Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#418 ⋊ C2 |
| Sylow | C3 ⋊ #3041 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#418 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #418 | D3 |
| hypocenter | C3 | #3041 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #418 | |
526 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 526 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) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(10,17)(13,21)(18,23)(20,22);(2,3)(9,14)(11,19)(13,20)(15,16)(18,23)(21,22);(2,3)(4,5,6,7)(8,10)(9,13)(11,18)(12,17)(14,20)(15,22)(16,21)(19,23);(5,7)(8,9)(10,13)(11,14)(12,16)(15,19)(17,21)(18,20)(22,23)Polycyclic presentation:
2^7,3|b^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=dg,h^b=h^2,f^c=fg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10832565, #10832575, #10843042, … |
| maximal normal subgroup of | #10832565, #10832575, #10843042, … |
| maximal quotient of | #10780346, #10780355, #10780364, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11520);