Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#585 ⋊ C2 |
| Sylow | C3 ⋊ #3018 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #953 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#585 | C2 |
| socle | C2×C3 | C2×#120 |
| hypercenter | #585 | D3 |
| hypocenter | C3 | #3018 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #585 | |
724 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 724 conjugacy classes of subgroups — too many to draw. Only its 161 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:
(1,2,3)(4,6,8,13)(5,10,11,17)(7,12,14,18)(9,16,15,19);(7,14)(9,15)(12,18)(16,19);(4,5)(6,10)(7,15)(8,11)(9,14)(12,19)(13,17)(16,18);(2,3)(5,9)(7,14)(10,16)(11,15)(12,18)(17,19)Polycyclic presentation:
2^7,3|d^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^2,e^b=eg,f^b=fg,e^c=eg66 conjugacy classes — view the full character table.
| maximal subgroup of | #10712506, #10712511, #10712516, … |
| maximal normal subgroup of | #10712506, #10712511, #10712516, … |
| maximal quotient of | #10767293, #10767344, #10767395, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16697);