Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C3×#491 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
384 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 200 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2)(3,4,5);(6,7)(8,11)(9,10)(12,13)(14,15,20,23)(16,18,22,19)(17,25,28,24)(21,26,29,27);(7,10)(11,13)(14,16,18,23,20,22,19,15)(17,25,26,21,28,24,27,29);(1,2)(6,8,9,12)(7,11,10,13)(14,17,18,26,20,28,19,27)(15,21,16,24,23,29,22,25)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17574);