Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#3185 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C23×C3 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
358 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 358 conjugacy classes of subgroups — too many to draw. Only its 184 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) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,17,18,24)(12,21,19,15)(14,22,23,20)(25,26,27);(2,5)(6,8)(11,18)(14,23)(17,24)(20,22);(9,11,19,22,13,18,12,20)(10,14,21,24,16,23,15,17);(1,3,4,7)(2,6,5,8)(11,18)(14,23)(17,24)(20,22)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17625);