Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#2239 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #481 |
| commutator, Frattini, cosocle | C2×C8 | C23×C11 |
| prime radical | C23×C11 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ C2×C11×M16 ◁ G | |
124 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 124 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)(12,13,17,21)(14,22,23,40)(15,26,27,43)(16,31,29,20)(18,33,34,24)(19,36,37,30)(25,38,32,41)(28,35,42,39);(15,27)(19,37)(22,40)(24,33)(26,43)(28,42)(30,36)(38,41);(12,14,16,32,17,23,29,25)(13,18,20,39,21,34,31,35)(15,28,30,33,27,42,36,24)(19,38,26,40,37,41,43,22)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 971);