Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#461 ⋊ C2 |
| Sylow | C3 ⋊ #3036 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#461 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #461 | D3 |
| hypocenter | C3 | #3036 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #461 | |
548 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2)(9,13)(12,14);(9,12)(10,15)(13,14);(4,5,6,7);(2,3)(5,7)(8,9)(10,12)(11,13)(14,15)Polycyclic presentation:
2^7,3|b^2=e,f^2=g,b^a=be,c^a=cf,d^a=dg,f^a=fg,h^a=h^2,f^c=fg,h^d=h^251 conjugacy classes — view the full character table.
| maximal subgroup of | #10771516, #10771518, #10771520, … |
| maximal normal subgroup of | #10771516, #10771518, #10771520, … |
| maximal quotient of | #10778966, #10778977, #10778985, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 13004);