Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1745 |
| 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 | #1745 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C4 ◁ C43 | |
272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(8,10,11,14)(9,13,12,15)(17,20)(21,23);(4,5,6,7)(9,12)(13,15);(2,3)(8,9)(10,13)(11,12)(14,15)(16,17,18,21,19,20,22,23)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=d,e^2=g,b^a=bd,c^a=cg,h^a=h^2120 conjugacy classes — view the full character table.
| maximal subgroup of | #10681732, #10681733, #10681785, … |
| maximal normal subgroup of | #10681732, #10681733, #10681785, … |
| maximal quotient of | #10685613, #10685620, #10685624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1965);