Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#3363 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C4×Q8 ◁ G | |
336 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 192 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(2,5)(6,7)(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18)(25,26,27);(1,2,4,5)(3,7,8,6);(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17);(1,3,4,8)(2,6,5,7)105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17803);