Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C23 |
| direct product | C11×#1814 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #121 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C22×C4 | C23×C11 |
| socle | C2×C11 | #380 |
| prime radical | C2×C11×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C22×C4×C11 ◁ G | |
256 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 116 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)(14,23)(15,28,30,35)(16,31)(17,21)(18,22)(19,25)(20,29,36,32)(24,33,40,43)(26,34)(27,38)(37,39,42,41);(12,14,16,25,18,27,17,26)(13,19,21,23,22,34,31,38)(15,24,29,39,30,40,32,41)(20,33,35,37,36,43,28,42);(12,15)(13,20)(14,24)(16,29)(17,32)(18,30)(19,33)(21,35)(22,36)(23,37)(25,39)(26,41)(27,40)(28,31)(34,43)(38,42)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 546);