Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#2194 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23×C3 |
| prime radical | C3×#576 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#131 ◁ G | |
216 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 64 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:
(1,2,7,13)(3,14,17,30)(4,18,22,29)(5,12,6,11)(8,24,27,15)(9,23,28,16)(10,20,25,21)(19,31,32,26)(33,34,35);(1,3)(2,8)(4,19)(5,16)(6,23)(7,17)(9,25)(10,28)(11,26)(12,31)(13,27)(14,18)(15,21)(20,24)(22,32)(29,30);(1,3)(2,9)(4,15)(5,23)(6,16)(7,17)(8,25)(10,27)(11,30)(12,14)(13,28)(18,31)(19,21)(20,32)(22,24)(26,29)69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5485);