Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#1515 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C23×C3 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
208 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 88 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,32)(29,31)(33,34,35);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,23)(22,31)(24,28)(25,32)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4806);