Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#399 ⋊ C2 |
| Sylow | C3 ⋊ #2842 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#399 | C2 |
| prime radical | C24×D3 | C22 |
| hypercenter | #399 | D3 |
| hypocenter | C3 | #2842 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #399 | |
548 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(6,10)(8,11)(14,19)(17,18)(21,24)(22,26);(5,9)(8,11)(12,14,15,19)(13,17,16,18)(20,22)(21,25)(23,26)(24,27);(4,6)(5,8)(7,10)(9,11)(13,16)(14,19)(20,21,23,24)(22,25,26,27);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13)(14,18)(15,16)(17,19)(21,24)(25,27)Polycyclic presentation:
2^7,3|b^2=eg,c^2=g,b^a=be,c^a=cf,h^a=h^2,c^b=cg,d^b=def,d^c=de48 conjugacy classes — view the full character table.
| maximal subgroup of | #10707406, #10707409, #10707992, … |
| maximal normal subgroup of | #10707406, #10707409, #10707992, … |
| maximal quotient of | #10725982, #10725983, #10725984, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10704);