Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2582 |
| 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 | #2582 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #384 | |
484 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 484 conjugacy classes of subgroups — too many to draw. Only its 181 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(8,11,12,10)(9,14,15,13);(2,3)(5,7)(16,18,19,23)(17,20,21,22);(8,10,12,11)(9,13,15,14)(16,17,19,21)(18,22,23,20);(2,3)(4,5)(6,7)(8,9,12,15)(10,14,11,13)Polycyclic presentation:
2^7,3|a^2=e,b^2=eg,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=de,h^a=h^2,c^b=cg,h^c=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | #10697495, #10697498, #10697522, … |
| maximal normal subgroup of | #10697495, #10697498, #10697522, … |
| maximal quotient of | #10717603, #10718574, #10718870, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6919);