Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#450 ⋊ C2 |
| Sylow | C3 ⋊ #3194 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#450 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #450 | D3 |
| hypocenter | C3 | #3194 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #450 | |
466 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 466 conjugacy classes of subgroups — too many to draw. Only its 113 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:
(1,2,3)(5,9)(8,10);(6,11)(8,10)(13,18)(14,15)(17,19);(2,3)(4,6)(5,8)(7,11)(9,10)(12,14,16,15)(13,17,19,18);(2,3)(4,5)(6,10)(7,9)(8,11)(12,13)(14,17)(15,18)(16,19)Polycyclic presentation:
2^7,3|b^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,h^a=h^2,c^b=ceg,h^b=h^2,f^c=fg45 conjugacy classes — view the full character table.
| maximal subgroup of | #10769338, #10769340, #10769406, … |
| maximal normal subgroup of | #10769338, #10769340, #10769406, … |
| maximal quotient of | #10804258, #10804276, #10805475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12729);