Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#1790 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #120 |
| commutator | C23 | C22×C3×C4 |
| Frattini, cosocle | C24 | C23×C3 |
| prime radical | C2×C3×H(5,𝔽2) | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C24×C3 ◁ G | |
670 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 670 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3,4)(5,6)(7,11,12,9)(8,10)(13,14,15);(5,7)(6,9)(8,12)(10,11);(5,6)(7,9)(8,10)(11,12)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5081);