Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#397 ⋊ C2 |
| Sylow | C3 ⋊ #2486 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#397 | C2 |
| prime radical | C22×#67 | C22 |
| hypercenter | #397 | D3 |
| hypocenter | C3 | #2486 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #397 | |
624 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 624 conjugacy classes of subgroups — too many to draw. Only its 151 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(2,3)(8,10,11,9);(1,2,3)(5,7)(14,18)(16,19);(8,9,11,10)(12,14)(13,16)(15,18)(17,19);(4,5)(6,7)(9,10)(12,13,15,17)(14,16,18,19)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=cg,h^d=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10705741, #10706204, #10767616, … |
| maximal normal subgroup of | #10705741, #10706204, #10767616, … |
| maximal quotient of | #10717089, #10717414, #10717969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10061);