Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#2155 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #106 |
| commutator | C8 | C22×C4×C11 |
| Frattini, cosocle | C2×C8 | C23×C11 |
| socle | C2×C11 | C2×#106 |
| prime radical | C23×C11 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C22×C4×C11 ◁ C22×C8×C11 ◁ G | |
144 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 100 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)(14,23,26,37)(15,20)(16,29)(17,32)(18,22)(19,25,34,39)(21,30)(24,36,27,41)(28,35)(31,42)(33,38,43,40);(12,14,16,25,17,27,30,40,18,26,31,39,15,24,28,38)(13,19,21,23,20,33,29,36,22,34,35,37,32,43,42,41);(12,15,18,17)(13,20,22,32)(14,24,26,27)(16,28,31,30)(19,33,34,43)(21,29,35,42)(23,36,37,41)(25,38,39,40)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 887);