Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#1812 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #117 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C2×C3 | #380 |
| prime radical | C2×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×M16 ◁ G | |
260 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 116 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)(3,11)(4,15,17,8)(5,18)(6,10)(7,13)(9,19)(12,25,27,23)(14,24)(16,22,31,30)(20,26,29,32)(21,28)(33,34,35);(1,3,5,13,6,14,19,28)(2,7,9,11,10,21,18,24)(4,12,16,26,17,27,31,32)(8,20,22,23,15,29,30,25);(1,4)(2,8)(3,12)(5,16)(6,17)(7,20)(9,22)(10,15)(11,23)(13,26)(14,27)(18,30)(19,31)(21,29)(24,25)(28,32)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5103);