Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4656 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C3×#117 ⋊ C2 |
| Sylow | C3 ⋊ #2392 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Frattini | C23 | C23×D3 |
| Fitting | C2×C3×#117 | C2 |
| prime radical | #4960 | C2 |
| hypercenter | C2×#117 | D3 |
| hypocenter | C3 | #2392 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#117 | |
696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(12,14,15,18)(13,17,16,19);(2,3)(5,7)(9,11);(8,9)(10,11)(13,16)(17,19);(4,5,6,7)(12,13)(14,17)(15,16)(18,19)Polycyclic presentation:
2^7,3|a^2=f,d^2=e,b^a=be,c^a=cf,c^b=cg,h^c=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | #10690284, #10690286, #10690289, … |
| maximal normal subgroup of | #10690284, #10690286, #10690289, … |
| maximal quotient of | #10716863, #10716882, #10716923, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 14849);