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) = C22 |
| direct product | C3×#2170 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C16 | D4 |
| commutator | C4 | C22×C3×C8 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C2×C3 | C8×D4 |
| prime radical | C3×#135 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C16 ◁ C2×C3×C16 ◁ G | |
146 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 92 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 — 3 generators on 35 points:
(1,2,6,12,7,11,25,15,8,13,26,30,5,23,24,31)(3,14,17,22,16,9,27,19,18,28,32,4,10,20,29,21)(33,34,35);(1,3)(2,9)(4,15)(5,10)(6,17)(7,16)(8,18)(11,28)(12,19)(13,20)(14,23)(21,30)(22,31)(24,29)(25,27)(26,32);(1,4,6,20,7,21,25,14,8,22,26,9,5,19,24,28)(2,10,12,29,11,3,15,17,13,16,30,27,23,18,31,32)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5461);