Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C8 · Schur multiplier (H2) = C22 |
| direct product | C3×#1378 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | C22⋊C4 |
| commutator | C22 | C3×C4×C8 |
| Frattini, cosocle | C22×C8 | C22×C3 |
| socle | C2×C3 | #336 |
| prime radical | C23×C3 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ C22×C3×C8 ◁ G | |
132 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 84 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 — 2 generators on 35 points:
(1,2,4,8,5,10,15,22,6,9,16,21,17,23,28,31)(3,11,13,24,14,25,27,32,12,7,18,19,26,20,29,30)(33,34,35);(1,3)(2,7,9,11)(4,13)(5,14)(6,12)(8,19,21,24)(10,20,23,25)(15,27)(16,18)(17,26)(22,30,31,32)(28,29)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 514);