Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#458 ⋊ C2 |
| Sylow | C3 ⋊ #3025 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×#120 |
| commutator, cosocle | C23×C3 | C24 |
| Frattini | C23 | C23×D3 |
| Fitting | C3×#458 | C2 |
| socle | C2×C3 | C2×#120 |
| hypercenter | #458 | D3 |
| hypocenter | C3 | #3025 |
| LCS | G ▷ C23×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #458 | |
766 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 766 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) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(4,7,8,14)(5,10,11,17)(6,12,13,18)(9,16,15,19);(4,6)(5,9)(7,12)(8,13)(10,16)(11,15)(14,18)(17,19);(4,5)(6,9)(7,10)(8,11)(12,16)(13,15)(14,17)(18,19);(2,3)(9,15)(16,19)Polycyclic presentation:
2^7,3|d^2=g,b^a=be,c^a=cf,h^a=h^2,f^b=fg,e^c=eg,h^d=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10949207, #10949209, #10949235, … |
| maximal normal subgroup of | #10949207, #10949209, #10949235, … |
| maximal quotient of | #10767457, #10767459, #10767466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12884);