Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1795 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C25 |
| direct product | C11×#1796 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #379 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C22×C4 | C23×C11 |
| prime radical | C2×C11×2−1+4 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C22×C4×C11 ◁ G | |
536 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 536 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,22)(15,26,28,19)(16,29)(17,21)(18,24)(20,31)(23,37,39,36)(25,35)(27,34,43,41)(30,33)(32,38,40,42);(12,14,17,25)(13,18,21,33)(15,23,28,39)(16,30,31,24)(19,32,26,40)(20,35,29,22)(27,42,43,38)(34,37,41,36);(12,15)(13,19)(14,23)(16,27)(17,28)(18,32)(20,34)(21,26)(22,36)(24,38)(25,39)(29,41)(30,42)(31,43)(33,40)(35,37)319 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 528);