Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#2212 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C23×C11 |
| prime radical | C11×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ C4×C8×C11 ◁ G | |
154 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 154 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)(14,22)(15,27)(16,31)(17,33)(18,23)(19,25)(20,24)(21,32)(26,40)(28,39)(29,43)(30,35)(34,36)(37,41)(38,42);(12,14,16,25,18,26,32,17)(13,19,21,22,23,33,31,40)(15,24,28,41,30,42,29,34)(20,35,36,39,38,27,37,43);(12,15)(13,20)(14,24)(16,28)(17,34)(18,30)(19,35)(21,36)(22,39)(23,38)(25,41)(26,42)(27,33)(29,32)(31,37)(40,43)253 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 944);