Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C25×C4 |
| direct product | C11×#3027 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | C2×#120 |
| commutator, Frattini, cosocle | C23 | C24×C11 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C22×C4×C11 ◁ G | |
772 conjugacy classes of subgroups; only the 212 normal subgroups are shown.
This group has 772 conjugacy classes of subgroups — too many to draw. Only its 212 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) = 176 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(15,22)(17,23)(18,25)(20,26);(12,13)(14,17)(15,18)(16,19)(20,24)(21,23)(22,25)(26,27);(12,14)(13,17)(15,20)(16,21)(18,24)(19,23)(22,26)(25,27);(12,15,16,22)(13,18,19,25)(14,20,21,26)(17,24,23,27)319 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1759);