Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| Sylow | C3 ⋊ #2375 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting, prime radical | C2×C3×#120 | C2 |
| socle, cosocle | C23×C3 | C24 |
| hypercenter | C2×#120 | D3 |
| hypocenter | C3 | #2375 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#120 | |
688 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 213 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 — 4 generators on 23 points:
(1,2,3)(9,13)(10,15)(17,21)(20,23);(10,15)(12,14)(17,21)(18,22);(8,10)(9,12)(11,15)(13,14)(16,18)(17,20)(19,22)(21,23);(2,3)(4,5,6,7)(8,9)(10,14)(11,13)(12,15)(16,17)(18,20)(19,21)(22,23)Polycyclic presentation:
2^7,3|a^2=g,b^a=be,c^a=cf,d^a=def,h^a=h^2,c^b=cef,d^b=de66 conjugacy classes — view the full character table.
| maximal subgroup of | #10711952, #10711956, #10712459, … |
| maximal normal subgroup of | #10711952, #10711956, #10712459, … |
| maximal quotient of | #10716805, #10716818, #10716826, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15156);