Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#454 ⋊ C2 |
| Sylow | C3 ⋊ #3021 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×#120 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#454 | C2 |
| socle | C2×C3 | C2×#120 |
| hypercenter | #454 | D3 |
| hypocenter | C3 | #3021 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #454 | |
736 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 736 conjugacy classes of subgroups — too many to draw. Only its 133 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) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(4,7,9,8)(5,10,12,15)(6,13,14,16)(11,17,19,18);(2,3)(4,6)(5,11)(7,13)(8,16)(9,14)(10,17)(12,19)(15,18);(4,5)(6,11)(7,10)(8,15)(9,12)(13,17)(14,19)(16,18);(5,10)(7,8)(11,18)(12,15)(13,16)(17,19)Polycyclic presentation:
2^7,3|d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fg,e^c=eg,h^c=h^2,h^d=h^245 conjugacy classes — view the full character table.
| maximal subgroup of | #10949384, #10949386, #10949398, … |
| maximal normal subgroup of | #10949384, #10949386, #10949398, … |
| maximal quotient of | #10767579, #10767626, #10767655, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12862);