Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#2009 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #392 |
| commutator, Frattini, cosocle | C22×C4 | C23×C11 |
| prime radical | C11×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C22×C4×C11 ◁ G | |
348 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 88 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)(14,24)(15,28)(16,22)(17,21)(18,23)(19,26)(20,30)(25,33)(27,38)(29,37)(31,36)(32,34)(35,41)(39,40)(42,43);(12,14)(13,19)(15,25)(16,32)(17,26)(18,27)(20,35)(21,38)(22,24)(23,34)(28,39)(29,42)(30,40)(31,41)(33,37)(36,43);(12,15,18,31)(13,20,23,37)(14,25,27,41)(16,30,17,29)(19,33,34,35)(21,28,22,36)(24,39,38,43)(26,42,32,40)253 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 741);