Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#1881 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #120 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C22×C4 | C23×C11 |
| socle | C2×C11 | #386 |
| prime radical | C11×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C22×C4×C11 ◁ G | |
310 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 108 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)(14,23,26,34)(15,27)(16,21)(17,31)(18,22)(19,25,35,30)(20,28)(24,39,42,33)(29,36)(32,37)(38,40,43,41);(12,14,16,30,18,26,17,25)(13,19,21,34,22,35,31,23)(15,24,28,41,29,42,32,40)(20,33,36,43,37,39,27,38);(12,15)(13,20)(14,24)(16,28)(17,32)(18,29)(19,33)(21,36)(22,37)(23,38)(25,40)(26,42)(27,31)(30,41)(34,43)(35,39)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 613);