Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2517 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#397 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | #397 | D3 |
| hypocenter | C3 | #2517 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #397 | |
412 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 412 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) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(2,3)(13,16)(17,19)(20,22,23,26)(21,25,24,27);(1,2,3)(6,10)(8,11)(12,14,15,18)(13,17,16,19)(21,24)(25,27);(4,6)(5,8)(7,10)(9,11)(13,16)(17,19);(4,5,7,9)(6,8,10,11)(12,13)(14,17)(15,16)(18,19)(20,21)(22,25)(23,24)(26,27)Polycyclic presentation:
2^7,3|a^2=g,c^2=e,d^2=f,b^a=be,c^a=cf,d^a=de,c^b=cg,h^d=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10707473, #10707477, #10708059, … |
| maximal normal subgroup of | #10707473, #10707477, #10708059, … |
| maximal quotient of | #10717247, #10718454, #10719106, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10074);