Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×#350 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | D4 |
| commutator | C4 | C22×C3×C8 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C22×C3 | C22⋊C8 |
| prime radical | C2×C3×#41 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ C22×C3×C8 ◁ G | |
224 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2)(3,4,5);(6,7,9,14,10,15,21,28,11,16,20,29,22,30,33,37)(8,13,18,26,19,27,32,36,17,12,23,24,31,25,34,35);(7,12)(8,17)(9,20)(13,16)(14,26)(15,25)(19,31)(21,33)(24,29)(27,30)(28,36)(35,37)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5407);