Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#3293 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ G | |
444 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 184 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 — 4 generators on 35 points:
(1,2,8,13)(3,14,9,24)(4,17,19,16)(5,10,22,28)(6,12,7,11)(15,30,25,26)(18,20,32,31)(21,29,23,27)(33,34,35);(1,3,8,9)(2,4,13,19)(5,20,22,31)(6,17,7,16)(10,26,28,30)(11,24,12,14)(15,23,25,21)(18,29,32,27);(1,4,8,19)(2,9,13,3)(5,18,22,32)(6,14,7,24)(10,25,28,15)(11,17,12,16)(20,27,31,29)(21,26,23,30);(1,5)(2,10)(3,15)(4,18)(6,23)(7,21)(8,22)(9,25)(11,29)(12,27)(13,28)(14,30)(16,31)(17,20)(19,32)(24,26)87 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17733);