Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C23 |
| direct product | C11×#2201 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C23×C11 |
| prime radical | C11×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C22×C11 ◁ C2×D4×C11 ◁ G | |
226 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 56 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)(14,22)(15,19)(16,20)(17,21)(18,24)(23,39)(25,38)(26,37)(27,33)(28,34)(29,36)(30,40)(31,41)(32,42)(35,43);(12,14,17,25)(13,18,21,31)(15,26,28,23)(16,24,29,41)(19,32,34,30)(20,22,36,38)(27,42,43,40)(33,37,35,39);(12,15,17,28)(13,19,21,34)(14,23,25,26)(16,27,29,43)(18,30,31,32)(20,35,36,33)(22,37,38,39)(24,40,41,42)187 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 933);