Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#399 ⋊ C2 |
| Sylow | C3 ⋊ #2668 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#399 | C2 |
| hypercenter | #399 | D3 |
| hypocenter | C3 | #2668 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #399 | |
816 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 816 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(5,7)(13,17)(14,19);(5,7)(9,11);(2,3)(8,9)(10,11)(12,14)(13,16)(15,19)(17,18);(2,3)(4,5)(6,7)(12,13)(14,18)(15,17)(16,19)Polycyclic presentation:
2^7,3|b^a=be,c^a=cf,d^a=def,h^a=h^2,c^b=cg,d^b=de,h^b=h^2,h^d=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10707792, #10707796, #10708276, … |
| maximal normal subgroup of | #10707792, #10707796, #10708276, … |
| maximal quotient of | #10722476, #10722593, #10722963, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10688);