Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2202 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#2203 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C23×C3 |
| prime radical | C3×#454 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ G | |
216 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 56 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)(4,16)(5,8)(6,11)(7,13)(9,19)(10,21)(14,29)(15,27)(17,25)(18,26)(20,30)(22,28)(23,31)(24,32)(33,34,35);(1,3,6,15)(2,7,11,25)(4,17,19,13)(5,20,21,14)(8,23,10,24)(9,27,16,12)(18,31,22,32)(26,29,28,30);(2,8)(5,21)(7,23)(9,28)(10,11)(12,29)(14,20)(16,26)(18,22)(24,25)(27,30)(31,32)51 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5494);