Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#2145 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #102 |
| commutator | C8 | C22×C3×C4 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C2×C3 | C2×#102 |
| prime radical | C2×C3×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
224 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 100 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)(3,12,15,29)(4,9)(5,17)(6,21)(7,11)(8,14,24,32)(10,19)(13,26,22,30)(16,27)(18,23,31,25)(20,28)(33,34,35);(1,3)(2,8)(4,13)(5,18)(6,22)(7,15)(9,23)(10,26)(11,24)(12,28)(14,16)(17,29)(19,32)(20,31)(21,25)(27,30);(1,4,7,6)(2,9,11,21)(3,13,15,22)(5,16,20,19)(8,23,24,25)(10,17,27,28)(12,26,29,30)(14,31,32,18)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5436);