Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#2214 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C23×C3 |
| prime radical | C3×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×C4×C8 ◁ G | |
150 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(1,2,7,12)(3,13,6,24)(4,17,21,32)(5,22,14,10)(8,23,11,16)(9,27,30,19)(15,29,25,28)(18,31,20,26)(33,34,35);(1,3,14,23,7,6,5,16)(2,8,22,24,12,11,10,13)(4,15,20,19,21,25,18,27)(9,26,29,17,30,31,28,32);(1,4)(2,9)(3,15)(5,18)(6,25)(7,21)(8,26)(10,28)(11,31)(12,30)(13,32)(14,20)(16,27)(17,24)(19,23)(22,29)69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5505);