Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×#2178 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | D8 |
| commutator | C8 | C22×C3×C4 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C2×C3 | C4×D8 |
| prime radical | C3×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ C2×C3×C8 ◁ C3×C4×C8 ◁ G | |
146 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,5,11,7,12,6,21)(3,13,16,29,17,30,22,28)(4,9,18,26,20,27,19,10)(8,14,23,31,25,32,24,15)(33,34,35);(1,3)(2,8)(4,14)(5,16)(6,22)(7,17)(9,13)(10,28)(11,23)(12,25)(15,19)(18,31)(20,32)(21,24)(26,29)(27,30);(1,4,5,18,7,20,6,19)(2,9,11,26,12,27,21,10)(3,14,16,31,17,32,22,15)(8,13,23,29,25,30,24,28)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5469);