Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| Sylow | C3 ⋊ #1726 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C22×D3 |
| commutator | C2×C3 | C43 |
| Fitting | C3×#376 | C2 |
| socle, prime radical | C23×C3 | C22×C4 |
| cosocle | C22×C3×C4 | C23 |
| hypercenter | #376 | D3 |
| hypocenter | C3 | #1726 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C4 ◁ #376 | |
328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(8,9,10,11)(14,20)(17,23)(21,26)(24,27);(4,5,6,7)(12,14)(13,17)(15,21)(16,20)(18,24)(19,23)(22,26)(25,27);(2,3)(12,13,15,18,16,19,22,25)(14,17,21,24,20,23,26,27)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,c^2=f,d^2=g,h^a=h^2,c^b=cg120 conjugacy classes — view the full character table.
| maximal subgroup of | #10685625, #10685629, #10685633, … |
| maximal normal subgroup of | #10685625, #10685629, #10685633, … |
| maximal quotient of | #10684541, #10684544, #10684550, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2010);