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×#3291 |
| 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 | |
460 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 460 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) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,7,12)(3,13,14,15)(4,9,8,16)(5,11,6,10)(17,18,19);(1,3)(2,8)(4,12)(5,16)(6,9)(7,14)(10,15)(11,13);(1,4)(2,3)(5,15)(6,13)(7,8)(9,10)(11,16)(12,14);(1,4)(2,9)(3,10)(5,13)(6,15)(7,8)(11,14)(12,16)87 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17731);