Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#2082 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #397 |
| commutator, Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C23×C3 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
184 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,6,11,7,12,5,10)(3,13,16,8,17,24,15,25)(4,18,20,26,21,27,22,9)(14,29,30,32,19,23,31,28)(33,34,35);(1,3,7,17)(2,8,12,25)(4,19,21,14)(5,16,6,15)(9,23,26,29)(10,24,11,13)(18,32,27,28)(20,30,22,31);(1,4,5,20,7,21,6,22)(2,9,10,18,12,26,11,27)(3,14,15,30,17,19,16,31)(8,23,13,28,25,29,24,32)69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5373);