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×#1663 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C11 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C11 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C42×C11 ◁ G | |
172 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 172 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) = 1408 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,18,22)(14,23,25,26)(15,24,19,27)(16,21,17,20)(28,29,33,31)(30,36,39,43)(32,40,42,35)(34,41,37,38);(12,14)(13,19)(15,22)(16,27)(17,24)(18,25)(20,26)(21,23)(28,30,33,39)(29,34,31,37)(32,41,42,38)(35,43,40,36);(12,15,18,19)(13,14,22,25)(16,26,17,23)(20,24,21,27)(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, 395);