Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2794 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Fitting | C3×#384 | C2 |
| hypercenter | #384 | D3 |
| hypocenter | C3 | #2794 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #384 | |
344 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 344 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) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(12,15,16,14)(13,17,19,18)(21,24)(25,27);(6,10)(8,11)(12,14,16,15)(13,18,19,17)(20,22,23,26)(21,25,24,27);(2,3)(4,6)(5,8)(7,10)(9,11)(12,14,16,15)(13,17,19,18)(20,21)(22,25)(23,24)(26,27);(4,5,7,9)(6,8,10,11)(12,13)(14,17)(15,18)(16,19)(21,24)(22,26)Polycyclic presentation:
2^7,3|a^2=g,b^2=ef,c^2=f,d^2=ef,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,h^b=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10707523, #10707527, #10708109, … |
| maximal normal subgroup of | #10707523, #10707527, #10708109, … |
| maximal quotient of | #10723278, #10723391, #10724044, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6928);