Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #1767 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C22×D3 |
| commutator | C22×C3 | C2×C42 |
| Fitting | C3×C43 | C2 |
| socle, prime radical | C23×C3 | C22×C4 |
| cosocle | C22×C3×C4 | C23 |
| hypercenter | C43 | D3 |
| hypocenter | C3 | #1767 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C4 ◁ C43 | |
292 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 165 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) = 384 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,6,7,5)(13,16)(17,19);(2,3)(9,11)(12,13,14,17,15,16,18,19);(4,5,7,6)(8,9,10,11)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=de,f^2=g,b^a=bd,c^b=cg,h^b=h^2120 conjugacy classes — view the full character table.
| maximal subgroup of | #10681729, #10681730, #10681743, … |
| maximal normal subgroup of | #10681729, #10681730, #10681743, … |
| maximal quotient of | #10686316, #10686326, #10686349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1967);