Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#1858 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #120 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C2×C3 | #385 |
| prime radical | C2×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ G | |
252 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 108 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:
(1,2,6,10,7,11,5,18)(3,12,14,8,15,23,19,22)(4,9,16,25,17,26,20,24)(13,28,31,32,30,27,29,21)(33,34,35);(1,3,6,14,7,15,5,19)(2,8,10,23,11,22,18,12)(4,13,16,31,17,30,20,29)(9,21,25,28,26,32,24,27);(1,4)(2,9)(3,13)(5,20)(6,16)(7,17)(8,21)(10,25)(11,26)(12,27)(14,31)(15,30)(18,24)(19,29)(22,32)(23,28)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5149);