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×#2148 |
| 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 | |
220 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 220 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:
(1,2,6,11,7,12,4,9)(3,13,14,26,16,30,22,29)(5,18,20,10,21,27,17,28)(8,19,23,15,25,31,24,32)(33,34,35);(1,3)(2,8)(4,14)(5,19)(6,22)(7,16)(9,23)(10,26)(11,24)(12,25)(13,27)(15,17)(18,30)(20,32)(21,31)(28,29);(1,4,7,6)(2,9,12,11)(3,14,16,22)(5,17,21,20)(8,23,25,24)(10,18,28,27)(13,26,30,29)(15,31,32,19)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5439);