Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#395 ⋊ C2 |
| Sylow | C3 ⋊ #2821 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#395 | C2 |
| prime radical | C23×D3 | C23 |
| hypercenter | #395 | D3 |
| hypocenter | C3 | #2821 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #395 | |
430 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(10,14)(12,15)(16,19,20,26)(17,23,21,29)(18,24,25,27)(22,30,28,31);(5,7)(9,13)(12,15)(16,18,20,25)(17,22,21,28)(19,27,26,24)(23,31,29,30);(2,3)(4,5)(6,7)(8,10)(9,12)(11,14)(13,15)(17,21)(19,26)(22,28)(24,27);(2,3)(8,9,11,13)(10,12,14,15)(16,17)(18,22)(19,23)(20,21)(24,30)(25,28)(26,29)(27,31)Polycyclic presentation:
2^7,3|a^2=f,c^2=e,d^2=e,b^a=be,c^a=cf,h^a=h^2,c^b=cg,d^b=def,h^b=h^2,d^c=de48 conjugacy classes — view the full character table.
| maximal subgroup of | #10770669, #10770673, #10770691, … |
| maximal normal subgroup of | #10770669, #10770673, #10770691, … |
| maximal quotient of | #10725874, #10725876, #10725882, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 9852);