Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×#2155 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #106 |
| commutator | C8 | C22×C3×C4 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C2×C3 | C2×#106 |
| prime radical | C23×C3 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
144 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2)(3,12,15,26)(4,8)(5,18)(6,11)(7,14,21,30)(9,16)(10,17)(13,25,29,27)(19,23)(20,28,22,32)(24,31)(33,34,35);(1,3,5,14,4,13,16,28,6,15,19,30,17,29,31,32)(2,7,9,12,10,22,18,27,11,21,24,26,8,20,23,25);(1,4,6,17)(2,8,11,10)(3,13,15,29)(5,16,19,31)(7,20,21,22)(9,23,24,18)(12,25,26,27)(14,28,30,32)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5446);