Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C3×#391 ⋊ C2 |
| Sylow | C3 ⋊ C4×#124 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#391 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C22×D12 | C22 |
| hypercenter | #391 | D3 |
| hypocenter | C3 | C4×#124 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #391 | |
652 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 652 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) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(4,6,7,5);(2,3)(8,10,11,9)(13,17)(16,19);(2,3)(8,9,11,10)(12,14,15,18)(13,16,17,19);(4,5,7,6)(9,10)(12,13)(14,16)(15,17)(18,19)Polycyclic presentation:
2^7,3|a^2=g,b^2=f,c^2=e,d^2=g,b^a=be,c^a=cf,h^b=h^2,h^c=h^2,h^d=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | #10683155, #10683228, #10683301, … |
| maximal normal subgroup of | #10683155, #10683228, #10683301, … |
| maximal quotient of | #10716342, #10716370, Dic3×#391, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 9056);