Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#1915 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #386 |
| commutator | C23 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C2×C3×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
308 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 308 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:
(1,2,5,10,6,9,11,24)(3,7,14,21,13,22,18,25)(4,15,17,26,16,8,19,23)(12,20,28,31,27,30,29,32)(33,34,35);(1,3,11,14,6,13,5,18)(2,7,10,25,9,22,24,21)(4,12,17,29,16,27,19,28)(8,20,26,31,15,30,23,32);(1,4)(2,8)(3,12)(5,19)(6,16)(7,20)(9,15)(10,26)(11,17)(13,27)(14,29)(18,28)(21,32)(22,30)(23,24)(25,31)87 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5206);