Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#2135 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C2×#99 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C2×Q8 | C23×C3 |
| prime radical | C2×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×#41 ◁ G | |
230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(1,2,5,10,6,11,21,28)(3,12,14,29,15,23,19,31)(4,8,17,26,18,9,20,27)(7,22,24,16,25,30,32,13)(33,34,35);(1,3,6,15)(2,7,11,25)(4,16,18,13)(5,19,21,14)(8,12,9,23)(10,24,28,32)(17,22,20,30)(26,29,27,31);(1,4,6,18)(2,8,11,9)(3,13,15,16)(5,17,21,20)(7,23,25,12)(10,26,28,27)(14,22,19,30)(24,31,32,29)69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5426);