Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×#2180 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #442 |
| commutator | C8 | C22×C3×C4 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| prime radical | C3×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#97 ◁ G | |
142 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 142 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:
(1,2,5,9)(3,11)(4,16,18,29)(6,20,17,10)(7,22)(8,26,27,19)(12,28)(13,25)(14,23)(15,31)(21,30)(24,32)(33,34,35);(1,3)(2,7)(4,12)(5,15)(6,21)(8,23)(9,25)(10,28)(11,29)(13,26)(14,17)(16,31)(18,32)(19,22)(20,24)(27,30);(1,4,17,26,5,18,6,19)(2,8,20,29,9,27,10,16)(3,12,14,13,15,32,21,22)(7,23,24,11,25,30,28,31)69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5471);