6,7)(8,9)(10,16)(11,15)(12,14)(13,17)(18,22)(19,21)(20,23)Polycyclic presentation:
2^7,3|a^2=g,e^2=f,b^a=bf,d^a=df,c^b=cf,h^b=h^2,h^c=h^2,h^e=h^2102 conjugacy classes — view the full character table.
| maximal subgroup of | #10795058, #10796001, #10796020, … |
| maximal normal subgroup of | #10795058, #10796001, #10796020, … |
| maximal quotient of | #10953400, #10953415, #10953419, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18448);