Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#424 ⋊ C2 |
| Sylow | C3 ⋊ #3045 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D12 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#424 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #424 | D3 |
| hypocenter | C3 | #3045 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #424 | |
504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 504 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) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(8,10,12,17)(9,14,16,21)(11,18,19,23)(13,20,15,22);(2,3)(9,13)(11,19)(14,20)(15,16)(18,23)(21,22);(4,5,6,7)(10,17)(14,21)(18,23)(20,22);(2,3)(5,7)(8,9)(10,14)(11,13)(12,16)(15,19)(17,21)(18,20)(22,23)Polycyclic presentation:
2^7,3|b^2=e,d^2=g,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^2,d^b=dg,f^c=fg,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10833223, #10833226, #10833238, … |
| maximal normal subgroup of | #10833223, #10833226, #10833238, … |
| maximal quotient of | #10780581, #10780592, #10780603, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11664);