Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#1902 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #386 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C2×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
304 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 108 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)(4,16,19,26)(5,20)(6,10)(7,11)(8,14)(9,18,25,17)(13,28,30,22)(15,23)(21,24)(27,29,32,31)(33,34,35);(1,3,5,14,7,15,6,21)(2,8,10,12,11,24,20,23)(4,13,18,31,19,30,17,29)(9,22,26,27,25,28,16,32);(1,4)(2,9)(3,13)(5,18)(6,17)(7,19)(8,22)(10,26)(11,25)(12,27)(14,31)(15,30)(16,20)(21,29)(23,32)(24,28)87 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5193);