Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#2231 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #476 |
| commutator, Frattini, cosocle | C2×C8 | C23×C11 |
| prime radical | C11×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ C4×C8×C11 ◁ G | |
120 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 120 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) = 352 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,18,23)(14,24,27,22)(15,28)(16,32,25,21)(17,19,34,36)(20,37)(26,40)(29,42)(30,41)(31,35)(33,38)(39,43);(12,14,25,34,18,27,16,17)(13,19,32,22,23,36,21,24)(15,26,30,43,31,37,29,33)(20,35,39,41,40,28,38,42);(12,15)(13,20)(14,26)(16,29)(17,33)(18,31)(19,35)(21,38)(22,41)(23,40)(24,42)(25,30)(27,37)(28,36)(32,39)(34,43)253 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 963);