Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#1664 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C11 |
| prime radical | C2×D4×C11 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C42×C11 ◁ G | |
184 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 704 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,17,21)(14,22,24,27)(15,20,16,19)(18,25,26,23)(28,29,33,31)(30,36,39,43)(32,40,42,35)(34,41,37,38);(12,14)(13,18)(15,25)(16,23)(17,24)(19,27)(20,22)(21,26)(28,30,33,39)(29,34,31,37)(32,41,42,38)(35,43,40,36);(12,14)(13,18)(15,23)(16,25)(17,24)(19,22)(20,27)(21,26)(28,31,33,29)(30,37,39,34)(32,35,42,40)(36,41,43,38)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 396);