Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2374 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#386 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C22×C3×D4 | C22 |
| hypercenter | #386 | D3 |
| hypocenter | C3 | #2374 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #386 | |
552 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 185 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)(10,15)(12,14)(16,19,20,18)(17,21,22,23);(2,3)(9,13)(12,14)(16,18,20,19)(17,21,22,23);(2,3)(4,6,7,5)(8,10)(9,12)(11,15)(13,14);(4,5,7,6)(8,9)(10,14)(11,13)(12,15)(16,17)(18,23)(19,21)(20,22)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,c^2=ef,d^2=ef,b^a=be,c^a=cf,d^b=de,h^b=h^2,h^c=h^2,h^d=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10711944, #10711948, #10712451, … |
| maximal normal subgroup of | #10711944, #10711948, #10712451, … |
| maximal quotient of | #10716651, #10716741, #10716776, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 7257);