Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #1437⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | #1437⋊C3 |
| Fitting, Sylow | #1437 ⋊ C3 |
| derived | C22×C42 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22×C42 | C2×C3 |
| Frattini, socle | C24 | C2×A4 |
| Fitting | #1437 | C3 |
| LCS | G ▷ C22×C42 | |
| UCS | C1 | |
168 conjugacy classes of subgroups; only the 13 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 13 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2)(3,7,8,5,4,6)(9,10,18)(11,16,20)(12,13,14)(15,19,17);(2,6)(5,7)(9,11,12,17)(10,13)(14,19,18,20)(15,16)Polycyclic presentation:
2,3,2^6|c^2=h,d^2=g,b^a=bh,c^a=ce,d^a=dfg,c^b=deg,d^b=cdef,e^b=fg,f^b=efh,g^b=gh,h^b=g24 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 584);