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×#1905 |
| 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 | |
310 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 310 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:
(1,2,6,10,7,12,5,11)(3,13,16,23,17,26,15,8)(4,18,20,9,21,24,19,25)(14,22,29,31,30,32,28,27)(33,34,35);(1,3)(2,8,12,23)(4,14)(5,15)(6,16)(7,17)(9,22,25,32)(10,13,11,26)(18,27,24,31)(19,28)(20,29)(21,30);(1,4,7,21)(2,9,12,25)(3,14,17,30)(5,20,6,19)(8,22,23,32)(10,18,11,24)(13,27,26,31)(15,29,16,28)87 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5196);