Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#1663 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
172 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 384 |
A minimal faithful permutation representation — 3 generators on 35 points:
(3,9)(4,11)(6,14)(7,12)(10,13)(15,16)(17,18,22,20)(19,25,28,32)(21,29,31,24)(23,30,26,27)(33,34,35);(1,2,5,8)(3,6,10,15)(4,12,11,7)(9,16,13,14)(17,19,22,28)(18,23,20,26)(21,30,31,27)(24,32,29,25);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,11)(12,14)(17,20,22,18)(19,26,28,23)(21,24,31,29)(25,30,32,27)60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4954);