Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#382 ⋊ C2 |
| Sylow | C3 ⋊ #2820 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#382 | C2 |
| prime radical | C22×D12 | C22 |
| hypercenter | #382 | D3 |
| hypocenter | C3 | #2820 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #382 | |
464 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 133 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)(10,14)(12,15)(16,19,20,18)(17,21,22,23);(2,3)(5,7)(9,13)(12,15)(17,22)(18,19);(2,3)(4,5)(6,7)(8,10)(9,12)(11,14)(13,15)(16,18,20,19)(17,21,22,23);(8,9,11,13)(10,12,14,15)(16,17)(18,23)(19,21)(20,22)Polycyclic presentation:
2^7,3|a^2=e,b^2=ef,d^2=ef,b^a=bef,c^a=cf,c^b=cg,d^b=de,h^b=h^2,d^c=def,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10696471, #10696475, #10696490, … |
| maximal normal subgroup of | #10696471, #10696475, #10696490, … |
| maximal quotient of | #10725859, #10725881, #10725901, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6831);