Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#2123 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C2×#99 |
| commutator | C23 | C22×C3×C4 |
| Frattini, cosocle | C2×D4 | C23×C3 |
| prime radical | C2×C3×#41 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×#41 ◁ G | |
234 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 234 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)(3,11)(4,8)(5,18,19,9)(6,10)(7,13,21,28)(12,25)(14,26)(15,29)(16,31,32,23)(17,24)(20,27,22,30)(33,34,35);(1,3,6,14)(2,7,10,21)(4,15,17,12)(5,13,19,28)(8,22,24,20)(9,11,18,26)(16,30,32,27)(23,29,31,25);(1,4,6,17)(2,8,10,24)(3,12,14,15)(5,16,19,32)(7,20,21,22)(9,23,18,31)(11,25,26,29)(13,27,28,30)69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5414);