Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#441 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C2×D4 |
| commutator | C4 | C23×C3×C4 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
480 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 480 conjugacy classes of subgroups — too many to draw. Only its 296 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(3,4,5);(6,7,9,8)(11,15)(12,18)(13,20)(16,23)(17,21)(22,24);(10,11,14,17)(12,16,18,24)(13,21,20,15)(19,23,25,22);(1,2)(6,8,9,7)(10,12)(11,16)(13,19)(14,18)(15,22)(17,24)(20,25)(21,23)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17381);