Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#386 ⋊ C2 |
| Sylow | C3 ⋊ #2709 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#386 | C2 |
| prime radical | C2×#940 | C2 |
| hypercenter | #386 | D3 |
| hypocenter | C3 | #2709 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #386 | |
612 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 137 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) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(10,15)(12,14)(17,21)(20,22);(2,3)(5,7)(9,13)(10,15)(16,18,19,23)(17,20,21,22);(2,3)(4,5)(6,7)(8,10,11,15)(9,12,13,14);(8,9)(10,14)(11,13)(12,15)(16,17)(18,22)(19,21)(20,23)Polycyclic presentation:
2^7,3|b^2=e,c^2=ef,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,h^b=h^2,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10714152, #10714154, #10714629, … |
| maximal normal subgroup of | #10714152, #10714154, #10714629, … |
| maximal quotient of | #10722734, #10723280, #10723340, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 7396);