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) = C22×C4 |
| direct product | C11×#1976 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #121 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C22×C4 | C23×C11 |
| socle | C2×C11 | #390 |
| prime radical | C2×C11×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C22×C4×C11 ◁ G | |
284 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 284 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)(14,18,23,32)(15,25)(16,28)(17,20)(19,26)(21,29)(22,34,35,43)(24,38,37,31)(27,39)(30,41,42,36)(33,40);(12,14)(13,18,20,32)(15,22)(16,24)(17,23)(19,30,33,42)(21,31,28,38)(25,34,39,43)(26,36)(27,35)(29,37)(40,41);(12,15,16,26,17,27,29,40)(13,19,21,25,20,33,28,39)(14,22,24,36,23,35,37,41)(18,30,31,34,32,42,38,43)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 708);