Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#1814 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #121 |
| 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 ◁ C22×C3×C4 ◁ G | |
256 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 256 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)(3,12)(4,17,19,24)(5,20)(6,10)(7,11)(8,14)(9,18,25,21)(13,22,29,32)(15,23)(16,27)(26,28,31,30)(33,34,35);(1,3,5,14,7,16,6,15)(2,8,10,12,11,23,20,27)(4,13,18,28,19,29,21,30)(9,22,24,26,25,32,17,31);(1,4)(2,9)(3,13)(5,18)(6,21)(7,19)(8,22)(10,24)(11,25)(12,26)(14,28)(15,30)(16,29)(17,20)(23,32)(27,31)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5105);