Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C23 |
| direct product | C11×#2049 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #394 |
| commutator, Frattini, cosocle | C22×C4 | C23×C11 |
| prime radical | C2×C11×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C22×C4×C11 ◁ G | |
254 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 254 conjugacy classes of subgroups — too many to draw. Only its 80 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)(14,24,27,39)(15,28,32,37)(16,22,17,21)(19,33,35,26)(20,30,38,31)(25,36,29,34)(40,43,42,41);(12,14)(13,19)(15,29)(16,33)(17,26)(18,27)(20,36)(21,39)(22,24)(23,35)(25,32)(28,42)(30,41)(31,43)(34,38)(37,40);(12,15,16,30,18,32,17,31)(13,20,21,37,23,38,22,28)(14,25,26,41,27,29,33,43)(19,34,24,40,35,36,39,42)253 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 781);