Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#2889 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22×C22⋊C4 |
| commutator | C22 | C23×C3×C4 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C2×C3×2−1+4 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×#41 ◁ G | |
704 conjugacy classes of subgroups; only the 340 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 340 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 — 4 generators on 35 points:
(1,2,5,10,6,9,18,23)(3,11,13,24,14,7,17,20)(4,8,16,22,15,21,28,31)(12,25,27,32,26,19,29,30)(33,34,35);(1,3,6,14)(2,7,9,11)(4,12,15,26)(5,17,18,13)(8,19,21,25)(10,24,23,20)(16,29,28,27)(22,32,31,30);(1,4)(2,8)(3,12)(5,16)(6,15)(7,19)(9,21)(10,22)(11,25)(13,27)(14,26)(17,29)(18,28)(20,30)(23,31)(24,32);(4,15)(8,21)(12,26)(16,28)(19,25)(22,31)(27,29)(30,32)123 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17329);